Derive the integrated rate equation for a first order reaction. Draw the graphs that give straight lines, and write the form used for a gas phase reaction A(g) → B(g) + C(g).

1. 1 · Write the rate law. For R → P: −d[R]/dt = k[R].
2. 2 · Separate and integrate. d[R]/[R] = −k dt, so ln[R] = −kt + I.
3. 3 · Find the constant. At t = 0, I = ln[R]₀, so ln([R]₀/[R]) = kt and k = (2.303/t) log([R]₀/[R]).
4. 4 · Draw the graphs. ln[R] vs t: straight line, slope −k. log([R]₀/[R]) vs t: straight line through origin, slope k/2.303.
5. 5 · Gas phase form. p(A) = 2pᵢ − pₜ, so k = (2.303/t) log[pᵢ/(2pᵢ − pₜ)].
6. 6 · Give an example. Radioactive decay, used in carbon-14 dating of old objects.
7. 7 · Close in one line. So the unit of k for a first order reaction is s⁻¹.

Scoring tip: Write 2.303 when you change ln to log; many students lose a mark by dropping it.