Characteristics

rate = k[A] — rate directly proportional to concentration.

Integration Derivation

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

$\int_{[A]_0}^{[A]} \frac{d[A]}{[A]} = -k\int_0^t dt$

$$\ln[A] - \ln[A]_0 = -kt \quad \Rightarrow \quad [A] = [A]_0 $e^{-kt}$$$

Key Features

1. ln[A] vs t: Straight line; slope = −k; intercept = ln[A]_{0}
2. log[A] vs t: Straight line; slope = −k/2.303; intercept = log[A]_{0}
3. Half-life: t_{1}/{2} = 0.693/k — INDEPENDENT of [A]{0}
4. All successive half-lives are EQUAL (unique to first order)
5. Never theoretically reaches zero (asymptotic approach)

Half-Life Multi-step Pattern

$[A] = [A]_0 \cdot \left(\frac{1}{2}\right)^n \quad \text{where } n = \frac{t}{t_{1/2}}$

Examples of First-Order Reactions

• Radioactive decay: ^{238}U → ^{234}Th + ^{4}He (t_{1}/_{2} = $4.5 \times 10^{9}$ yr)
• $H_{2}O_{2}$ decomposition: 2$H_{2}O_{2}$ → $2H_{2}O$ + $O_{2}$ (SMILES: OO → O)
• $N_{2}O_{5}$ decomposition: 2$N_{2}O_{5}$ → 4N$O_{2}$ + $O_{2}$