The decomposition of formic acid on gold surface follows first order kinetics. If the rate constant at 300 K is 1.0 × 10⁻³ s⁻¹ and the activation energy Eₐ = 11.488 kJ mol⁻¹, the rate constant at 200 K is ____________ × 10⁻⁵ s⁻¹. (Round off to the Nearest Integer). (Given : R = 8.314 J mol⁻¹ K⁻¹)

Solution

Chemistry ma’am · 4 steps

The Arrhenius exponent comes out to exactly ln 10, so cooling to 200 K makes k ten times smaller.

1. Compare two temperatures. Write k = A e^(-Eₐ/RT) at 300 K and at 200 K, then divide one by the other. A cancels, so Eₐ alone sets the ratio. Call the 300 K values k₁, T₁ and the 200 K values k₂, T₂.

   ln k₁/k₂ = Eₐ/R(1/T₂-1/T₁)

2. Put in numbers. The bracket is 1/200-1/300 = 1/600 K⁻¹. With Eₐ in joules, 11488/(8.314 × 600) = 1381.8/600 = 2.303.
3. Recognise the number. 2.303 = ln 10, so k₁/k₂ = 10. That gives k₂ = (1.0 × 10⁻³)/10 = 1.0 × 10⁻⁴ s⁻¹.
4. So. 1.0 × 10⁻⁴ s⁻¹ = 10 × 10⁻⁵ s⁻¹, so the answer is 10.

The common slip: 1. Stopping at k₂ = 1.0 × 10⁻⁴ s⁻¹ and entering 1 misses the scale. The blank is in units of 10⁻⁵, so the value to enter is 10.

The idea to keep. Use the two-temperature Arrhenius equation with Eₐ in joules. If the exponent comes out as 2.303, the ratio of the two rate constants is exactly 10.

Answer from the NTA final answer key. Solution by PrepWiser.