Arrhenius Equation and Temperature Dependence of Rate Constant

To determine the correct statements, we analyze the Arrhenius equation, which expresses the dependence of the rate constant $k$ on temperature $T$ and activation energy $E_a$:

$k = A e^{-E_a / RT}$

where:

• $k$ is the rate constant,
• $A$ is the pre-exponential (frequency) factor,
• $E_a$ is the activation energy,
• $R$ is the universal gas constant,
• $T$ is the absolute temperature in Kelvin.

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Analysis of Options:

1. Option (A): A high activation energy $E_a$ means the magnitude of $-\frac{E_a}{RT}$ is a large negative quantity. As a result: $e^{-E_a / RT} \ll 1 \implies k \text{ is small}$ A small rate constant $k$ corresponds to a slow reaction, not a fast reaction.
   $\therefore$ Statement (A) is incorrect.

2. Option (B): The term $e^{-E_a / RT}$ represents the fraction of total molecular collisions that possess energy greater than or equal to the activation energy $E_a$. As temperature $T$ increases, the ratio $\frac{E_a}{RT}$ decreases, which causes $e^{-E_a / RT}$ to increase. Consequently, a greater number of collisions have sufficient energy to cross the activation energy barrier, leading to an increase in $k$.
   $\therefore$ Statement (B) is correct.

3. Option (C): Taking the natural logarithm of the Arrhenius equation: $\ln k = \ln A - \frac{E_a}{RT}$

   Differentiating with respect to temperature $T$: $\frac{d(\ln k)}{dT} = \frac{E_a}{RT^2}$

   The fractional change in rate constant per unit temperature change, $\frac{d(\ln k)}{dT}$, is directly proportional to $E_a$. Therefore, a higher activation energy $E_a$ implies a steeper slope/change in $\ln k$ with temperature, meaning the rate constant has a stronger temperature dependence.
   $\therefore$ Statement (C) is correct.

4. Option (D): According to collision theory, the pre-exponential factor $A$ is given by: $A = P \cdot Z_{AB}$ where $Z_{AB}$ is the total collision frequency (number of collisions per unit time per unit volume) and $P$ is the steric/probability factor. The factor $A$ accounts for the total frequency of collisions regardless of whether the colliding molecules possess energy $E \ge E_a$ (which is separately accounted for by $e^{-E_a/RT}$).
   $\therefore$ Statement (D) is correct.

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Conclusion:

The correct statements are (B), (C), and (D).