Most chemical reactions are accelerated by an increase in temperature. The temperature dependence of the rate of a chemical reaction is explained by the Arrhenius equation
\[k = A\,e^{-E_a/RT}\]
where \(A\) is the Arrhenius factor (also called the frequency factor or pre-exponential factor) and \(E_a\) is the activation energy, expressed in J mol−1. A plot of ln \(k\) against \(1/T\) is a straight line with slope \(-E_a/R\). The rate constants \(k_1\) and \(k_2\) of a reaction at temperatures \(T_1\) and \(T_2\) are related as
\[\log\frac{k_2}{k_1} = \frac{E_a}{2.303R}\left[\frac{T_2 - T_1}{T_1T_2}\right]\]
For a chemical reaction, a rise in temperature by 10 K nearly doubles the rate constant, because the fraction of molecules having energy equal to or greater than the activation energy increases.
The rate constant of a first order reaction is 2.0 × 10−4 s−1 at 300 K. The activation energy of the reaction is 45.96 kJ mol−1. What is its rate constant at 400 K? (2.303R = 19.15 J K−1 mol−1)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2.0 × 10−2 s−1
The Arrhenius equation written at two temperatures links the two rate constants through Ea. Here k1, T1 and Ea are known, so the equation is solved for k2. Ea must be in J mol−1, the unit of R.
\( \displaystyle \log\frac{k_2}{k_1} = \frac{E_a}{2.303R}\left[\frac{T_2 - T_1}{T_1T_2}\right] \)
\( \displaystyle \log\frac{k_2}{k_1} = \frac{45960\ \mathrm{J\,mol^{-1}}}{19.15\ \mathrm{J\,K^{-1}mol^{-1}}}\left[\frac{400 - 300}{300 \times 400}\right]\mathrm{K^{-1}} \)
\( \displaystyle \log\frac{k_2}{k_1} = 2400 \times \frac{1}{1200} = 2.000 \)
\( \displaystyle \frac{k_2}{k_1} = 10^{2} = 100 \)
\( \displaystyle k_2 = 100 \times 2.0 \times 10^{-4}\ \mathrm{s^{-1}} = 2.0 \times 10^{-2}\ \mathrm{s^{-1}} \)
The rate constant becomes 100 times larger for this 100 K rise, as expected, since k increases with temperature.
NCERT (2026-27 edition):
3.4 Temperature Dependence of the Rate of a Reaction – “The first order rate constant for the decomposition of ethyl iodide by the reaction C2H5I(g) → C2H4 (g) + HI(g) at 600K is 1.60 × 10–5 s–1. Its energy of activation is 209 kJ/mol. Calculate the rate constant of the reaction at 700K.”
\( \displaystyle \log k_2 - \log k_1 = \frac{E_a}{2.303R}\left[\frac{1}{T_1} - \frac{1}{T_2}\right] \)