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## Half-life in 1st order reactions

The half-life can be calculated from the law of speed. Again we consider a simple reaction:

$$\begin{array}{ccc}\hfill \text{A.}& \to & \text{B.}\hfill \end{array}$$

These reactions take place in the 1st order:

- $$-\frac{\text{d}\left[\text{A.}\right]}{\text{d}\mathit{t}}=\mathit{k}\cdot \left[\text{A.}\right]$$

By changing (variable separation) one obtains:

- $$-\frac{\text{d}\left[\text{A.}\right]}{\left[\text{A.}\right]}=\mathit{k}\cdot \text{d}\mathit{t}$$

The equation is now integrated, with the starting situation before the start of the reaction being chosen as the zero point. ${\left[\text{A.}\right]}_{\text{0}}$ denotes the initial concentration of A at $\mathit{t}$ = 0.

- $$\underset{{\left[\text{A.}\right]}_{0}}{\overset{\left[\text{A.}\right]}{\int}}\frac{\text{d}\left[\text{A.}\right]}{\left[\text{A.}\right]}=\underset{0}{\overset{\mathit{t}}{\int}}-\mathit{k}\cdot \text{d}\mathit{t}$$

It follows:

- $$ln\left(\frac{\left[\text{A.}\right]}{{\left[\text{A.}\right]}_{0}}\right)=-\mathit{k}\cdot \mathit{t}$$

This equation applies to the entire course of a first-order reaction. We now consider the half-life ${\mathit{t}}_{\mathrm{\xbd}}$. After a half-life, the concentration of A has decreased by half: [A] = ${\text{\xbd}\left[\text{A.}\right]}_{\text{0}}$. This is now plugged into the equation, ${\left[\text{A.}\right]}_{\text{0}}$ can be shortened.

- $$ln\left(\frac{\frac{1}{2}{\left[\text{A.}\right]}_{0}}{{\left[\text{A.}\right]}_{0}}\right)=ln\left(\frac{1}{2}\right)=-k\cdot {\mathit{t}}_{\mathrm{\xbd}}$$

It applies $ln(1/2)=-ln\left(2\right)$. By transformation we get an expression for the half-life:

- Half-life

- $${\mathit{t}}_{\mathrm{\xbd}}=\frac{ln\left(2\right)}{\mathit{k}}$$

The half-life is therefore independent of the concentration of the educt. This is characteristic of first-order reactions. Radioactive decay also obeys a rate law 1. This is used when determining the age, for example using the C14 method.