Chemical Kinetics: Step-by-Step Worked Examples (8)

Question

The decomposition of $N_2O_5$ is first order with rate constant $k = 6.93 \times 10^{-3}\,\text{s}^{-1}$ at $300\,\text{K}$.

(a) Find the half-life. (b) Calculate the time required for $75\%$ of the $N_2O_5$ to decompose.

Solution — Step by Step

Why This Works

For first-order reactions, the half-life is independent of starting concentration. Each half-life chops the concentration in half — so 75% decomposition (1/4 remaining) takes exactly 2 half-lives. This is the cleanest signature of first-order kinetics.

That is why the answer is exactly $2 t_{1/2} = 200$ s. Memorise these multiples:

• 50% decomposed = 1 half-life
• 75% decomposed = 2 half-lives
• 87.5% decomposed = 3 half-lives
• 99.9% decomposed ≈ 10 half-lives

Alternative Method

For (b), recognise the half-life pattern directly: 75% decomposed = 2 half-lives = 200 s. No log needed.

Common Mistake