The rate constant, activation energy and the Arrhenius parameter at 27°C are \( 2.0 × 10^{-5}\, s^{-1},\, 100\, kJ\, mol^{-1} \) and \( 6.0 × 10^{14}\, s^{-1} \), respectively. The value of the rate constant at a very high temperature (T→∞) approaches to:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(6.0 \times 10^{14}\, s^{-1}\)
The Arrhenius equation relates the rate constant to temperature:
\( \displaystyle k = A\,e^{-E_a/RT} \)
Here A is the Arrhenius factor (pre-exponential or frequency factor), and \( e^{-E_a/RT} \) is the fraction of molecules with energy equal to or greater than \( E_a \). This fraction can never be more than 1.
As \( T \to \infty \):
\( \displaystyle \frac{E_a}{RT} \to 0 \)
\( \displaystyle e^{-E_a/RT} \to e^{0} = 1 \)
\( \displaystyle k \to A = 6.0 \times 10^{14}\ \mathrm{s^{-1}} \)
At a very high temperature practically every collision has enough energy, so the rate constant approaches the Arrhenius factor, \( 6.0 \times 10^{14}\ \mathrm{s^{-1}} \). The values of k at 27 °C and of \( E_a \) are not needed.
Why the other options are incorrect
Option (1) \(2.0 \times 10^{-5}\, s^{-1}\) — Incorrect. This is the rate constant at 27 °C (300 K); k rises with temperature, so it cannot stay at this value.
Option (2) Infinite — Incorrect. The exponential factor approaches 1 and no more, so k has an upper limit, A; it does not grow without bound.
Option (4) \(4 \times 10^{-5}\, s^{-1}\) — Incorrect. This is only twice the value at 27 °C; a moderate rise in temperature can give such a value, but the limit at very high temperature is A, about \( 10^{19} \) times larger.