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Chemistry Notebook

First-order reaction kinetics

From a differential rate law to concentration as a function of time.

The rate law

Section titled “The rate law”

For a first-order disappearance of AA in a constant-volume system:

dCAdt=−kCA.\frac{dC_A}{dt}=-kC_A.

The rate constant kk has units of inverse time.

Integrating the equation

Section titled “Integrating the equation”

With CA=CA0C_A=C_{A0} at t=0t=0, separate variables and integrate:

∫CA0CAdCC=−k∫0tdt,CA(t)=CA0e−kt.\int_{C_{A0}}^{C_A}\frac{dC}{C}=-k\int_0^t dt, \qquad C_A(t)=C_{A0}e^{-kt}.

Half-life

Section titled “Half-life”

Setting CA=CA0/2C_A=C_{A0}/2 gives t1/2=ln⁡2/kt_{1/2}=\ln 2/k. For this model, the half-life is independent of the initial concentration.

A numerical check

Section titled “A numerical check”

If k=0.10 min−1k=0.10\,\mathrm{min^{-1}}, the half-life is about 6.93 min6.93\,\mathrm{min}. After two half-lives, one quarter of the initial concentration remains.