If the activation energy of a reaction is 80.9 kJ mol⁻¹, the fraction of molecules at 700 K, having enough energy to react to form
products is e⁻ˣ. The value of x is __________. (Rounded off to the nearest integer) [Use R = 8.31 J K⁻¹ mol⁻¹]

Solution

Chemistry ma’am · 3 steps

The fraction of molecules that can react is e^(-Eₐ/RT), so x is just Eₐ/RT.

1. Boltzmann factor. In the Arrhenius equation k = A e^(-Eₐ/RT), the factor e^(-Eₐ/RT) is the fraction of molecules that have energy equal to or greater than Eₐ. Compare it with e⁻ˣ to get x = Eₐ/RT.

   x = Eₐ/RT = (80900 J mol⁻¹)/(8.31 J K⁻¹ mol⁻¹ × 700 K)

2. Put in numbers. First convert Eₐ to joules: 80.9 kJ mol⁻¹ = 80900 J mol⁻¹. The denominator is 8.31 × 700 = 5817 J mol⁻¹, so x = 80900/5817 = 13.91.
3. So. x ≈ 13.91, which rounds to 14.

The common slip: 0. Leaving Eₐ in kJ while R is in joules gives x = 0.0139, which rounds to 0. Convert 80.9 kJ to 80900 J first.

The idea to keep. The fraction of molecules with energy at least Eₐ is e^(-Eₐ/RT). Put Eₐ and RT in the same energy unit before you divide.

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