Lindemann mechanism at a given pressure

UKChO Round 2 · IChO · unreviewed

Lindemann mechanism with rate constants known; the gas pressure and temperature are given. Find the half-life.

Worked example

Lindemann parameters: k₁ = 1.2 × 10⁷ dm³ mol⁻¹ s⁻¹, k₋₁ = 5.5 × 10⁹ dm³ mol⁻¹ s⁻¹, k₂ = 3.2 × 10⁵ s⁻¹. What is the half-life of A at 424 K and 27 kPa? (Assume an ideal gas and that M is the total gas.)

1. Multiply R × T → RT
= 3.525 kJ mol⁻¹
2. Divide p ÷ RT → [M]
= 0.007659 mol dm⁻³
3. Multiply k₋₁ × [M] → k₋₁[M]
= 4.212 × 10⁷ s⁻¹
4. Add k₋₁[M] + k₂
= 4.244 × 10⁷ s⁻¹
5. Multiply k₁ × [M] → k₁[M]
= 91910 s⁻¹
6. Multiply k₁[M] × k₂ → k₁k₂[M]
= 2.941 × 10¹⁰ s⁻²
7. Divide k₁k₂[M] ÷ (k₋₁[M] + k₂) → k_uni
= 692.9 s⁻¹
8. Half-life (first order): solve for t½
$t_{1/2} = \dfrac{\ln 2}{k}$
= 0.001000 s

Answer: t½ = 0.00100 s

Method

[M] = p/RT. Then k_uni = k₁k₂[M]/(k₋₁[M] + k₂), which is first order in A, so t½ = ln 2/k_uni.

Common mistakes

That uses the high-pressure limit k₁k₂/k₋₁. At this pressure you need the full Lindemann expression.

Formulas used

Half-life (first order)
$t_{1/2} = \dfrac{\ln 2}{k}$

Practise this with fresh numbers

Lindemann mechanism at a given pressure

Worked example

Method

Common mistakes

Formulas used

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