First-Order Reactions

• First-Order Reactions are vital in drug stability, absorption, and pharmacokinetic studies.
• Definition: Rate of degradation is proportional to the concentration of the drug.
• Rate law:

Derivation:

Separate variables:

$\text{Rate} = -\frac{d[A]}{dt} = k[A]$

Integrate both sides:

$\frac{d[A]}{[A]} = -k \, dt$

• Integrated form:

$\int_{[A]_0}^{[A]} \frac{d[A]}{[A]} = -k \int_{0}^{t} dt$

$\ln [A] – \ln [A]_0 = -kt$

$\ln\!\left(\frac{[A]}{[A]_0}\right) = -kt$

• Graph: $\text{Plot of } \ln [A] \text{ vs. time = straight line with slope } -k$

$\ln [A]_t = \ln [A]_0 – kt \quad \text{or} \quad [A]_t = [A]_0 e^{-kt}$

Features:

• Constant fraction of drug degrades per unit time.
• Half-life is constant:

$t_{1/2} = \frac{0.693}{k}$

• Shelf life:

$t_{90} = \frac{0.105}{k}$

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