Integrated Rate Law

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For a first-order reaction, the integrated rate law is:

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For a **First-Order Reaction**:
$-\frac{d[A]}{dt} = k[A]$
Separating variables and integrating:
$\int_{[A]_0}^{[A]_t} \frac{d[A]}{[A]} = - \int_0^t k dt$
$\ln[A]_t - \ln[A]_0 = -kt \implies \ln[A]_t = \ln[A]_0 - kt$
- This can also be written in exponential form: $[A]_t = [A]_0 e^{-kt}$.
- Or in common logarithms: $t = \frac{2.303}{k} \log \frac{[A]_0}{[A]_t}$.

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Integrated first-order rate law (A → products) is:

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