Consider a complex reaction taking place in three steps with rate constants k₁, k₂ and k₃ respectively. The overall rate constant k is given by the expression k=√((k₁ k₃)/k₂). If the activation energies of the three steps are 60, 30 and 10 kJ mol⁻¹ respectively, then the overall energy of activation in kJ mol⁻¹ is _________ . (Nearest integer)

Solution

Chemistry ma’am · 4 steps

Take ln of the rate constant expression: the powers on each k become the weights on each Ea.

1. Arrhenius form. Write every rate constant as k = A e^(-Eₐ/RT). Taking ln gives ln k = ln A - Eₐ/RT, so each Eₐ sits in the same place as ln k.
2. Take ln. From k = ((k₁ k₃)/k₂)^(1/2) you get ln k = 1/2(ln k₁ + ln k₃ - ln k₂). Match the -1/RT terms on both sides.

   Eₐ = 1/2(Eₐ₁ + Eₐ₃ - Eₐ₂)

3. Put in numbers. Eₐ = 1/2(60 + 10 - 30) = 1/2 × 40 kJ mol⁻¹.
4. So. The overall activation energy is 20 kJ mol⁻¹.

The common slip: 40. Dropping the square root, that is the factor 1/2, gives 60 + 10 - 30 = 40, not 20.

The idea to keep. When k is built from products, quotients and powers of step constants, Eₐ follows the same pattern: multiply means add, divide means subtract, and a power multiplies.

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