Arrange the following reactions in increasing order of the value of the ratio of rate constants (\( k_{310}/k_{300} \))
(A) \( A+B → C: E_a = 70\, kJ \)
(B) \( P+Q→R: E_a = 45\, kJ \)
(C) \( G+H→I: E_a = 95\, kJ \)
(D) \( X+Y→Z: E_a = 100\, kJ \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (A), (C), (D)
The effect of temperature on the rate constant is given by the Arrhenius equation. For two temperatures:
\( \displaystyle \log\frac{k_2}{k_1} = \frac{E_a}{2.303R}\left(\frac{T_2 - T_1}{T_1 T_2}\right) \)
For all four reactions the temperatures are the same (300 K and 310 K), so the bracket is a common positive number. The ratio \( k_{310}/k_{300} \) therefore increases with the activation energy: a reaction with a higher \( E_a \) is more sensitive to temperature.
\( \displaystyle \frac{T_2 - T_1}{T_1 T_2} = \frac{10}{300 \times 310} = 1.075 \times 10^{-4}\ \mathrm{K^{-1}} \)
(B) \( E_a = 45\ \mathrm{kJ} \): \( \log(k_2/k_1) = 0.253 \), ratio about 1.8.
(A) \( E_a = 70\ \mathrm{kJ} \): \( \log(k_2/k_1) = 0.393 \), ratio about 2.5.
(C) \( E_a = 95\ \mathrm{kJ} \): \( \log(k_2/k_1) = 0.533 \), ratio about 3.4.
(D) \( E_a = 100\ \mathrm{kJ} \): \( \log(k_2/k_1) = 0.562 \), ratio about 3.6.
In increasing order of \( k_{310}/k_{300} \):
\( \displaystyle \text{(B)}\ 45 \lt \text{(A)}\ 70 \lt \text{(C)}\ 95 \lt \text{(D)}\ 100\ \ (E_a/\mathrm{kJ}) \)
Shortcut: No calculation is needed: arrange the reactions in increasing order of \( E_a \). Only Option (2) starts with (B), the lowest activation energy.