First-order reactions

For first order, rate = −d[R]/dt = k[R]. Integration gives ln([R]₀/[R]) = kt, or [R] = [R]₀e^(−kt).

In common logs: k = (2.303/t) log([R]₀/[R]).

A plot of ln[R] against t is a straight line of slope −k; a plot of log([R]₀/[R]) against t is a straight line through the origin with slope k/2.303.

Between two times t₁ and t₂: k = 2.303/(t₂ − t₁) × log([R]₁/[R]₂).

Examples: hydrogenation of ethene (rate = k[C₂H₄]), natural radioactive decay such as that of radium, and the decomposition of N₂O₅ and N₂O.

Worked number: [N₂O₅] falls from 1.24 × 10⁻² to 0.20 × 10⁻² mol L⁻¹ in 60 min at 318 K. k = (2.303/60) log 6.2 = 0.0304 min⁻¹.

For a gas-phase reaction A(g) → B(g) + C(g) at constant volume, pressure follows the progress. If the initial pressure is pᵢ and the total pressure at time t is pₜ, then p_A = 2pᵢ − pₜ and k = (2.303/t) log(pᵢ/(2pᵢ − pₜ)).

First-order decay is exponential: the reactant never quite reaches zero, but falls by the same fraction in every equal time interval.