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- $$\mathit{r}=\frac{{\mathit{r}}_{\mathit{Max}}\cdot \left[\text{S.}\right]}{{K}_{\text{M.}}+\left[\text{S.}\right]}$$

Multiply the Michaelis-Menten equation by $\frac{{K}_{\text{M.}}+\left[\text{S.}\right]}{\left[\text{S.}\right]}$ leads to:

- $$\begin{array}{rcl}\frac{{K}_{\text{M.}}+\left[\text{S.}\right]}{\left[\text{S.}\right]}\cdot \mathit{r}& =& {\mathit{r}}_{\mathit{Max}}\\ \left(\frac{{K}_{\text{M.}}}{\left[\text{S.}\right]}+1\right)\cdot \mathit{r}& =& {\mathit{r}}_{\mathit{Max}}\\ \frac{{K}_{\text{M.}}}{\left[\text{S.}\right]}\cdot \mathit{r}+\mathit{r}& =& {\mathit{r}}_{\mathit{Max}}\\ \mathit{r}& =& -\frac{{K}_{\text{M.}}}{\left[\text{S.}\right]}\cdot \mathit{r}+{\mathit{r}}_{\mathit{Max}}\\ \mathit{r}& =& -{K}_{\text{M.}}\cdot \frac{\mathit{r}}{\left[\text{S.}\right]}+{\mathit{r}}_{\mathit{Max}}\end{array}$$

The equation is in the form of a linear equation.

## Canon mathematics

### Table of contents

### Preamble

#### Prehistory

In 1990 the German Rectors' Conference published a catalog of basic knowledge in mathematics. This stipulates which substance is assumed to be known at the start of the study. After the high school diploma reform in 1995, the catalog was revised by DMK, CRM and a commission from ETH Zurich in 1997 and, after extensive consultation, recommended for implementation throughout Switzerland. Since the turn of the millennium, the framework conditions have changed again. School time has been shortened in many places, pocket calculators with graphic capabilities and CAS (Computer Algebra Systems) as well as geometry software offer new possibilities in high school mathematics lessons. Studies such as EVAMAR II have also shown that problems arise at the transition from grammar school to university, which are due, among other things, to the extremely heterogeneous number of hours in the basic subject of mathematics. At the conference and transition from high school to university in October 2010 in the Centro Stefano Franscini, it was therefore suggested that the catalog be redesigned. The Gymnasium-Universit & uumlt commission then commissioned DMK to form a broad-based group of representatives from DMK / CRM / CMSI and the universities to carry out the work and submit the result to a general consultation.

#### Objectives of this catalog

On the one hand, in grammar schools one often does not know exactly which mathematical knowledge and skills the universities require in the basic lectures, on the other hand in universities one often does not know exactly what level of education one can expect from the high school graduates. The present catalog is intended to provide orientation and information for both sides. However, an undesirable degree of standardization of content and curricula should be avoided. Freedom of teaching and diversity are valuable assets, without which no school development is possible. Historically grown or cultural differences should also have their place. Nevertheless, it should be clear what knowledge, what abilities and skills and what understanding high school graduates should bring with them in the subject of mathematics with a general university entrance qualification. At the same time, excessive heterogeneity is to be counteracted by using the catalog as a guide when revising cantonal or internal school curricula. This catalog thus also indirectly makes the expectation of the grammar school clear about the level of training when transitioning from lower secondary level. As a whole, the catalog shows which skills and specialist knowledge are expected of students at the university. This also includes topics that have to be taught before the upper secondary level. For example, term transformations, equations, elementary geometry, stereometry and stochastic considerations should be discussed at an early stage and should later be a subject of instruction again at high school level at a higher level of abstraction.

#### Framework

This catalog is designed for the basic subject of mathematics. It is based on a total of 16 weekly lessons for a four-year high school. Schools that are below this number of lessons have to accept compromises, because mathematics takes time. Regardless of this, the grammar schools must inform their students that, depending on the desired course of study, the choice of the supplementary subject *Applications of Mathematics* or the major *Physics and Applications of Mathematics* can decisively promote their academic success.

#### What's new?

Internet and computers permeate our lives, permeate every subject taught at high school. However, no subject is more affected by it than mathematics, notably through its very own invention, the computer. Internet, visualization and geometry software as well as CAS and other aids have their place in every modern math lesson. These new instruments offer the opportunity to teach mathematics more comprehensively. Topics such as the discussion of curves or the finding of antiderivatives for tricky integrands have lost their importance in the age of computer algebra systems. However, the use of electronic aids must be targeted and well-considered and must not lead to a loss of arithmetic skills: The universities still expect sufficient manual skills, for example with term transformations, solving equations and performing longer calculations To master your studies. Lessons that integrate CAS must therefore not be limited to dealing with traditional tasks with a calculator. Rather, he has to reorient himself and align the use of a CAS sensibly and critically questioning the content, without neglecting the manual skills.

The actual requirements in the course vary considerably and depend on the chosen field of study, the examination subject and the respective university. At some universities, pocket calculators of any kind are not permitted for most basic exams after the first year of study. On the one hand, however, they are not necessary, because the examination tasks are oriented towards understanding and require skills in the manual handling of mathematical objects and, on the other hand, obvious legal reasons speak against electronic aids for examinations. At the same time, the use of mathematical software is a central element of numerous courses. The universities are therefore obliged to present the requirements for the individual fields of study in brochures and on websites. Conversely, grammar schools, teachers and school leavers must inform themselves about the requirements in order to ensure adequate preparation for the course.

Due to the heterogeneous situation, the catalog cannot provide any general information on dealing with CAS, but strives for a balance between procedural skills (syntax), understanding-oriented knowledge (semantics) and independent research (exploration). This approach is reflected in the content of the catalog below.

#### General objectives of math class

Mathematics is a huge field of knowledge and a cultural asset that has grown over millennia. Your applications form the basis of our high-tech society. It provides basic tools for all quantitatively working and logically reasoning sciences. However, its importance as an educational goal in the classroom goes far beyond science as a study goal. There must be time in school lessons

- for questions, for the search for answers, for reasons, for discovering rules,
- for explaining, living, learning and repeating,
- for exploring, playing, for story (s) and for competition.

Mathematics is modern and lively, it is constantly changing. This concerns the internal mathematical further development of this science, the image of mathematics from the outside, the instruments used and the ever new applications. This dynamic has to be reflected in the school. Mathematics is curiosity and creativity, the lessons should awaken the joy of it. A concept for the lesson therefore contains the elements:

- Mathematics as a field of knowledge: history and stories, applications and overview are important for motivation and as an educational goal for everyone
- Basic tools
- Mathematics as science

From this point of view, there are now general consequences for teaching.

#### Aspects not related to the topic

In mathematics the facts are proven, and these truths then hold universally and forever. No other subject can convey this experience. The intensive occupation with mathematics demands and promotes a certain rigor in the mind and teaches tenacity and perseverance in the face of difficult tasks. Self-confidence and the courage to try things out are basic prerequisites for students to be open to mathematical topics and to want to discover, justify and prove for themselves. In doing so, they should also experience that the clear presentation of a solution and plausibility considerations are a prerequisite for being able to discover and correct errors. A positive attitude to work and a consistent self-assessment are important tools for a successful start in your studies.

*Logical reasoning:* Dealing with knowledge and information requires, not only in mathematics, the ability to link different elements logically with one another. Internal mathematical reasons also make it necessary to deal with elementary logical structures, because by combining them one finally arrives at the central concept of proof. What a proof is and why you need it should be a topic in the class as well as special proof methods (direct and indirect proof, complete induction). The mathematical working method requires that students are familiar with the terms definition, sentence (requirement, assertion), proof, necessary and sufficient condition, as well as negation, inversion and counterposition. It is necessary that these terms are practiced using clear examples.

*Modeling:* Mathematics is a language and a tool for other sciences. The tremendous benefit of mathematics for our daily life remains mostly hidden from the superficial observer. It is based on the one hand on the fact that mathematics allows models of real conditions to be created and on the other hand on examining these models with mathematical methods and drawing conclusions about reality from the results. Central questions are: What is a model? How do you get from a real phenomenon to a model? Which factors are negligible, which are relevant? What are the limits of models? What is the relationship between model and reality? How can forecasts, simulations and optimizations be implemented with the help of models? These questions should come up in class.

*Skills by hand:* Just as mastering a language requires a basic vocabulary and familiarity with grammar, you can only be mathematically active if you have a certain degree of manual skills and techniques. This also includes the safe and correct use of mathematical notation. These manual skills should be constantly practiced using transparent examples. Only sufficient arithmetic skills make it possible to become mathematically expressive.

*Algorithmic thinking:* The algorithm is a central term for mathematics and emphasizes its constructive and dynamic side. It also forms a bridge to IT. For example, recursions, loops and loop invariants, or iterative and numerical processes, play an important role in many areas of mathematics.

*Geometric imagination:* Numerous terms of geometry such as dimension, symmetry, conservation quantity or mapping, but also spatial imagination, are essential for a deeper understanding of large parts of mathematics. What can be vividly experienced in geometry can be transferred to more abstract objects. The geometric visualization can promote the understanding of deeper-lying aspects.

*Use of new technologies:* The students should get a first contact with mathematical software in the analytical, numerical and statistical treatment of mathematical problems. In particular, fears of contact with electronic aids should be reduced and an initial approach to current technologies should be maintained, also with a view to more advanced use at the university level. On the one hand, this should enable contact to realistic problems in simple applications, on the other hand, it should also support and promote the discovery of more complex mathematical phenomena through experiment, simulation and numerical calculation. This should also make the use of visualization methods and the handling of data accessible. At the same time, it must be possible to experience the problem that uncritical use of computers can lead to wrong conclusions.

As part of the framework didactics and the black-box / white-box principle, the calculator also enables weaker students to have mathematical success. Because routine syntactic calculations can also be delegated to the computer, you can gain time in class to make space for theoretical and understanding-oriented deepening. However, this should not lead to neglecting manual calculations.

*Scientific mindset:* Mathematics is a scientific discipline with its own working methods that should be reflected in the classroom. A hypothesis is created on the basis of random or systematic observations, generalizations, or consideration of special cases. A distinction must be made between what is given and what is sought, and attention must be paid to the use of appropriate terms. Building on a corresponding core idea, an attempt is then made deductively, based on existing knowledge, to prove the thesis or, if necessary, to modify it in order to develop new knowledge and to rearrange the knowledge. This also includes the discussion of the solution through plausibility considerations or rough calculations. This process, which requires a high degree of creativity, gives rise to new points of view, relationships, possibilities for interpretation and new questions.

#### Interdisciplinary

With MAR 95 and the creation of key subjects such as *Physics and Applications of Mathematics* or *biology and chemistry* the interdisciplinary approach has become more important in high school teaching. The catalog provides information as to where the mathematical content can be linked to other subjects. This also shows the usefulness of mathematics as a structural science and provides additional motivation. However, interdisciplinarity must not be made the sole principle: Without well-founded disciplinary knowledge, interdisciplinary work cannot be done. Mathematical content is not infrequently the most understandable first in its pure form. It is precisely then that they show their own aesthetics and the justification of mathematics as an independent science and also as a central subject.

#### The areas of matter

The mathematics curriculum is based on the classic pillars: *geometry*, elementary *algebra*, *Analysis*, Applications of mathematics, in particular *Stochastics*. School mathematics thus traces the historical development of science, following the genetic principle.

the *algebra* at the grammar school opens the way for students to get out of the arithmetic of numbers and to understand variables and to calculate with them. This creates an important prerequisite for the concept of function and analysis. The fundamental algebraic rules of calculation (commutative laws, associative laws, distributive law) are developed in school by analogy with numerical calculations or geometrically and form the basis of term transformations and the theory of simple equations. The study of simple functions (power, root, exponential and logarithmic functions) and the technical handling of them is traditionally also assigned to algebra in grammar school mathematics. The language of algebra opens up a wide field of applications and simple models, which further support analysis. Aspects of higher algebra, such as the close connection with geometry, can hardly be found in the basic subject, but the historical framework is able to exemplify the development of mathematics as a science.

the *Analysis* deals with the functional relationships between science and mathematics. It provides a language to represent this and at the same time develops computational-analytical and graphical-visual tools and methods for researching them. The computing techniques (syntax) must be subordinated to a deeper insight into the meaning of the concepts (semantics). Technology can support this endeavor well. One of the traditional core topics of high school analysis, the curve discussion, is therefore no longer up-to-date and does not appear as an independent topic.Many of the most significant and fundamental scientific laws and models, from physics to meteorology, biology, chemistry, medicine and economics, are based on differential and integral calculus and are formulated in the form of differential equations. In order that such models and laws can only be partially understood, it is essential that the concept and a qualitative understanding of a differential equation are also conveyed in the basic subject. This is also possible with simple mathematical methods and without the need for analytical solution procedures.

In the field of *geometry* Historically, the foundations of the methodology of mathematical activity were laid. Thus, in the elements of Euclid, the structure of mathematical texts that is still common today was practiced for the first time: definitions, axioms, propositions and proofs. This structure has proven to be fundamental in mathematics and can be understood and made tangible in geometry lessons using comparatively clear examples. The further development of geometry in the context of mapping geometries, transformation groups and invariants has led to far-reaching generalizations, such as topology or discrete geometry, and to an inner unity with algebra. The students should also experience geometry as a modern tool, which has changed with technical developments and has turned to modern issues. The numerical recording and description of figures and bodies (shape, size and position) and their representation is reflected in a wide variety of sciences. Vector geometry, in particular, offers the possibility of going beyond the purely mathematical subject area and applying geometric concepts in physics, chemistry, biology, geography, economics, etc.

the *Stochastics* has found numerous applications in all quantitative sciences in the recent past and has thus gained in importance: This also applies to many fields of study outside of natural science and technology, such as medicine, economics, psychology, sociology or economics and law. Stochastics has become increasingly important for the relevant fields of study, which is reflected in the content of the material catalog. Furthermore, stochastics are now part of general education: we are all confronted daily with chance, statistics, risk and uncertainty. It is not about anticipating the content of statistical lectures at the universities, but about conveying an elementary insight into the way of thinking and the concepts of stochastics. The everyday relevance of stochastics creates a special motivation for mathematics for many students. Frequently, the related questions also offer them access to mathematics that is independent of their previous teaching experience. Stochastics is very well suited to illustrate aspects of mathematical modeling. Stochastic models form an important complement or a contrast to the deterministic models of analysis.

For reasons of time alone, other areas of mathematics besides the four pillars mentioned above cannot be reflected in schools to the same extent. Nevertheless, the areas of numerics or discrete mathematics that are important in connection with the computer should, if possible, be addressed in the classroom.

### Subject-specific content

The content-related part is shown for each mathematical area as a table with three columns. Each table is supplemented by references to further in-depth topics, cross-connections to the other, internal mathematical areas or to other specialist disciplines, and to applications.

The three columns *Understanding-oriented, content-based knowledge (semantics)*, *Procedural, algorithmic skills (syntax)* and *Understanding-oriented exploration / deepening (exploration)* should be equally reflected in the classroom. For example, it is not enough to have a syntactic mastery of the derivation rules if the term derivation is not associated with a content concept or if the practical relevance of the derivation as a tool in applications, removed from the pure derivation term, remains untreated. This approach also means that exams may not only ask about procedural skills. The gap *Understanding-oriented exploration / deepening (exploration)* Takes on a special role insofar as it can be used either for an exploratory introduction or for an in-depth discussion of a topic. The universities can assume that the contents of the two columns semantics and syntax have been covered in the classroom. With regard to the contents of the third column, the teachers are free to set their own priorities, which means that there is also the option of only touching on another or leaving out another due to an in-depth treatment of a certain topic. The other in-depth topics and cross-connections listed are to be seen as enriching additional elements and can be replaced by other meaningful content.

- Forms of representation
- Exact and approximate representation of numbers
- Absolute amount of a number

- Calculating with fractions and root terms

- Difference rational - irrational

- Geometric illustration of term transformations
- Recognize term structures

- Commutative, associative, distributive law
- Basic operations with fraction terms
- Double fractions, root terms, factoring

- Binomial theorem (Pascal's triangle)

- Detection in applications

- Recognition of other functional dependencies

- Derivation of the power laws for natural exponents
- Terms
*n*-th root and logarithm

- Power laws (for rational exponents) and logarithmic set logarithmic scale

- Derivation of the power laws for negative whole and rational exponents
- Logarithmic scale

- Concept of the solution of an equation, geometric interpretation of the solution set (straight line, circle)
- & Equivalent conversion

- Apply equivalent transformations
- Different types of equations solve: linear equations, quadratic equations, equations which refer to the shape
*x a*=*b*,*a x*=*b*, or lied_{a}*x*=*b*can be brought - Appointment control

- Solving inequalities

- Linear and non-linear systems of equations
- Geometric interpretation of linear systems of equations in R 2

- Use different solution methods

- Special cases with linear systems of equations
- Concept of the amount of solution

##### Time requirement

30 to 35 weeks & agrave 4 lessons (secondary level II) plus 20 to 40 weeks (secondary level I).

##### Further exploratory in-depth topics

- Equivalence of equations
- Complex numbers
*Cryptology*: RSA method, Diffie-Hellman key exchange protocol*Numerics*: know different methods to solve equations approximately*Set theory*: Cardinality of sets, Cantor's diagonal method

##### Cross connections to the other areas

Algebraic transformations are used in all subject areas of high school mathematics lessons. In particular, the following should be addressed:

*Analysis:*Terms - Equations - Functions*Geometry:*Reshape area and volume formulas*Stochastics:*Binomial distribution

##### Applications and links to other specialist disciplines

###### Computer science

###### Physics

- Laws like Ohm's law
*s*=*v t*, 3. Kepler's law - Lens equation
- dB scale Straight in parametric representation and straight movement

###### Chemistry

###### Biology

- Weber-Fechner law (perception of brightness and volume)
- Henry coefficient (mass transfer)

###### Geography

###### Economics

###### Philosophy

- Discrete models (e.g. cell division, compound interest, cooking curve)
- Explicit and recursive form
- Intuitive concept of limit values

- Sequences, finite and infinite sums (series)
- Dealing with summation symbols
- Arithmetic sequences, geometrical sequences and series
- Calculate elementary limit values

- Complete induction
- Applications in financial mathematics (interest and annuities)

- Discrete vs. continuous models
- Function concept, difference between term, equation and function rule
- Graph of a function

- Evaluation of functions
- Functions as a black box

- From data to function (interpolation and approximation)

- Monotony, symmetry, periodicity, limitation, zeros
- Clear understanding of asymptotic behavior, the concept of limit values, continuity, invertibility

- Basic functions and their graphs (linear and quadratic functions, power functions, polynomial functions, exponential and logarithmic functions, trigonometric functions)

- Bijectivity, injectivity, surjectivity
- Inverse functions (e.g. root, arc functions)

- Logical analysis of function terms (e.g. being able to recognize linked functions)

- Operations with functions: sum, product, quotient, concatenation
- Transformations (stretching, shifting, mirroring)

- Measure of change: mean and current rate of change over time using applications
- Linearization, tangent, slope (e.g. monotony)
- Extreme and turning points
- Optimization applications

- Difference quotient
- Differential quotient / derivative
- Derivative function
- Basic properties of the derivation (linearity)
- Derivation of the basic functions
- Derivation of
*f*(*ax + b*) - Extremal problems

- Linear approximation, Newton method
- Applications in the natural and social sciences as well as economics (kinematics, growth and decay)
- Derivative lime (product, quotient, chain rule)

- Riemann sum using applications

- Subtotal sequences
- Definite integral
- Basic properties of the definite integral (linearity, additivity)

- Applications from geometry (e.g. surfaces and bodies of revolution) and physics
- Aspects of numerical integration

- Intuitive justification

- Antiderivative, indefinite integral
- Antiderivatives of the basic functions

- Applications from physics (e.g. path and speed, work, energy, momentum)
- Integration lime & uumll

- Derivation (product, quotient, chain rule) and integration calculus
- Applications in physics (e.g. path and speed, work, energy)

- Basic concepts of modeling with differential equations
- Current rate of change of a function
- Concept of the solution of a differential equation

- Discretization using the Euler method
- Directional fields and graphic solutions

- Applications (e.g. population models in biology, free fall in physics)

##### Time requirement

40 to 50 weeks & agrave 4 lessons.

##### Further exploratory in-depth topics

- Limit values, rule of Bernoulli-de l'H & ocircpital
- Numerical differentiation and integration
- Deepening the approximation (e.g. Taylor series), regression (least squares method)
- Deepening integration: integration methods, improper integrals (e.g. probability densities, integral criterion for series, potential and gravitational fields)
- Center of gravity, moment of inertia, Steiner's theorem, Guldin's rules
- Elementary, analytical solution methods for differential equations (exponential approach, separation)

##### Cross connections to the other areas

###### Stochastics

- Data and functions (interpolation, regression)
- Probability densities, expected value, median

###### Algebra

- Numerical equation solver: bisection, Heron, specialization Newton-Raphson
- Recursion and iteration

###### Geometry

- Fractals and attractors in iterations (e.g. Newton-Raphson method)
- Geometric probabilities (e.g. Buffon's problem)

##### Applications and links to other specialist disciplines

###### Physics

- Specialization in mechanics: path, speed, acceleration functions, energy and work, fields (potential, gravitation), movement of particles (simple differential equations, Galileo's law of inertia, free fall), oscillation processes
- Exponential decay (radioactivity)
- Newton's law of cooling

###### Biology

- Discrete and continuous growth models (linear, exponential (e.g. cell division), logistic, Fibonacci numbers)
- Simple systems of differential equations for population interactions: R & aumluber prey (e.g. Lotka-Volterra), competition and symbiosis

###### Philosophy

###### Finance and Economics

- Interest and rents, consumer and producer surplus
- Marginal and total functions
- Models for competition

###### Medicine

- Bateman functions and geometric series for modeling continuous resp. discreet medication intake
- Pumping ability of the heart and Hagen-Poiseuille law of laminar flow in blood vessels
- Epidemic models with differential equations (e.g. SIR model)

###### Sports

- Evolute and involute (e.g. starting line of the 1500 meter run)
- Modeling and optimizing the free throw in basketball

- Triangles, squares, circles, angles, congruence, similarity / radiance, sentence group of Pythagoras, Thaleskreis, arc of arcs
- Finding and explaining connections in geometric situations
- Images (translation, rotation, mirroring, centric stretching)

- Calculate missing sizes in geometric figures

- Constructions with compasses and ruler
- Regular polygons
- Derive formulas for area contents
- Prove the congruence and resemblance of figures

- Right triangle calculations
- Definition of the trigonometric functions
- Radians
- Sine and cosine law

- Right-angled and general triangle calculations
- Recognize relationships between definitions in the unit circle and the graph of the trigonometric functions
- Trigonometric equations of the form trig (
*ax*+*b*)=*c*solve

- Recognize relationships between definitions in the unit circle and the graph of the trigonometric functions
- Relationships between the trigonometric functions
- Nonlinearity of the trigonometric functions (addition theorems)

- To be able to interpret an oblique image

- Clear presentation of spatial situations

- Volume and surface (cube, cuboid, prism, pyramid, tetrahedron, octahedron, cylinder, cone, sphere) Derive volume formulas

- Cavalieri's principle

- Coordinate system
- Concept of the vector
- Collinearity, Complanarity
- Meaning of the elementary vector operations, the scalar and vector product
- Forms of description of straight lines (in plane and space) and planes through parameter and coordinate equations
- Mathematical solution of incidence problems

- Add, subtract and scalar multiply vectors graphically and mathematically
- Length of a vector
- Scalar and vector product
- Solve incidence problems arithmetically

- Process more complex tasks with different solutions
- Calculate missing points in a geometric figure or body
- Calculate area contents and volume

- Terms "vector" and "vector space", linear (in) dependence, base and dimension (it is better not to define vector space properly)
- Know concept of group

- Check linear (in) dependency
- Linear combinations
- Decompositions on a standard basis

- multi-dimensional spaces (> 3)

- Homogeneous and inhomogeneous linear systems of equations

- Different solution methods (Gauss method) can be used

- Geometric interpretation
- Relationship with matrices and linear mappings (core)

- Terms "matrix" and "linear mapping"

- Know arithmetic operations with 2x2 matrices (product and inverse)

Calculate determinant - Solving a system of equations using an inverse coefficient matrix
- Description of a linear mapping through a matrix

##### Time requirement

30 to 40 weeks & agrave 4 lessons (secondary level II) plus 15 to 30 weeks (secondary level I).

##### Further exploratory in-depth topics

- Golden cut
- Harmonic oscillations
- Platonic bodies
- Euler's polyhedron substitute
- Circle, sphere and tangent plane
- Late product
- Conservation factors for parallel projections
- Affinity
- View of the central perspective

##### Cross connections to the other areas

###### Analysis

###### Algebra

- Position of lines and planes and systems of linear equations
- Intersection straight line - circle / sphere and quadratic equations

##### Applications and links to other specialist disciplines

###### Physics

- Addition of speeds and forces
- Scalar product and physical work
- Vector product and torque
- Magnetic part of the Lorentz force
- Especially with parameters as time and direction vector as speed vector
- Harmonic oscillations

###### Geography

###### Technology

###### Astronomy

- Historical distance determinations: Earth circumference according to Eratostenes, distance earth-moon according to Aristarchos of Samos, Lalande and Lacaille, sizes and distances of the moon and sun to Aristarchos of Samos
- Definition of 1 parsec

- Graphic / numerical representations (histogram, box plot, scatter plot)
- Measures of position and scatter (arithmetic mean, median, standard deviation, quartile difference)
- Scatter diagram and basic idea of the correlation

- Create calculations / graphics with IT resources

- Recognize misleading representations
- Difference in correlation / causal relationship effects of variable transformations
- Linear regression

- Random experiment
- Finite base space (or event space, sample space)
- Concept of probability
- Additivity of probability
- Stochastic independence
- Conditional probabilities

- Calculations in the Laplace model (combinatorics)
- Exploiting P (
*A.*C) = 1 - P (*A.*) - Representation of multi-stage random experiments as a tree or four-field table
- Application of the path rules

- Examples of paradoxes
- Geometric probabilities
- Distinction between random and constructed binary sequences (musts)
- Bayes' formula (tree version)

- Stochastic independence
- Conditional probabilities

- Representation of multi-stage random experiments as a tree or multi-field table
- Application of the path rules
- Tree version

- Bayesian formula
- Medical tests
- Simpson's paradox

- Random variable, expected value (interpretation "in the long term")
- Fair play
- Binomial distribution (occurrence, requirements)

- Calculation in examples
- Probability of intervals [
*k*_{1},*k*_{2}] with the binomial distribution (calculation with a pocket calculator or with Sigma rules) - Dealing with summation symbols

- Apparent contradiction between the lack of memory of chance and the interpretation "in the long term"
- Shape and spread of the binomial distribution as a function of
*n*and*p* - Differences between binomial and hypergeometric distribution
- Approximation of the binomial distribution by the normal distribution

- Differences:
- Sample / population
- relative frequency / probability

- Calculations in the binomial distribution (quantiles)

- Rejection area for the two-sided binomial test (determined with a pocket calculator or with the Sigma rule)

- Opinion polls, questionnaires (systematic and random errors) sign tests

- Types of errors in testing
- Addition principle
- Multiplication principle
- Urn model (pulling), fan model (distributing)
- Number of permutations, variations and combinations
- Facult
- Binomial coefficients
- Use different counting methods
- Recursive calculations
- Unfavorable spectra for the primary valences only lead to a small gamut in image reproduction, but perfect reproduction of the colors is possible within this small gamut (the triangle becomes small).
- Unsuitable spectral sensitivities of the primary colors of an image recording device lead to mostly uncorrectable color errors (one bends the horseshoe).
- It is not possible to build a monitor that can display all the colors that humans can perceive.
- The dead and hot pixels of a camera can be mapped out, but this is not easily possible for a display.
- In this video I show you how to calculate the acid constant for a given degree of dissociation and a known concentration.
- For the general dissociation reaction you can calculate the dissociation constant as follows: c (AB), c (A) and c (B) stand for the concentrations of the substances AB, A and B in mol / l. stands for the rate constant of the forward reaction (= dissociation)
- Example. Acetic acid has a dissociation constant of α = 1.8 ⋅ 10 - 5. This means that a degree of dissociation of 10% is only exceeded below a concentration of approx. 2 mmol L-1
- silver (I) complex has reacted
- Dissociation constant. Author: Hans Lohninger The dissociation constant specifies the tendency of a substance A x B y to split reversibly into two smaller parts A and B in a solution:. A x B y xA + yB. The dissociation constant is denoted by K d and is calculated as follows:. where [A], [B], and [A x B y] are the molar concentrations of the molecules / ions A, B, and A x B y.
- The dissociation is an equilibrium reaction, so that an equilibrium constant, here referred to as the dissociation constant Kc, can also be calculated for it using the law of mass action. The degree of dissociation a is defined as the quotient of the amount of substance x in moles that is dissociated in ions to the total amount of dissolved substance a in moles
- The solution from knowing the pH value is easier than from the dissociation constant of the acid and the initial concentration. Solution of a known pH or pOH. Check whether the information includes the pH or the pOH of the solution. Calculate the concentration of hydrogen ions by setting the pH negative to 10. For example for a solution of pH.
- Naturally, this does not work with the weak electrolyte CH 3COOH; the detour described in the basic section must be chosen here. Typical values of molar equivalent conductance values in the Kohlrausch plot. For acetic acid, the degree of dissociation α and the dissociation constant Kc must also be calculated and presented as a function of the concentration. Degree of dissociation α of.
- For example, the K M value calculation is to be carried out using a Hanes plot. Eadie-Hofstee Plot With the Eadie-Hofstee plot, too, the problem of the irregular distribution of the measuring points and the increase in errors is overcome by directly plotting the working speed on the y-axis. For the x-axis, the working speed is determined by the.
- LK only: Fill out the table for the dissociation constant completely. 1. Set up the reaction scheme for the decomposition of dinitrogen tetroxide to nitrogen dioxide and calculate the reaction enthalpy of the reaction. Explain what an enthalpy of reaction is and how it can basically be measured. 2. Calculate.
- Starting materials available (in the previous example: N 2 and H 2). Therefore, initially only the forward reaction begins, and at a relatively high speed. As soon as the first particles of the reaction products have formed (in the previous example: NH 3), the reverse reaction begins, albeit initially at a very slow rate. As the reaction progresses, arise.
- Dissociation constant K. - With the Nernst equation z. B. calculate the membrane potential. Examples: With an H + ion concentration of 103O -2, the pH value of the solution is 2. If an acid has a Ks value of 10 6, its pKs value is −6, which means that it is a strong acid. In addition, the logarithm can also be found for all exponential processes.
- When developing, an alkaline, The reciprocal value 1 / K is the so-called complex disintegration or dissociation constant. The larger the value for K, the further the equilibrium is to the right and the more stable the complex is. When it comes to the stability of complexes, a distinction is made between thermodynamic and kinetic stability. In the.
- It can be calculated from the concentration using the following equation: p osm = [A] · R · T. R = 0.0831 l · bar / mol · K. A one-molar glucose solution, i.e. a solution of 1 mol of glucose in 1 l of water, provides an osmotic pressure of 24.8 bar, a solution of 1 mol of glucose in 24.8 l a pressure of 1 bar (at 25 ° C)
- Teaching qualification 1a summer semester 2010 4 Measuring principle: With the control resistances R 1 and R 2 it is achieved that I B = 0. Then the following applies for the resistance of the measuring cell: RZ = R 3 · (R 1 / R2) Hence: κκκκ = 1 / ρ = 1 / Rz · (ℓ / A) = k / R Z electrical conductivity
- My book also discusses the concentration of $ & # 92ce
$ calculated where $ & # 92ce <[HIn] = [In ^ <->]> $ is calculated. Apparently this is the concentration of $ & # 92ce $ at which the indicator changes color and thus the end point of a titration can be identified, but how is this the point at which the indicator changes color. I don't mean that. - Calculate the constant of stability for the [Cu (NH3) 4] ² + ion. Can someone help me? EtOH Registration date: 10/22/2004 Posts: 3367 Residence: leverkusen / köln: Posted: Jan 24, 2005 7:18 PM Title: hmmm. So now just know that the complex formation constant (and you have a complex there). is the reciprocal of the dissociation constant. and this (the complex formation constant.
- E = enzyme, I = inhibitor, EI = enzyme-inhibitor complex, ES = enzyme-substrate complex, EIS = enzyme-inhibitor-substrate complex P = product, k = rate constant, K = dissociation constant For the allosteric / non- In competitive inhibition, the inhibitor does not bind directly to the active site, but to another point on the enzyme
- The Bronsted Lowry definition of an acid and a base is that an acid donates hydrogen ions while a base receives the hydrogen ions.
- Solving from acid dissociation constant (Ka) and amount A hydrogen ion concentration in a solution results from the addition of an acid. Strong acids give a higher concentration of hydrogen ions than weak acids, and it is possible to calculate the resulting hydrogen ion concentration by taking either the pH or the strength of the acid in a solution.
- The first dissociation constant of sulfuric acid says that the first dissociation stage of sulfuric acid corresponds to that of a strong acid (K 1 & gt & gt 1). This means that in the first dissociation step all protons are completely released (H 2 SO 4 becomes HSO 4 -).The concentration of protons corresponds to the initial concentration of sulfuric acid

In the calculus of probabilities, not only Laplace models should be dealt with.

The combinatorics can also be dealt with separately, especially if it is discussed in more detail (see table below).

##### Time requirement

25 to 30 weeks & agrave 4 lessons.

Parts of descriptive statistics, occasionally also combinatorics, are already dealt with in lower secondary level.

## Story

Continuum mechanics is based on mechanics, physics, differential geometry, differential and integral calculus whose historical development can be looked up there. Some stages in the development of fluid mechanics also run parallel to continuum mechanics. At this point the specific continuum mechanical development with emphasis on the mechanics of solid bodies is to be sketched.

Archimedes (287–212 BC) already dealt with fundamental mechanical questions concerning solids and fluids, over 1500 years before Leonardo da Vinci (1452–1519) devised solutions to numerous mechanical problems.

Galileo Galilei (1564–1642) discussed strength problems in his Discorsi and thus founded the theory of strength at a time when solid bodies were mostly modeled as undeformable. Edme Mariotte (1620–1684) made contributions to problems of liquids and gases and established the first constitutive laws, which Robert Hooke (1635–1703) also did for elastic solids with Hooke's law 1676 named after him.

Isaac Newton (1643–1727) published his Principia in 1686 with the laws of gravity and motion. The members of the Bernoulli family made contributions to mathematics, fluid mechanics and - through Jakob I. Bernoulli (1654–1705) - to beam theory. Leonhard Euler (1707–1783) gave important impulses for the mechanics of rigid and deformable bodies as well as for hydromechanics. Jean-Baptiste le Rond d’Alembert (1717–1783) introduced Euler's approach, derived the local mass balance and formulated the d’Alembert principle. Joseph-Louis Lagrange (1736–1813) set up his fundamental work in 1788 *Mécanique analytique* the mechanics consistently mathematically.

The fundamental concepts of stress and strain tensor in continuum mechanics were introduced by Augustin-Louis Cauchy (1789–1857). Other fundamental insights were brought in by Siméon Denis Poisson (1781–1840), Claude Louis Marie Henri Navier (1785–1836), Gabrio Piola (1794–1850) and Gustav Robert Kirchhoff (1824–1887).

As a result of industrial and practical needs, technical issues dominated the science that was pursued in France, among others, in the École polytechnique, which was shaped by Cauchy, Poisson and Navier. Their model spread across Europe, in Germany as a technical university. Engineering disciplines such as plasticity theory, creep theory, strength theory, elasticity theory and civil engineering emerged. As a result of this fragmentation, research and teaching in the sub-areas developed independently of one another and the continuum mechanical relationships were lost. Fluid mechanics also developed independently.

David Hilbert (1862–1943) gave a new impulse to thought with his 1900 list of 23 mathematical problems awaiting a solution. The sixth problem, “How can physics be axiomatized?” Is still unsolved at the beginning of the 21st century, but interdisciplinary work on continuum mechanics, especially by Georg Hamel (1877–1954), arose before the Second World War. After the war, intensive interdisciplinary basic research began, in which Clifford Truesdell (1919–2000) and Walter Noll (* 1925) provided impetus.

From the middle of the 20th century, computer hardware and software and numerical methods for solving equations developed so far that solutions for complex, practical, continuum mechanical problems can be found with their help. & # 911 & # 93 & # 912 & # 93

## Summary

A method for determining thermodynamic acidity or basicity constants is described. The potentiometric titration curve is linearized with iterative inclusion of the activity coefficient and with an acidity constant calculated without an activity coefficient. The most probable straight line determined by the method of the minimum sum of the squares of errors is indirectly derived from the acidity or basicity constant. The method is also suitable for mixed solvent systems if the dielectric constant is large enough to prevent associative formation. The accuracy of the method (as standard deviation) is approx. 0.01 pK units.

## Table of contents

### Generation edit

### Edit density function

### Distribution function edit

The distribution function of the logarithmic normal distribution appears on logarithmically divided probability paper as a straight line.

### Multidimensional log-normal distribution edit

### Edit quantiles

### Median, multiplicative expectation edit

### Multiplicative standard deviation edit

Since the multiplicative or geometric mean of a sample of lognormal observations (see “Parameter estimation” below) is itself lognormally distributed, its standard deviation can be given, it is (σ ∗) 1 / n < displaystyle ( sigma ^ <*>) ^ <1 / < sqrt

### Expected value edit

The expected value of the logarithmic normal distribution is

### Variance, standard deviation, coefficient of variation edit

### Skew editing

d. that is, the log-normal distribution is skewed to the right.

The greater the difference between the expected value and the median, the more pronounced i is. a. the skewness of a distribution. Here these parameters differ by the factor e σ 2/2 < displaystyle mathrm

### Edit moments

All moments exist and the following applies:

The moment-generating function and the characteristic function do not exist in explicit form for the log-normal distribution.

### Entropy edit

The entropy of the logarithmic normal distribution (expressed in nats) is

### Multiplication of independent, log-normally distributed random variables

### Edit limit set

### Expected value and covariance matrix of a multidimensional log-normal distribution

The expectation vector is

### Relationship to normal distribution edit

### Heavy-Margined Distribution Edit

### Parameter estimation edit

The parameters are estimated from a sample of observations by determining the mean value and (squared) standard deviation of the logarithmic values:

### Edit statistics

In general, the simplest and most promising statistical analysis of log-normally distributed quantities is carried out in such a way that the quantities are logarithmized and the methods based on the normal normal distribution are used for these transformed values. If necessary, the results, for example confidence or prediction intervals, are then transformed back into the original scale.

Such asymmetrical intervals should therefore be shown in graphical representations of (untransformed) observations. [2] [3]

Variation in many natural phenomena can be described well with the log-normal distribution. This can be explained by the idea that small percentage deviations work together, i.e. the individual effects multiply. This is particularly obvious in growth processes. In addition, the formulas for most of the fundamental laws of nature consist of multiplications and divisions. Additions and subtractions then result on the logarithmic scale, and the corresponding Central Limit Theorem leads to the normal distribution - transformed back to the original scale, i.e. to the log-normal distribution. This multiplicative version of the limit theorem is also known as Gibrat's law. Robert Gibrat (1904–1980) formulated it for companies. [4]

In some sciences it is customary to give measured quantities in units that are obtained by taking the logarithm of a measured concentration (chemistry) or energy (physics, technology). The acidity of an aqueous solution is measured by the pH value, which is defined as the negative logarithm of the hydrogen ion activity. A volume is specified in decibels (dB), that = 10 log 10 (E) < displaystyle = 10 log _ <10> (E)>, where E < displaystyle E> is the ratio of the sound pressure level to a corresponding reference value is. The same applies to other energy levels. In financial mathematics, logarithmic values (prices, rates, income) are also often used for calculations, see below.

For such "already logarithmized" quantities, the usual normal distribution is often a good choice, so if one wanted to consider the originally measured quantity, the log-normal distribution would be suitable.

In general, the log-normal distribution is suitable for measured quantities that can only have positive values, i.e. concentrations, masses and weights, spatial quantities, energies, etc.

The following list shows with examples the wide range of applications of the log-normal distribution.

## Use

The RGB color space is also used for self-luminous (color-displaying) systems that are subject to the principle of additive color mixing *Light mixing* designated. According to Graßmann's laws, colors can be defined by three specifications, in the RGB color space these are the red, green and blue components. The specific form of the color space depends on the specific technical system for which the respective color space was determined.

*sRGB* (Standard-RGB) was developed for monitors whose coloring base is three phosphors (luminescent substances). Such a substance emits a spectrum of light when electrons hit it; suitable phosphors are those with narrow-band emissions at wavelengths in the range of the perceptual qualities blue, green, red. The viewer gets the color impression defined in the RGB color space (with sufficient distance from the screen the pixels merge additively). The intensity of the excitation beam corresponds to the triple in the RGB color space and can be specified, for example, as a decimal fraction (0 to 1 or 0 to 100 & # 160%) or discretely with 8 bits per channel (0 ... 255) (8-bit TIFF). Depending on the type of application, certain representations of values are preferred.

With larger storage media, tone levels of 16 bits per channel became possible. So three times from 0 to 65535 ($ 2 ^ <16> $) are possible, so a total of 281 trillion colors, for example with 16-bit TIFF and 16-bit PNG. Good technical output systems can reproduce more colors than humans can distinguish, even trained people only come up with around 500 & # 160000 color nuances. & # 912 & # 93 For special applications, however, 16-bit values make perfect sense. In this way, more precise observations are possible for evaluations in X-ray diagnostics.

The color reproduction in cases such as color pictures from PC printers, color photos on a silver halide basis, the printing of magazines, color pictures in books is done by remission on the presenting surface. The laws of subtractive color mixing, for which the CMY color space was developed, apply here because of the color depth, usually with black for color depth as the CMYK color space.

The representation of the RGB color space takes place (less clearly than with other color spaces) in the Cartesian coordinate system as a cube. The figure shows the view of the rear wall on the left, the view in the middle, and a view of the interior on the right. Red, green and blue components follow the axes in the corners are yellow, magenta, cyan. At the coordinate origin with R = G = B = 0 there is black, along the spatial diagonal gray up to the corner point in white.

### Use of RGB color spaces for image reproduction

RGB color spaces as additive color spaces serve as the basis for displaying color images by means of image reproduction devices that additively combine colors from three or more colors. In addition to CRT and TFT displays, these are also video projectors.

It is irrelevant how the individual color channels are controlled, whether by an analog or a digital signal with 5, 8, 10 or 16 bits per color channel.

The three basic colors red, green and blue are usually used for representation. However, more colors can also be used to enlarge the gamut or the maximum brightness. In this way, colors covered by the polygon can be represented better, at least with lower brightness. There is no restriction to the RGB triangle enclosed by the horseshoe. To increase the maximum brightness, white can also be used as an additional basic color. Greater brightnesses can thus be represented, but with a further loss of gamut. Both options are used with DLP projectors.

However, in these cases further processing of the RGB data of the graphics card by the output device is necessary. In the case of multi-color projection, a suitable working color space of the graphics card is necessary in order to be able to use the advantages.

The corner points of the RGB chromaticity triangle can be chosen arbitrarily, they are *not* limited by the availability of fluorescent crystals. There is no inseparable relationship to the three (basic) light colors that the luminescent materials of the output device can produce. Color values outside the triangle defined by the corner points cannot be displayed. In a picture tube, for example, many of the strong, rich green and blue tones that occur in nature are missing, and the spectrally pure red and violet are also missing in the RGB space.

If the luminescent materials of a screen are used by LEDs or similar elements for red, green, blue, the color effect remains unchanged compared to this description, provided they can cover the RGB space used. For example, flat screens do not have a picture tube and generate the colors through electrical field excitation. Other phosphors require a different position of the RGB triangle (shown on the xy color sole). The technical requirement is to adapt the position of the diagram corner points for LC displays to the position in picture tubes as much as possible. If this does not succeed, a mathematical conversion has to be carried out, whereby colors can be omitted, since the coordinates cannot have negative values. If the conversion is also omitted, the colors will be displayed distorted. It is possible that color nuances between red and (yellow-orange) are displayed noticeably differently on different devices.

### Use of RGB color spaces for image acquisition

Although at first glance it looks as if the image recording is subject to the same principles as the image reproduction, there are fundamental differences between the image recording and the image reproduction:

## Calculating the dissociation constant Example

The use of the negative logarithm of the dissociation constant Ks leads to the following relationship between the pKa value of an acid and the pH value of an aqueous solution of this acid: Accordingly, the pKa value of an acid is the pH value at which 50% of the acid is (HA) is dissociated to A, that is, if [A-] = [HA] The dissociation constant, abbreviated to K d, is a special case of the equilibrium constant from the law of mass action for the dissociation of an electrolyte (acid, base, salt) in one Solvent (liquid e.g. water). The constant indicates the relationship between the dissociated and undissociated form. 2 calculation **Therefore, every dissociation can also be calculated using the law of mass action**. In the case of a dissociation, however, we do not speak of an equilibrium constant (which is derived from the law of mass action), but rather as a dissociation constant. Note: as mentioned above, most people know the degree of dissociation in acids and bases. In this case one does not speak of that. Dissociation constant pharmacodynamics. synonymous: KD, Ki definition and properties. In order for a drug to be able to exert its pharmacological effects, the active substances it contains usually have to bind to molecular target structures in the organism, the so-called drug targets, such as receptors, enzymes, transporters and ion channels

*The dissociation constant of the enzyme-inhibitor complex (Ki) is calculated using the following formula: The process is shown schematically using the example of the LDH reaction: 49 NADH has an absorption maximum at 260 nm and an additional maximum at 340 nm, that of the oxidized form, the NAD +, is missing*. In practice, the measurement is often carried out at 366 nm instead of 340 nm. (c (H 3 O +) ⋅ c (A -) c (H A) ⋅ 1 l m o l) The smaller the pK s value, the stronger the acid. For example, nitric acid (HNO 3, degree of dissociation of 96% at a concentration of 1 mol / L) has a pK a value of −1.32, acetic acid (degree of dissociation of 0.4% at a concentration of 1 mol / L) has a pK s of 4.75

### What is the acid dissociation constant and how is calculated

Dissociation, pH value and buffer The molar concentration (molar concentration) c of a substance is shown in square brackets in this text, e. B. [CH3COOH] instead of cCH3COOH or c (CH3COOH). 1 Electrolytes Electrolytes are substances that dissociate in water: acids, bases, salts. HCl → H + + Cl- NaOH → Na + + OH- NaCl → Na + + Cl As always with the law of mass action, you have to make sure that the stoichiometric coefficients are taken into account! In the next step you can set the activity of your solid, i.e. equal to 1. Then you replace the activities with concentrations, because this is a good approximation for very dilute solutions. Then you will get the. The dimensionless degree of dissociation α (also called degree of protolysis) gives the ratio of the dissociated acid or dissociated. Base particles for the formal initial concentration of the undissociated acid or base in an aqueous solution. The degree of dissociation of an acid or base depends on its acid constant (or base constant), its concentration and the existing pH value. **Dissociation constant** K D (M) • The higher the affinity of a protein for its ligand, the lower the **Dissociation constant** (K D) of the complex association constant **Dissociation constant**. Determination of the binding affinity of a peptide for TAP 0 100 200 300 400 0 5 10 15 C4F peptide (nM) BC-II practical WS 06/07 Experiment 1C nM bound peptide fit using one-site binding. When comparing with calculated binding energies, the zero point energy and temperature-dependent corrections must be taken into account. Dissociation energy: The dissociation energy D '' of small molecules AB in the ground state and D 'in an electronic excited state can be determined with the help of the convergence point in the band spectrum. E at and E mol are the excitation energies of the.

Example: HCl c = 0.01 mol / l. Æ pH = - lg (0.01) = 2. Calculating the pH value of weak acids: Weak acids have a weak degree of dissociation. Approximation: c (HA) @ initial concentration. In addition, the following applies: c (H3O +) = c (A-) Ks = () 3 () * () c HA + c HO c A - = () 3 () 2 c HA c HO Æ c (HO +) = 3. Ks c HA * () Æ pH = 2 1 × (pKs - lg c (HA)) Example: acetic acid pKs = 4. Dissociation is the term used to describe the splitting of molecules or ions into smaller components. It can be homolytic or heterolytic. The dissociation of substances is triggered by various processes. It occurs when substances are dissolved in water or other solvents, but can also be caused by high temperatures, electrical currents or electromagnetic This is the case, for example, when water is both the reactant and the solvent (e.g. in ester hydrolysis). In this case, the reaction rate follows the laws of a first-order reaction. However, since it is still a bimolecular reaction, one speaks of pseudo first order (or apparently first order) reactions. A. When dissolving exactly 100.0 g of 98% sulfuric acid in exactly 962.7 g of distilled water, the concentration of the sulfuric acid is exactly 1 mol / L. If you take 106.38 g of the sulfuric acid solution and dissolve it in 900 mL of distilled water, this solution has a concentration of 0.1 mol / L, if you take 10.64 g of the first solution and dissolve it in 990 mL of distilled water. *In the case of acetic acid, the dissociation constant is 1.8 · 10-5 [1]*. The following table shows some acids according to their degree of dissociation, i.e. according to their strength. The strength of the acid decreases from top to bottom

to represent. The calculation of the rate constant k (according to the Arrhenius equation) includes the frequency factor A (also known as the pre-exponential factor), the activation energy EA (unit: J / mol), the universal gas constant R (= 8.314 J / (K mol)) and the absolute temperature T (in Kelvin K):. The empirical Arrhenius equation assumes that the frequency factor A. Limiting conductivity calculates the dissociation constant of the acid. You will be given the molar conductivity limit of the acid to be examined (see label on the bottle). You can also approximate this value by extrapolating from your own experimental data. 2 Measurements / calculations to be carried out 1. Determination of the cell constants. **Inferences about the climate of the past**. For example, water containing the isotope O-16 evaporates more easily than water containing the isotope O-18. Ice ages, during which large amounts of water are de-formed as an ice sheet

The total magnification of the microscope is calculated from the number of scales of the objective, multiplied by the magnification of the eyepiece and, if necessary, multiplied by intermediate magnifications. There is a difference between magnification and reproduction scale. While the magnification always relates to the impression of the eye, the image scale is always a measurable variable. *Law of mass action, abbr*.MWG, law on the equilibrium concentrations of a chem. Reaction, that is: The product of the concentrations of the end substances divided by the product of the concentrations of the starting substances in the chem. Equilibrium is a constant, the equilibrium or mass action constant K c. The stoichiometric coefficients appear as exponents of the.

### Dissociation constant - DocCheck Flexiko

Dissociation constant as equilibrium constant Kc: + this is what one makes with the colorless one. p-Nitrophenol takes advantage of the fact that the phenolate anion is colored yellow. The relationship between concentration. c. i. of the substance. i and the measured absorbance E. i, λ. as a measure of the light absorption is according to the Lambert-Beer law. E. i, λ = log (I. 0 / I) = ε. i, λ ⋅ c. The ratio of the concentration of dissociated particles (protons and acid residue ions) to the concentration of the starting acid is called the degree of dissociation (dissociation constant) α. The degree of dissociation depends on the concentration. The dissociation equilibrium shifts to the right with increasing dilution. Strong acids are over 60. The reciprocal value that describes the dissociation of complexes is called the dissociation constant. Calculation of the complex decay constant. The complex formation constant is calculated from the law of mass action for the overall reaction: KK = c (Ni (CN) 4) to the power of 2 divided by C (Ni to the power of 2 +) times c4 (CN-) the complex dissociation is the reverse of the complex formation. Complexes that are in aqueous. Calculating the dissociation constant Ks of an acid In chemistry, we are supposed to calculate Ks of a weak acid HX with 0.26 mol / l and a pH value of 2.86 and I can neither find any useful instructions on how this works on the Internet nor in the chemistry book. I really need a clear explanation. Above all, I can't do anything with the equation [H +] * [X-] / [HX] because. V6 dissociation constant of a weak acid. presented by. Fabian Hecker Roland Schulz Bastian Blume. The aim of the experiment is to determine the dissociation constant of p-nitrophenol. For this purpose, p-nitrophenol solutions with different concentrations are examined photometrically. The extinctions can be used to determine the.

### The degree of dissociation in chemistry - learning location-MIN

### PharmaWiki - dissociation constant

The equilibrium constant is called the dissociation constant K. Logarithmic transformation results in the HENDERSON-HASSELBALCH equation from the mass action equation: [A-] pH = pK + log ----- [HA] If the concentration of [A-] = [HA], the logarithm has the value zero and then pH = p. According to Arrhenius, all acids dissociate with the help of water to form hydrogen ions and the corresponding acid residue ions :. Since all acids have the same (or similar) properties (red color of the universal indicator, acidic tasting or caustic), there must also be a particle that is responsible for all of these properties

### Acid constant - chemistry school

Dissociation constant pharmacology. The dissociation constant is an equilibrium constant that indicates where the equilibrium of this process lies and indicates the ratio of bound and unbound active ingredient. It is defined as follows (C = concentration, in mol / L): K D = C (W) · C (R) / C (WR) The unit of K D is mol / L and can also be specified with M Die. Example: Formic acid HCOOH, KS = 10-4 mol / L -4 S -2 0 0-4 S -1 0 -2 0 K10 c = 1 mol / L: = = = 10 1% c1 K10 c = 0.01 mol / L: = = = 10 10% c 10 α α ≡ ≡ Task: A 0.1 molar HAc is protolyzed to 1.34%. Calculate KS and pK S. KS = α 2c 0 = (0.0134) 2 × 0.1 mol / L = 1.796 × 10-5 pK S = -log KS = -log 1.796 × 10-5 = 4.74 This means that we can write down a dissociation level for every proton given off. This is shown in Figure 5 using the example of sulfuric acid. Figure 5: Dissociation stages of the biprotonic acid H2SO4 (sulfuric acid) One proton is given off by the acid for each dissociation stage The complex dissociation represents the inverse of the complex formation, accordingly the dissociation constant KD is the reciprocal of the complex formation constant K K. KD = c (Ni 2 +) c 4 (CN -) c [Ni (CN) 4] 2- = 1 K K. Since a gradual addition or dissociation of the ligands is possible, it makes sense to determine individual formation constants for the individual partial reactions. For the tetracyanonickelate (II) complex, four different from one another can be identified. Suitable eye protection (protective goggles) must be worn when handling hazardous substances. Protective goggles must be worn at all times in areas where chemical substances are used. The contact of hazardous substances with the skin or clothing is to be avoided as a matter of principle. When working with hazardous substances is suitable.

### Calculation of the acid constant Ks (acetic acid) - YouTub

### Dissociation • Definition, degree of dissociation · [with video

**Calculate the specific and molar activity of the enzyme from vmax**. To do this, you need the concentration of the enzyme in the measurement approach. 4. 5.4 Determination of KI The inhibitor constant KI is to be determined with equation (10). To do this, measure v at five different concentrations of the inhibitor K2HPO4 at a constant [S] = 2.5.10-3 mol / l. Use yours for the calculation. From the test results of D03 (example) one obtains (HAc). = 423 S. cm2 / mol + 91 S. cm2 / mol - 122 S. cm2 / mol = 392 S. cm2 / mol. Literature value: 390.7 p. cm2 / mol. Determination of the dissociation constant of acetic acid 1. Calculation of the degree of dissociation From the theoretical considerations (worksheet D00 - page 5 equation 8) the degree of dissociation can be calculated as a quotient from the.

I am supposed to calculate dissociation constant K and give it in mol / L. An example: I have a buffer of 44.2ml 0.1M citric acid and 75.8ml 0.2M Na2HPO4, and a 0.01 molar p-nitrophenol solution. I put 1 ml of this in a 100 ml flask and fill it up to the mark with the buffer. On the colorimeter I read the absorbance 0.15 and concentration 29, no idea whether this is here. **When plotting cII against cI, a straight line with a slope of K should result**. Often this is not the case, however, since dissociation, hydrate formation or polymerization can occur in water or dimers (association) can be formed in non-polar solvents. As a result of these effects, the mean particle size is different in the two phases. The distribution of the substance. 3. Titration of acetic acid with sodium hydroxide solution - closer examination of a selected example. Calculation of the pH at different points of the titration curve Region 1: Region 2: Buffer area: Apply the Henderson - Hasselbalch equation for VB = 3.00 ml: for VB = 5.00 ml the following applies: Region 3: analogous to the calculation in Region 1: Region 4: pH value at VB = 12.00 ml: 4.

Examples of clinically relevant gene variants: CYP2D6 polymorphism. There are hyper- and hypoactive variants of the enzyme. Involved in the metabolism of many pharmaceuticals There are gender-specific differences in the metabolism via CYP2D6 known N-acetyltransferase polymorphism. There are hyper- and hypoactive variants of the enzyme, among other things isoniazid. Simple handling of the calculation using the input values in the tool. The enzyme reaction and its speed is used in medicine and biology and accordingly the calculation takes place here. Biological substrates and their effect on the human organism are considered and analyzed with these calculations in the enzyme kinetics. This is where the. Perhaps @dermarkus you could count on what you come up with and post it to me: dermarkus administrator Registration date: 12/01/2006 Posts: 14788 dermarkus Posted: May 12, 2008 14:48 Title: I only typed the last term because the makes the largest contribution to the overall error. I also get the 1.77, but I still have a unit behind it. The example of weak electrolytes shows that dissociation reactions are reversible. This statement is not suitable for strong electrolytes, since practically all molecules are broken down into ions. The tendency of the system to equilibrium is described by the electrolytic dissociation equation KxAu ↔ x • K + + y • A- and shows the dissociation constant Kd = [K. **Based on the dissociation constant of water (k Diss), the pH value for pure water and dilute aqueous solutions at 25 ° C is divided into the following scale: pH & lt7 - solution with acidic effect pH = 7 - absolutely pure water or neutral solution pH & gt7 - Solution with basic effect k Diss = c (H +) * c (OH-) = 10-14 mol 2**.

With a spherical rotor (tetrahedral or octahedral symmetry of the molecule, e.g. with CH4 or SF6), Ix = Iy = Iz = I. In general, Lx 2 + L y 2 + L z 2 = L2. During the transition from the classical to the quantum mechanical view, L2 is replaced by J (J + 1) h2, see Chapter 3.1.4, and the result is () (1 2 1 2 h B J J = + = + I J E J J h). (6.02) J no longer provides the. **example**: To **to calculate** is the molar concentration c in mol·l-1 of a concentrated ammonia solution (25%) with r = 0.907 g · ml-1. c = n * V-1, n = m * M-1, r = m * V-1 Þ c = r * M-1. c (NH3) = 25% r M-1 = 13.31 mol l-1. **example**: Determination of formulas from element analysis of a compound consisting of C, H and O: w (C) = 76.4%, w (H) = 6.40%, w (O) = 16.90%. What's that. 4 examples. Acid pK S Base pK B Perchloric acid HClO4 - 10: perchlorate ClO4-24 hydrochloric acid HCl - 6: chloride Cl-20 sulfuric acid H 2 SO 4 - 3: sulfate HSO 4-17 ammonia NH 3: 23: amide NH 2-- 9 see also: strength of acids and bases. Tags: base, base strength, equilibrium, law of mass action, acid, acid-base reaction, acid strength. Subjects: biochemistry, chemistry. More important. equilibrium constant conversion from kc to kp. 05 the law of mass action equilibrium constant kc kp with youtube as an example. determination of the equilibrium constant. preparation for the law of mass action and the calculation with kc youtube. question about the calculation of equilibrium constant chemistry biology biochemistry. relationship between free enthalpy and chemical equilibrium. 11 vo.

Examples of pH value calculation Please calculate the pH value a) a 2 molar HCl b) a NaOH (c = 0.05 mol / L) c) a 10-8 molar HCl d) an acetic acid with pKa = 4.75 c = 2 mol / L e) an ammonia solution with a concentration of 0.05 mol / l, Ks = 5.62 * 10-10 mol / L Sabine Willem. Please explain the relative position of KM / Ki (drug)! In the case of non-competitive inhibition, the inhibitor is not bound to the active center, but to another point on the enzyme (see Fig. 1). This reduces the activity of the enzyme. Because in this case the inhibitor is not. mol / l, mmol / l base capacity / dissociation constant of the base Fig. 2-9 Formation of the hydroxide ion using the example of water and ammonia. 18 Fig. 2-10 Hydration of sodium and chloride ions. 18 Fig. 2-11 pH value curve during the titration of a weak acid with a strong base, plotted for different pKa values. 22 Fig. 2-12 Existence areas of the solved. Check the translations of 'Dissociation Constant' to Dutch. Take a look at examples of dissociation constant translations in sentences, listen to the pronunciation and learn the grammar KCHA CH cs 5, CHA-Cg 'However, the accuracy of the constants calculated in this way is no longer as great as that of the dinitrophenol itself , of which one can see by calculating a few examples. According to the equations, the conditions are most favorable when the dissociation of the dinitrophenol is reduced as much as possible.

### Electrolytic conductivity - Chemgapedi

calculate. This value is not far from the heat of neutralization in an aqueous solution (13 700 cal.). The second method used to determine the dissociation constant of methyl alcohol was as follows. According to ARRHENlus', the dissociation constant Zeitschr. physics. Chem. 5, 16 (1890) Example Calculate the dissociation constant KS of a 1 molar weak acid with the pH value 3. So use the appropriate formula: [H 3O +] = KS ∙ [acid], into which we add First insert all the information and then resolve according to KS: The information pH = 3 gives us the [H 3O +] = 10−3, 1 molar means [acid] = 1 and therefore only KS remains under the root: 10 - 3 = KS or K. You also have constant dissociation learning materials? Then share them on Uniturm.de and help others to get through their studies more easily. This not only ensures good karma, but also secures you points that you can exchange for nice prizes in our bonus section! Search: Subjects Title of document uploaded Biosciences University of Potsdam »Mathematisch. calculate and use equation (7) to calculate the acid constant (dissociation constant) of acetic acid. As can be easily checked, this is almost constant within the given concentration range (pK s (HAc) at 25oC: 4.76). OSTWALD's law of dilution can be formulated differently with equation (8): 2 eq0 c eq c K (9) With this equation, weak electrolytes. KD dissociation constant λ wavelength LUV large unilamellar vesicle (large single-layer lipid vesicle) shown in Fig. 1.1 using the example of the sensory rhodopsin II from Natronobacterium pharaonis (NpSRII) (Chizov, 1998). From its ground state, the all-trans conformation, it changes to the 13-cis conformation when a photon is absorbed. It then turns under thermal relaxation.

Experimental Protocols E7 and T4 1. Introduction This experiment was about Ostwald's law of dilution, which describes the degree of dissociation of weak electrolytes or the proportion of free dissociated ions in the solution with the help of the law of mass action. According to this law, the degree of dissociation α increases with increasing dilution or with decreasing initial concentrate. Equivalence point (for calculating the concentration, not equal to the neutral point), half-equivalence point (buffer area), pKa value of the propanoic acid (literature value 4.87) • Determination of the content of the propanoic acid solution 5 7. Lower Saxony Ministry of Culture Page 2 of 2 Aufg. Expected student performance AFB assessment I II III 2.4 • Dealing with equilibrium reaction, shifting equals. Example 1: You want to produce 1000 mL 0.1 M Na citrate buffer. You weigh in 0.1 mol of citric acid (or acetic acid, formic acid, etc.) and make up to approx. 900 ml. You place the solution on the magnetic stirrer and fix the pH electrode. Now carefully add sodium hydroxide solution (2-10 M) until the desired pH is reached. All that remains is to prepare the buffer in several steps: Calculation of the components (concentration and amount) according to the desired use and target volume, weighing in of the components, dissolving the components, adjusting the pH value, filling up to the final volume, labeling, documentation of the results and direct use or storage for later use. Choosing a. Example 3 Find the pH when the H. + The concentration is 0.0001 moles per liter. Here it helps to rewrite the concentration as 1.0 x 10 -4 M, because this gives the formula: pH = - (- 4) = 4

### Solubility Product • Formula and Calculate · [with video

Law of mass action, dissociation constant With the help of the law of mass action (MWG) the position of the equilibrium of a reaction can be calculated in the form of an equilibrium constant. If it is an acid-base reaction and thus a dissociation, the equilibrium constant is also called the dissociation constant.From this, you calculate the degree of dissociation with the help of the ∞ values determined in Exercise 1 depending on and represent the relationship ( ) graphically. 3.) With the values obtained in exercises 1 and 2, calculate the dissociation constant of acetic acid Examples of some important buffers: Acetic acid acetate buffer Phosphate buffer Carbonic acid hydrogen carbonate Buffers Area of application / use of acid-base buffers. Buffer systems are necessary in the human body, for example to keep the pH of the blood constant. With the help of a buffer, pH value measuring devices can be recalibrated. - Formulate Hess' theorem and show its meaning with examples for the calculation of reaction enthalpies - For chemical reactions, calculate the molar standard reaction enthalpy from the molar formation enthalpies. 4 5.3 - Carry out an experiment to determine the molar enthalpy of reaction (calorimeter) and calculate it using specified and measured quantities 5.4 - the. Dissociation constant: HSO 4- + H 2 O 2 SO 4 - + H 3 O + Polybasic bases: Ca (OH) 2, H 2 NNH 2 1st protonation constant: H 2 NNH 2 + H 2 O + H 2 NNH 3 + OH -2. Protonation constant: H 2 NNH 3 + + H 2 O - H 3 NNH 3 2+ + OH 14. 15 The splitting of water is endothermic and requires 57.4 kJ / mol. H 2 O + H 2 O H + 57.4 kJ 3 O + + OH- When a neutralization reaction occurs, water becomes.

## Determination of constants

The typical procedure for determining the constants involves a series of running enzyme tests at various substrate concentrations and measuring the initial reaction rate. "Initially" means here that the reaction rate is measured after a relatively short time, during which it is assumed that the enzyme-substrate complex has formed, but the substrate concentration is kept approximately constant and thus the equilibrium or quasi-steady- State approximation remains valid. By plotting the rate of reaction against concentration and using a nonlinear regression of the Michaelis-Menten equation, the parameters can be obtained. V max < displaystyle V _ < max >> K M < displaystyle K _ < mathrm

Before computing options for non-linear regression were available, graphic methods were used that included a linearization of the equation. Some of these have been proposed, including the Eadie-Hofstee diagram, the Hanes-Woolf diagram, and the Lineweaver-Burk diagram. Of these, the Hanes-Woolf plot is the most accurate. Although useful for visualization, all three methods distort the error structure of the data and are inferior to nonlinear regression. Assuming a similar error on, an inverse representation leads to an error of on (propagation of uncertainty). If the values are not correctly estimated, linearization should be avoided. In addition, the least squares regression analysis assumes that errors are normally distributed, which is not valid after a value transformation. Nevertheless, their use can still be found in modern literature. dv 0 < displaystyle dv_ <0>> v 0 < displaystyle v_ <0>> dv 0 / v 0 2 < displaystyle dv_ <0> / v_ <0> ^ <2>> 1 / v 0 < displaystyle 1 / v_ <0>> dv 0 < displaystyle dv_ <0>> v 0 < displaystyle v_ <0>>

In 1997 Santiago Schnell and Claudio Mendoza proposed a closed solution for the time history kinetics analysis of Michaelis-Menten kinetics based on the solution of the Lambert W function. Namely,

whereby *W.* is the Lambert W function and

The above equation was used to estimate and derive time-lapse data. V max < displaystyle V _ < max >> K M < displaystyle K _ < mathrm

## Transformations for linearization according to Eadie-Hofstee - chemistry and physics

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I have given the following definition:

This definition is clear to me so far, I think.

However, I have now given a system of differential equations in x and y.

I have a rest position (0.0) and a conservation size

My assumption now would be that one would like to consider the behavior of the solution of the system, ie the integral curves.

These lie on H (x, y) = c and since one is interested in the point (0,0) and H (0,0) = 0, one continues to consider H (x, y) = 0.

Now comes my real problem, that I have no idea how to proceed. I have a solution, but it is not entirely clear to me.

In another example, a solution (global flow) y ((x, y) = (a (t), b (t)) was given, where one could see more clearly for which (x, y) the condition is met , if one has formed the limit for t.

I don't see it so clearly here. Do I have to look at the behavior of the initial system?

I hope someone can help me.

I'm going to post the solution. Here is as

If I transform H (x, y) = 0, I get:

That looks quite similar to the solution, but I don't understand where the signs come from, or my question would be whether you can

could specify even more precisely.

unfortunately it is not quite clear what your task is.

Should you e.g. describe the (global) invariant manifolds or should you give a Taylor approximation of the local invariant manifolds or something else?

The phase portrait looks something like this:

You have already correctly described that the invariant manifolds both lie in the level set H (x, y) = 0.

The (local) stable manifold has two branches, both of which are tangential to the corresponding eigenspace, i.e. tangential to y = -x,

the unstable manifold as well as two branches that are tangent to y = x. The restriction to x & gt0 in your solution is actually not correct, but should also apply to the local invariant manifolds

I thought I understood the definition itself, but I can't imagine anything about it.

How do you come from the conservation size to the two branches, or y = -x and y = x.

Am I thinking correctly that I can look at the system itself to distinguish between stable and unstable in order to determine the direction? And then see which part of the solution is stable or unstable manifold?

"The restriction to x & gt0 in your solution is actually not correct, but also for the local invariant manifolds should be | x | & lt delta."

You're just talking about the amount, aren't you?

Your comments are all correct: In principle, the stable manifold of a rest position should be imagined as the set of all initial conditions from which the solution for converges towards the rest position and, analogously, the unstable manifold as the set of those initial conditions from which the solution for converges towards the rest position.

In your example it is special that a Hamilton function / conservation quantity exists and the invariant manifolds lie in the level set of this function.This is generally not the case. As far as the direction is concerned, one can either look at the vector field itself or use the theorem that the (local) (in-) stable manifold is tangential to the (in-) stable eigenspace.

The role of the invariant manifolds for the long-term behavior can be seen quite well in the example. Compare the behavior of two solutions that start just inside and just outside the homoclinic orbit. That makes a big difference between periodic / bounded and unbounded, which is exactly separated by the stable manifold.

-----------------

"The best way to have a good idea is to have lots of ideas."

- Linus Pauling

I've tried yet another example and would be very grateful if you could see whether that fits.

EDIT: I just see that my preservation size is miscalculated. Everything that follows is wrong.

I have the only rest position (0.0).

Now stability and type were asked by means of linearization, ie I create the matrix of the derivatives and insert the point (0,0). Then I get lambda = + -1, which corresponds to an (unstable) saddle point.

Now one should further specify the manifolds.

I can calculate a conservation size:

I look again at H (x, y) = 0 and get y = + - sqrt (-6x).

In real life I have a parabola that is open to the left.

If you look at the original system, it runs towards the origin for y & gt0 and y & lt0.

So I would have said that my stable manifold

W_s = <(x, y): y = + - sqrt (x), abs (x) & lt delta> and there is no unstable manifold. (Can that be at all or is the system unstable for y & lt0? Then I don't see why.)

The whole thing seems a little strange to me. What happens to the complex solutions?

Is there a connection to the saddle point?

That looks pretty much the same as the result from the other example, only that the axes are swapped, ie. in the normal x-y coordinate system the "loop" would be above the x-axis.

If you had known that, I would have said that the manifolds are now swapped for stability.

But when I look at the system I don't really see it, since mMn for y & gt0 yes y '& lt0 is for all x.

How do I know that?

according to my calculation and the level set H (x, y) = 0

consists of the x-axis as an unstable manifold of (0,0) and a parabola as a stable manifold of (0,0).

P.S .: It might make sense if you also realize that the concept of the invariant manifold is generally not related to a conserved quantity. This is a rather rare special case.

-----------------

"The best way to have a good idea is to have lots of ideas."

- Linus Pauling

I tried to look at these kind of special cases because I only have examples using conserved quantities.

Now I want to try again differently.

You said that the manifolds are tangential to the corresponding eigenspace, did I understand that correctly?

In the second example, the linearization gives me a saddle point with x-axis and y-axis as eigenvectors.

Wouldn't I then have two parabolas tangential to them as manifolds, each on the axes?

Why is the unstable manifold the axis and the stable one the parabola?

Wouldn't the axes be the stable or unstable manifold for both?

It was the same with the first example.

There were the eigenvectors (1,1) and (1, -1).

I would be really happy if you could take the time again to resolve this confusion with me.

Thanks!

Played around a bit or made a polynomial approach. If that hadn't worked, I would have assumed that there is no explicitly representable conservation quantity.

You said that the manifolds are tangential to the corresponding eigenspace, did I understand that correctly?

In the second example, the linearization gives me a saddle point with x-axis and y-axis as eigenvectors.

Wouldn't I then have two parabolas tangential to them as manifolds, each on the axes?

You then have two curves that are tangential to the axes (or eigenvectors). In one case it is a straight line, in the other a parabola and in the general case usually a more complicated curve.

Why is the unstable manifold the axis and the stable one the parabola?

Wouldn't the axes be the stable or unstable manifold for both?

It was the same with the first example.

There were the eigenvectors (1,1) and (1, -1).

In the first example the invariant manifolds were not straight lines, but this "leaf".

In the situation here, where both invariant manifolds are one-dimensional, they consist of two solution curves of the DGL, one on each side of the equilibrium.

Because y '= - y, the x-axis is invariant, but the y-axis is not, so the stable manifold is not the y-axis, but only tangential to it.

-----------------

"The best way to have a good idea is to have lots of ideas."

- Linus Pauling

In the second example, I had the two axes. With what you said I would now brag:

Is it really + -? The conservation quantity only describes a parabola that is open to the left? Can you see it somewhere without knowing the preservation size?

If I go back to the first example, then the manifolds are tangential to (1, -1) and (1,1). But how can I tell from the system which side they are on?

For (1,1) it should be a parabola below and for (1, -1) a parabola above.

But where does the y = -x or y = x in the description of the manifolds come from? Do I always need that to describe that there are "several" curves?

According to your explanations, I understand both, but I still don't know how I would come up with it myself, or how I really specify the manifolds correctly.

only briefly because I have little time.

1) It is not possible to state the invariant manifolds only with the help of the eigenvalues and eigenvectors, because here the higher order terms in the vector field are decisive.

2) You write "I still don't know how I would come up with it myself, or how I really specify the manifolds correctly".

I would say that in most situations it is not so important to - state - the manifolds (in most cases they can only be calculated approximately numerically), but to know that they exist and what properties the solutions have within these manifolds.

-----------------

"The best way to have a good idea is to have lots of ideas."

- Linus Pauling

I have received other examples where the information always looks like this:

a.) Determine the rest positions of the differential equation. Study their stability and determine their type.

Here I always determined the type by means of linearization and made a small sketch for stability. So far that's not a problem.

b.) Determine the type of all rest positions and their stable and unstable manifolds.

This is where I fail, although I am a bit irritated that the type is asked twice.

Or could something different be meant at point a.)?

I assumed that I should give the manifolds in a form similar to that of

I don't have any more than the eigenvalues and eigenvectors?

## EBERHARD - KARLS - UNIVERSITY OF TÜBINGEN. Institute for Physical and Theoretical Chemistry

1 EBERHARD - KARLS - UNIVERSIT & AumlT T & UumlBINGEN Institute for Physical and Theoretical Chemistry Physico-chemical internship for beginners Internship administration: Ms. Doez, (0707) Bkinetics Test contents Test aim of the reaction of chymotumphenyl acetate, nitro-aryl-phenyl-ester or reaction of chymotumphenyl-acetyl or nitro-p-hydrolysis-ester spontaneously catalysed by chymotumphenyl -npa are examined spectroscopically in the ultraviolet (UV) and visible (Vis) range. Based on the measurements, the dynamic and thermodynamic fundamentals of the reaction kinetics are presented and worked out Educts and products are high, ie if the values for the activation energy are large, the reaction rate constant will depend significantly on the temperature and the reaction rate itself will be low The extent of the reaction or the position of the equilibrium, on the other hand, depends on the thermodynamic free reaction enthalpy (IUPAC: GIBBS energy) (energy difference between the starting materials and products) A typical example is enzyme catalysis, in which new reaction pathways are selectively opened that have a lower energy barrier The spontaneous hydrolysis of p-nitrophenyl acetate is a base-catalyzed reaction which, if there is excess water, takes place after a pseudo-first-order reaction Imidazole can be catalyzed, a model reaction for enzymatic ester cleavage The serine protease chymotrypsin can also catalyze the cleavage of p-npa Working in a transition complex All reactions can be followed by absorption spectroscopy in the ultraviolet and visible spectral range. The obtained extinction-time or concentration-time diagrams can be used to determine reaction rate constants, the activation energy of the reaction and, if chymotrypsin is involved, characteristic values such as the maximum reaction rate In addition, it can be checked whether one or more linearly independent reaction steps occur during the reaction.

2 2 BODENSTEIN hypothesis, MICHAELIS-MENTEN equation, diagrams according to LINEWEAVER and BURK, as well as according to EADIE and HOFSTEE, application according to GUGGENHEIM and SWINBOURNE, temperature dependence of the reaction rate, ARRHENIUS equation, kinetic and thermodynamically controlled reaction, complex theory, reaction coordinate , Theory according to EYRING Basics of UV / Vis absorption spectroscopy, as described in the experiment instructions for determining the pk values, in particular the relationship between extinction and concentration according to LAMBERT-BEER's law 4 Literature [] PW Atkins: Physikalische Chemie, 2 ed , VCH, Weinheim 996 [2] K Laidler: reaction kinetics, volumes I and II, Bibliographisches Institut, Mannheim 973 [3] H Mauser: Formal Kinetik, Bertelsmann Universit & aumltsverlag, D & uumlsseldorf 974 [4] G Gauglitz: GIT Fachz Lab 26, 205 ( 982), 26, 597 (982), 29, 86 (985) (available from Ms. Doez) [5] HU Bergmeier (editor): Method s of Enzymatic Analysis, Vol: Fundamentals, Vol 2: Samples, Reagents and Assessments of Results, 3rd edition, Verlag Chemie, Weinheim 983 [6] Bisswanger, H Enzymkinetik, 2nd edition, Wiley VCH, Weinheim Aids to be brought along for the experiment are a pocket calculator and concept paper to bring with you 2 Basics 2 Fundamental terms of kinetics Chemical kinetics describes the speed of a chemical reaction as well as all factors on which the speeds depend (concentration, temperature, pressure) The speed of a reaction can be defined in different ways assume a positive numerical value In order to simplify the formulas in these instructions, the following notation is chosen: Upper case letters: A, B, C, designation of the reactants Lower case letters: a, b, c, their concentrations (instead of c A or [A] etc.) 2 Reaction speed v The reaction speed can be defined & uumlb er the 2 temporal change in the amount of substance ni When considering a reaction (e.g. BAB), the reaction rate v can be described as a change in the amount of substance dn i per unit of time, whereby the index i can stand for the educts or the products: dna dnb v = = Eq ( 2) 22 Change in substance concentration ci over time The reaction rate v can also be defined via the substance concentration ci:

3 3 v da db = A = B Eq (22) The relationship is shown in Fig. Depending on the definition, the speed corresponds to the positive or negative slope of the time course of the substance concentration a 0 a or b db products Fig b 0 = 0 0 since educts time t changes in concentration for the simple reaction AB as a function of time, shown for an educt A or a product B Formula conversion normalized: dni d & xi = Eq (23) & nu The reaction rate v is defined as the change in the reaction sequence number d & xi per unit of time: i d & xi dni v = = & nu In general, therefore, for a complex reaction of the type i Eq (24) & nu A + & nu applies B & nu C + & nu DABCD using Eqs. (23) and (24) for the reaction rate d x dna dnb dnc nd v = = = = = & nu & nu & nu & nu ABCD Eq. (25) Using the stoichiometry of the reaction equation under consideration, an expression for the reaction rate can be directly formulated 24 Temporal change of the reaction variables x L & a reaction at constant volume V from , based on the reaction sequence number & xi, a quantity analogous to the substance concentration c can be defined, called the reaction variable x: d & xi dni dci dx = = = Eq (26) V & nu V & nu ii The reaction rate v under these conditions is dx d & xi dni dci v = = = = VV & nu & nu ii Eq (27)

4 4 and also results from the stoichiometry of reaction equation 22 Reaction rate constant k and reaction order The reaction rate v is generally dependent on the concentrations ci of the starting materials involved in the reaction i The rate constant k is introduced as a proportionality factor: dx nmv = = k ca cb Eq ( 28) The exponents n, m etc. indicate the order of the reaction in relation to the components A, B etc. The sum of these exponents represents the order of the reaction of the entire reaction The order of the reaction is a purely empirical quantity and sometimes only coincides with the stoichiometry by chance The theoretically formulated reaction equation cannot therefore reliably predict the order of the reaction. Conversely, however, the experimentally determined order of the reaction can be used to formulate several conceivable, albeit ambiguous, reaction mechanisms and equations actions of the zeroth order results from Eq. (28) that the reaction rate is equal to the reaction rate constant and is therefore a time and concentration-independent quantity 23 Reaction Molecularity The reactions often consist of several individual steps, the elementary reactions The number of particles involved in an elementary reaction a transition state (activated complex) is called reaction molecularity. In this context, mono- or bimolecular elementary reactions are used. The order of the reaction and the molecularity of the reaction must not be confused If the concentration changes linearly from one another over time, then all these elementary reactions can be converted into a single linearly independent part summarize step For example, in a parallel reaction that starts from the same starting material and whose partial steps have the same reaction order, both steps together form a linearly independent partial reaction (a linearly independent partial step) 2 F & uumlr A kk B, AC da da = ka = k2 a Eq (29a and b) since (2) = k + ka Eq. (20) 22 Time laws and their graphic evaluation in concentration-time curves 22 First and pseudo-first order reactions k Eq (29a) In applies to the reaction AB of this differential equation the rate constant k has the dimension s - k If the reaction is of the second order, as for A + BC, then da = kab, Eq and thus its concentration remains practically constant during the entire reaction, then with da = ka Eq. (22)

5 5 a pseudo-first order reaction, the reaction rate constant of which contains the concentration of B at the start of the reaction: k = kb 22 Spontaneous hydrolysis as a pseudo-first order reaction The hydrolysis of the ester of nitrophenyl acetic acid (p-npa) circulates according to O ON 2 OC CH 3 + H 2 O + OH ON 2 OH + HOOC CH 3 + OH, whereby the concentrations of OH as a catalyst and of H 2 O (used in large excess) are viewed as constant. This reduces the reaction order to that of a pseudo-first reaction Order 222 Evaluation of the concentration-time curves 222 Curve fitting of the exponential function Integration of Eq. (29a) between the limits t = 0 to t results in ata 0 ek = t Eq. (23) () () as an exponential relationship to the concentration-time Values can be adapted to a curve with computer programs (methods according to GAU & szlig-NEWTON or MARQUARDT) The size k can be obtained from this curve adaptation 2222 Linearization by logarithmizing Eq. (23) can be logarithmized and leads to In at In a 0 = kt Eq. (24) () () A corresponding graphical representation is shown in Fig ln a 0 ln (a / 2) 0 -kt & frac12 t Fig 2 Logarithmic representation of the concentration profile 2223 Linearization according to SWINBOURNE If the reaction rate is monitored by UV / Vis spectroscopy, the total absorbance E (t) of a reaction AB according to the LAMBERT- BEER's law (857 August Beer:) as () = & epsilon () + & epsilon () E td A at B bt or E () t = d (& epsilon & epsilon) b (t) + & epsilon a () BAA 0 Calculated at the start of the reaction the extinction E (0) is E (0) d & epsilon a (0) =, A Eq (25) Eq (26)

6 6 at the end of the complete reaction, the extinction E () is equal to E () d & epsilon a () = B 0 With a layer thickness of d = cm (usual inside dimensions of the cavity), during the course of the reaction, E t E 0 = & epsilon & epsilon bt Eq ( 27) () () () () or E () E () (& epsilon & epsilon) a () BBAA 0 = 0 Gl (28) Since the extinction is difficult to determine at the start of the reaction, E (0) is eliminated: E ( ) t E () = (& epsilonb & epsilon) b (t) a (0) Gl (29) db From this it follows with = ka () b (t) 0: () E () (& epsilon & epsilon) E t Gl (220 ) db = BA Eq (22) k To simplify the equations, a layer thickness d = cm (i.e. d =) is assumed below. By differentiating Eq. (26) we get E (t) db = (ε b ε A) Eq (222) and thus from Eq. (22) () E t By integrating this differential equation, we get () () = k E t E Eq (223) () t de (t) = k () () Eq (224) Et E t E = E (0) t 0 E t ln E () E () (0) E () E t E = EE kt or ln () () ln (0) () ln EE t = ln EE 0 kt or () () () () = kt Eq (225) Eq (226) Eq (227) For the evaluation according to SWINBOURNE Eq (225) becomes for the times t and (t +) de-logarithmized, where approximately the half-life (see Chap. 223) corresponds to: kt E () t E () = E (0) E () e, Eq. (228) () () () () k (t) E t + E = E 0 E e + E (0) E () can be eliminated from these equations: kt E () t E () ek (t +) (+) () = = E t E ee + kkk or E () t E (t) e E () (e) Eq (229) Eq (230) = + + Eq (23) If E (t) is plotted against E (t +), the rate constant k results from the gradient

7 Linearization according to GUGGENHEIM E () can be eliminated from Eqs. (228, (229)): kt (+) t kt k kt E (t +) E (t) = E (0) E () (ee) = E (0) E () (e) e Eq. (232) Taking the logarithm leads to a linearization of this equation: If ln E (t +) E (t) k (+) () = + () ln E t E tkt ln E (0) E () e Eq (233) plotted against time, the result is a straight line with the gradient k 2225 Principle of formal integration In the methods described so far, the solution of the differential equation is always necessary k k2 This already leads to a subsequent reaction ABC Because of the resulting inhomogeneous DGL for the temporal dependence of B on problems. The possibilities of modern computers often avoid this way and the original DGL is numerically integrated, a process that is also called formal integration according to H Mauser (literature [3]) that in the simplest case Eq (22) as a = a (t) da = ka () t Eq (234) a = a (0) t = 0 is written Now only the integral on the right side of the equation has to be determined numerically and plotted against different points in time.Especially in the case of thermal reactions, it is beneficial not to have to use the concentration values at point in time t = 0 for the evaluation Integration, the area under a curve is approximated by a finite number of trapezoids.The width of the trapezoids does not have to be constant for this purpose, as Fig 3 shows: tambxhxn Fig 3 Trapezoidal rule for numerical integration The area of a trapezoid is the product of the center line and Height: or with a = yi, b = y i + and h = x i + xi: A = mh = a + bh 2 () summation leads to: A = (y + y) (xx) i 2 i i + i + i

8 8 xn A = (y + y) (xx) f (x) dxn ges 2 i i + i + ii = Eq. (235) x 223 Half-lives The half-life indicates the time after which half of the initial concentration has reacted, i.e. a (t & frac12) = & frac12a (0) For a first-order reaction, Eq. (24) yields the half-life t & frac12 to a (0) ln = ln a (0) k t & frac12 2 Eq. (236) t & frac12 ln = = kk 23 Enzyme catalysis Enzyme catalysis occupies an intermediate position between homogeneous and heterogeneous catalysis, as enzymes have a diameter of approx. 0-00 nm (water: approx. 03 nm) The phenomenon of substrate saturation (see below) occurs in both enzymatic and heterogeneous catalysis active centers are occupied: the catalyst concentration limits the process In contrast, this effect cannot be observed in homogeneous catalysis. 23 Determination of the reaction rate of an enzyme-catalyzed reaction according to MICHAELIS-MENTEN Scheme k E + S ES ES + PE + P k k2 k3 2 + k4 SP P2 Gl (237) can be described with the inclusion of a MICHAELIS-MENTEN complex rate-determining step (i.e. in the slowest step of the overall reaction) further to the end product (or to several products) reacts with the release of the enzyme i.e. equal to the initial concentration e (0) of the enzyme minus the concentration of the enzyme bound in the complex es: e = e (0) es Gl (238) For the change in concentration of the complex over time, taking into account Gl (238) des = ks ( e (0) es) k es k2 es Gl (239) Shortly after the start of the reaction, a quasi-stationary state occurs, in which the concentration of the Complex assumes a low, largely constant value (BODENSTEIN hypothesis) (Max E A Bodenstein:): des = 0 Gl (240)

9 9 From Eqs (239) and (240) it follows (0) (0) keses = = kks + k k2 + kks Eq (24) The expression in the denominator is summarized in the so-called MICHAELIS constant KM: KM k + k2 = Eq. (242) k As long as no reverse reaction occurs (i.e. at the start of the reaction), the following applies: dp = k2 es Eq. (243) According to Eq. (27), dp / corresponds to the reaction rate v 0 at the start of the reaction. 24) e (0) s v0 = k2 Eq (244) K + s This initial speed becomes maximum when KM is small compared to the substrate concentration: M () v0, max = k2 e 0 Eq (245) With these values, the Establish the so-called MICHAELIS-MENTEN equation: vvsv 0, max 0, max 0 = = KKM + s M s + Eq (246) The course of the reaction rate v 0 of an enzyme-catalyzed reaction for different substrate concentrations is shown in Fig. 4 v 0 v 0 , max & frac12v 0, max Fig. 4 Description of the enzyme kinetics: Course of the initial velocity v 0 as a function of the substrate At concentration s In addition, the MICHAELIS constant can be determined via the value of v 0, max, as will be explained below. Using Eq. (246), two limiting cases for the initial speed can be distinguished: Denominator compared to the substrate concentration s in Eq. (246) can be neglected s can thus be shortened out and leads to () 0 2 0, max sv = ke 0 = v Eq (247) The initial speed v 0 corresponds to the maximum initial speed The reaction can be solved thus as a reaction of the zeroth order with respect to the substrate concentration s, ie the initial speed is completely independent of the substrate concentration

10 0 Proportionality to the substrate concentration For s & lt & lt KM, s can be neglected compared to KM in the denominator, according to Eq. (246) vv = Eq. (248) 0, max 0 s KM The initial velocity v 0 is proportional to the substrate concentration s The reaction is first Order from, as shown in Eq. (237) with k 4 23 Determination of KM and v 0, max In order to be able to determine the MICHAELIS constant KM and the maximum initial speed v 0, max from Eq. (246), the MICHAELIS-MENTEN -Equation linearized, ie converted into an equation of the form y = ax + b (with a: slope of the straight line and b: y-axis segment). In biochemistry, the following two methods are of interest: 23 Method according to Lineweaver-Burk KM = + Eq (249) vvsv 0 0, max 0, max In Fig. 5, the reciprocal value of the initial speed v 0 is plotted against the reciprocal value of the associated substrate concentration s inditude, from the slope the MICHAELIS constant KM 232 method according to EADIE-HOFSTEE In this case, Eq. (246) is transformed into v0 v0 = Km + v0, max Eq (250) s plotted from v 0 / s For different substrate concentrations a straight line results, from the slope of which the MICHAELIS constant and from the ordinate the maximum initial speed v 0, max can be determined / v 0 v 0 Fig. 5 Linearization of the MICHAELIS-MENTEN equation according to LINEWEAVER-BURK / sv 0 / s Fig 6 Linearization of the MICHAELIS-MENTEN equation according to EADIE-HOFSTEE 233 Direct determination from the MICHAELIS-MENTEN plot In Eq. (246) it can be shown that the MICHAELIS constant KM exactly then equals the Substrate concentration is when the initial speed is half the maximum initial speed. If the initial speed is plotted against the substrate concentration, as in Fig. 6, it approximates for h without substrate concentrations of the maximum initial speed asymptotically. The MICHA-ELIS constant can be derived from the intersection between half the maximum speed and the

11 Read off the curve on the abscissa 232 The serine protease chymotrypsin Although only proteins can be used as natural substrates for chymotrypsin, the enzyme can also cleave the ester bond, which is very similar to the peptide bond, if - as in the case of p-npa - there is a hydrophobic residue in the vicinity For this reason, the ester bond p-npa played a major role as an artificial substrate in the elucidation of the reaction mechanism of the serine proteases. At alkaline pH, however, the acetyl group is hydrolytically split off from the enzyme and the enzyme is regenerated in this way. The acetylation step is rapid, the deacetylation step determines the rate. The Michaelis-Menten equation describes this known as the ping-pong mechanism, ki Technically complex reaction mechanism only applies under certain boundary conditions: The initial rates are determined by neglecting the reverse reaction, in addition, water as the second substrate of the rate-determining step must not be present as a limiting factor for the course of the reaction 232 Enzyme catalysis using the example of the imidazole-catalyzed hydrolysis of imidazole-catalyzed hydrolysis of p-nitrophenyl acetate (p-npa) is an organic-chemical model reaction for enzymatic ester cleavage.The overall mechanism can be represented by the following scheme: However, the spontaneous hydrolysis is still running in parallel According to Arrhenius, the temperature dependence of the reaction rate constants k or in logarithmic form k = A e EAEA RT ln k = a, RT + EA lg k = + a Eq. (25) 2303R T Tr & auml If one g k against / T on, one can determine the activation energy from the rise of the straight line:

12 2 lgk EA tan α = =, 2303R EAT lgk = 2303 R While Arrhenius came to this relationship through thermodynamic considerations, Lewis derived the temperature dependency of the reaction rate constants via the shock theory and the thermodynamic rate constant from the equilibrium rate constant of the involved molecules from the equations of statistical thermodynamics Although these calculations are certainly not easy to carry out with the present reaction system, for small molecules they can be calculated according to the theory of the transition state of Eyring with a free activation enthalpy G, which corresponds to the equilibrium constant K and the equilibrium constant K Activation energy is related, according to kt B k = K h there is a relationship between the statistical variables and the desired reaction rate Find constants (details in the literature citations) The following relationships are used for further calculations: Activation enthalpy: H = E RT, free activation enthalpy: G = RTlnK = 2303RTlg K, K = A kh, k TBT Planck's constant: h = J s, Boltzmann -Constant: k B = JK with G = HTSHG follows for the activation entropy: S = T 3 devices and chemicals 3 devices and chemicals 3 solutions The following substances / solutions are issued by the supervisor: 4-nitrophenyl acetate (p-npa, stock solution with 20 mm ): A dilution series (see table) must be prepared for the enzyme kinetics! Chymotrypsin (frozen lyophilisate, mass indicated on storage vessel): Prepare a solution with 0 mg / ml, store on ice! S & OumlRENSEN phosphate buffer ph 7 (from Na 2 HPO 4, NaH 2 PO 4) imidazole buffer (Na phosphate buffer with 20 mm imidazole) Preparation of the 4-nitrophenyl acetate stock solution

13 3 Stock solution 20 mm stock solution 5 ml volume ethanol 543 mg initial weight p-npa (M: 85 g / mol) p-nitrophenyl acetate dilution series (from stock solution, dilute with ethanol) 20 mm stock solution stock solution 200 microns: 2 200 5 mm: 4 50 & microl 50 & microl 2 mm: 0 20 & microl 80 & microl mm: 20 0 & microl 90 & microl 05 mm: 40 0 & microl 390 & microl 02 mm: 00 0 & microl 990 & microl 32 K & uumlvetten The 3 ml double-sided cuvettes used are only allowed on the matted sides are touched K & uumlvetten are expensive and must not be placed directly on stone tiles or tiles (risk of scratching and splintering if you fall over) so please always put paper under the cuvette Before inserting into the spectrometer, the windows of the cuvette with paper are clean and dry to wipe 33 pipettes The EPPENDORF pipette (05-0 & microl) is adjusted by first pulling out the push button and then turning it After setting the desired amount, it is fixed by pressing the button. For handling, three pressure points must be observed: Sucking in the amount of liquid 2 Completely emptying the amount of liquid 3 Removing the pipette tip The volume is adjusted with a handle on the GILSON pipette Push buttons, one for filling and emptying with a total of two pressure points and one for removing the pipette tip 34 Spectrometers The instructions for the various UV / VIS spectrometers are available at the respective test site There are both single-beam and double-beam spectrometers available 4 Test implementation 4 Spontaneous hydrolysis of p-nitrophenyl acetate The spontaneous hydrolysis of p-nitrophenyl acetate in the S & OumlRENSEN phosphate buffer at 405 nm is monitored on the two-beam spectrometer Called zero adjustment: 2970 microl S & OumlRENSEN phosphate buffer is filled into two cuvettes and 30 microl pure ethanol is pipetted into it

14 4 The cuvettes are closed, shaken and placed in the cuvette holders of the spectrometer.After equilibrating in the spectrometer, the zero adjustment takes place (blank) The cuvette in the sample holder is emptied and used as a measuring cuvette (the other measuring cuvette can remain in the other experiment as a reference after the start of this measurement!) 2970 microl S & OumlRENSEN phosphate buffers are pipetted into the measuring cuvette and the cuvette is placed in the spectrometer for temperature compensation mixed The recording of the spectra is started by computer The computer measures the extinction at the time intervals given in the table for 80 min Stored & overview: Spontaneous hydrolysis of 4-nitrophenyl acetate Method Timedrive (Td) BC_4MTD Wavelength: Measurement duration: Interval: 405 nm 80 min 60 s Zero adjustment Reference S & oumlrensen buffer ph Blank S & oumlrensen buffer 2970 Buffer 2970 Buffer 2970 T & oumlrens-buffer S & oumlrens 2970 -npa 0 mm 30 temperature: 35 C as for zero adjustment Store the sample! 42 Imidazole-catalyzed hydrolysis of p-nitrophenyl acetate 42 Reaction spectrum The SP (spectrum) method and the BC_42 program are called up in the UV WinLab program The device is calibrated to zero (blank) and the measuring cuvette is emptied again Measurement: 2970 microliters of imidazole buffers are pipetted into the measuring cuvette. Then 30 microlos started The recording of the spectra is started by computer. To determine E (), the reaction solution is stored in a clean, labeled glass vessel after the end of the measurement

15 5 Overview: Imidazole-catalyzed hydrolysis of 4-nitrophenyl acetate (reaction spectra) Method Spectrum BC_42MSC Wavelength: nm Number of spectra: 20 Interval: 60 s Temperature: room temp. npa 0 mm 30 Store the sample! 422 E (t) curves at 405 nm An E (t) diagram at 405 nm is recorded for the imidazole-catalyzed reaction over 30 to 60 min at room temperature. The TD (time drive) method and the program BC_422 called Zero adjustment (blank): Two cuvettes are filled with 2970 microl imidazole buffer and 30 microl ethanol each and sealed. Then the device is calibrated to zero (blank) and the measuring cell emptied again. solution (0 mm) is pipetted in. The cuvette is closed, shaken, inserted into the device and the measurement started [t] curve) Method Timedrive (Td) BC_422MTD Wavelength: 405 nm Measurement duration: 30 or 45 min Interval: 30 s Zero adjustment reference imidazole buffer mm 2970 Blank imidazole buffer 2970 Measurement with reference imidazole buffer 2970 td_422 Imidazole buffer 2970 p-npa 0 mm 30 Temperature: room temperature as for zero adjustment. 43 Determination of E () The stored and labeled reaction solutions of the previous experiments are heated for min in a water bath to 70 ° C. After cooling, spectra (nm) are recorded for all samples, reference and blank sample are either S & OumlRENSEN phosphate buffer or imidazole buffer (which differences are to be expected depending on the solution used?)

16 6 Why do the reaction solutions have to be cooled down to device temperature or 20 C? Overview: Determination of E () method spectrum BC_43MSC Wavelength: nm Zero adjustment reference S & oumlrensen / imidazole buffer 2970 Blank S & oumlrensen / imidazole buffer 2970 Measurement with reference S & oumlrensen / imidazole buffer 2970 sample from sample of 43_spoimi sample from sp_43_timi2 from room temperature: with zero adjustment 44 chymotrypsin-catalyzed hydrolysis of p-nitrophenyl acetate A total of 7 reactions are carried out with different 4-nitrophenyl acetate solutions -Buffer, 30 microliter ethanol and 30 microl chymotrypsin solution are pipetted into two dry cuvettes. The cuvettes are sealed In solution and 30 microl of the corresponding p-npa solution are pipetted in. Then the cell is turned several times (do not shake!), placed in the spectrometer and the time-dependent measurement started Overview: Chymotrypsin-catalyzed hydrolysis of 4-nitrophenyl acetate Method Timedrive (Td) BC_44MTD Wavelength: 405 nm Measurement time: 3 min Interval: 5 s Chymotrypsin concentration: 0 mg / ml Zero adjustment Reference S & oumlrensenbuffer 29 Temperature: Room temperature as with zero adjustment

17 7 chymotrypsin 30 td_44_ S & oumlrensenpuffer 2940 Chymotrypsin 30 p-npa 02 mm 30 td_44_2 S & oumlrensenpuffer 2940 Chymotrypsin 30 p-npa 05 mm 30 td_44_3 S & oumlrensenpuffer 2940 Chymotrypsin 30 p-npa mm 30 td_44_4 S & oumlrensenpuffer 2940 Chymotrypsin 30 p-npa 2 mm 30 td_44_5 S & oumlrensenpuffer 2940 Chymotrypsin 30 p-npa 5 mm 30 td_44_6 S & oumlrensenbuffer 2940 Chymotrypsin 30 p-npa 0 mm 30 td_44_7 S & oumlrensenbuffer 2940 Chymotrypsin 30 p-npa 20 mm 30 5 Evaluation 5 Spontaneous hydrolysis F & uumlante should be determined according to the following methods: the spontaneous hydrolysis of spontaneous hydrolysis Linearization by taking the logarithm Evaluation: First of all, the values ln E () E (t) ln E () E (t) = ln E () E (0) kt Evaluation 2: be calculated for the linearized function Regression line determined with the help of the Origin program (if necessary increase decimal places to 9) Formal integration Evaluation 3: The formal integr Ation you can use the program Formal Integration under Windows by adapting an exponential function Evaluation 4: In Origin, the measured absorbance is plotted against time A function of the form f (x) = A (e Bx) + C is adapted to the curve The interpretation of A, B and C is important when specifying sensible starting parameters. B and C are estimated linearization according to SWINBOURNE For the application methods according to Swinbourne and Guggenheim you need value pairs E (t) and E (t +), where half the total measurement time corresponds

18 8 Evaluation 5: The values E (t) are your measured values up to half the total measurement period. For some of these values you have measured E (t +) values (from half the total measurement period). The other E (t +) values are obtained from Insertion of t + into the exponential fit function determined in the previous exercise. You enter these values according to the equation E () t = E (t +) e + E () (e) and determine the rate constant linearization using linear regression GUGGENHEIM evaluation 6: The previously determined values are plotted according to Eq. (233) k ln E (t +) E (t) = kt + ln (E (0) E () e) and the rate constant is determined 52 Imidazole-catalyzed hydrolysis 52 Interpretation of the Absorbance diagram evaluation 7: What do you observe in the reaction spectrum? In Origin, an extinction diagram (E diagram) for the two absorption maxima should be created from the reaction spectrum and interpreted with regard to the gradation of the rate-determining step (see pk value instructions) 522 Determination of the ARRHENIUS parameters Evaluation 8: The measured values of the E (t ) Curve at 20 C are entered in Origin. The rate constant is determined by adapting the above exponential function. The data is exported as an ASCII file and loaded into the Extinct program under Windows. A) and the product (C) (units!) Values for other temperatures can be simulated.After choosing a file name, the data is written directly to a file.These simulated data can be read into Origin as an ASCII file and displayed graphically Evaluation 9: Evaluation 0: By curve fitting w ground the rate constants determined for the simulated curves The activation energy can be determined according to ARRHENIUS from the temperature dependence of the rate constants (linear regression) 523 Determination of the rate constant over the half-life Evaluation: 53 Enzyme catalysis The E (t) diagram created in 522 becomes an The time is read from the point E = E () / 2 and the speed constant is calculated from the initial gradients. The MICHAELIS constant and maximum speed are to be determined using various methods and compared with each other Evaluation 2: Evaluation 3: MICHAELIS-MENTEN plot Calculate the concentrations of p-npa in the cells Using the respective reaction curves in the Origin program, adapt straight lines using linear regression (select a sensible range!), calculate the initial velocity v 0 from the initial slope by calculating the extinction coefficient of the product uktes (calculated in Exercise 8) and take the layer thickness into account

19 9 Evaluation 4: The values for v 0 and c 0 are entered in Origin and a graphic is created.A function of the form f (x) = is adapted to this A x curve from the half-value concentration (A / 2) or from the parameter B The MICHAELIS- B + x constant can be determined from the fitted curve, the maximum speed linearization according to LINEWEAVER and BURK evaluation 5 from parameter A. By plotting / v 0 against / c 0 and subsequent linear regression, KM and v max according to Lineweaver Burk determine linearization according to EADIE and HOFSTEE Evaluation 6: With the initial slopes determined above, KM and v max are determined by plotting v 0 against v 0 / c 0 and linear regression 54 Comparison of the results Evaluation 7: Compare the enzymatic results obtained with the various evaluation methods Large KM and v max, discuss the advantages and disadvantages of the evaluation method (cf. Lit [6]!) Use equation (247) to determine di e constant k 2 for the enzyme-catalyzed reaction (molecular weight chymotrypsin: 25 kda) and compare this with the rate constants of spontaneous hydrolysis and imidazole-catalyzed hydrolysis (table) What do you expect and will your expectations be confirmed? (Brief discussion of the results!) Are there substances that chymotrypsin can convert even faster?

20 20 Table of contents Experiment contents Experiment goal 2 Teaching content 3 Key words for preparation 4 Literature 2 5 Aids to be brought along 2 2 Basics 2 2 Fundamental terms of kinetics 2 2 Reaction speed v 2 2 Temporal change in the amount of substance ni 2 22 Temporal change in the substance concentration ci 2 23 Temporal change in the number of reactions & xi 3 24 Temporal change of the reaction variables x 3 22 Reaction rate constant k and reaction order 4 23 Reaction molecularity 4 24 Linear independent substep 4 22 Time laws and their graphic evaluation in concentration-time curves 4 22 First and pseudo-first order reactions 4 22 Spontaneous hydrolysis as a pseudo reaction -first order evaluation of the concentration-time curves curve adaptation of the exponential function linearization by logarithmizing linearization according to SWINBOURNE linearization according to GUGGENHEIM principle of formal integration half-life 8 23 enzyme catalysis 8 23 Determination of the reaction rate of an enzyme-catalyzed reaction according to MICHAELIS-MENTEN 8 23 Determination of KM and v 0, max The serine protease chymotrypsin 232 Enzyme catalysis using the example of imidazole-catalyzed hydrolysis of p-npa 3 Devices and chemicals 2 3 Devices and chemicals 2 3 Solutions 2 32 Cells 3 33 Pipettes 3 34 Spectrometers 3 4 Carrying out the experiment 3 4 Spontaneous hydrolysis of p-nitrophenyl acetate 3 42 Imidazole-catalyzed hydrolysis of p-nitrophenyl acetate 4 42 Reaction spectrum E (t) curves at 405 nm 5 43 Determination of E () 5 44 Chymotrypsin-catalyzed hydrolysis of p-nitrophenyl acetate 6 5 Evaluation 7 5 Spontaneous hydrolysis 7 52 Imidazole-catalyzed hydrolysis 8 52 Interpretation of the extinction diagram Determination of the ARRHENIUS parameters Determination of the rate constant over the half-life 8 53 Enzyme catalysis 8 54 Comparison of the results 9 6 Table of contents 20