Question

The number of given statement/s which is/are correct is ________ [JEE MAINS - ONLINE - 2023]

Answer 1

2

Answer 2

The relationship between the rate constant ($k$) and temperature ($T$) is described by the Arrhenius equation: $k = A e^{-E_a/RT}$ where $A$ is the pre-exponential factor, $E_a$ is the activation energy, and $R$ is the ideal gas constant. Taking the natural logarithm of both sides, we get: $\ln k = \ln A - \frac{E_a}{RT}$

This equation can be rewritten as $\ln k = \ln A + \left(-\frac{E_a}{R}\right) \frac{1}{T}$. This is a linear equation of the form $y = mx + c$, where $y = \ln k$, $x = 1/T$, $m = -E_a/R$, and $c = \ln A$.

Let's analyze each statement:

(A) The stronger the temperature dependence of the rate constant, the higher is the activation energy. The temperature dependence of the rate constant can be quantified by the relative change in $k$ with respect to $T$. Differentiating $\ln k$ with respect to $T$: $\frac{d(\ln k)}{dT} = \frac{d}{dT}\left(\ln A - \frac{E_a}{RT}\right)$ $\frac{1}{k}\frac{dk}{dT} = 0 - E_a R \left(-\frac{1}{T^2}\right) = \frac{E_a}{RT^2}$ The term $\frac{E_a}{RT^2}$ represents the relative rate of change of the rate constant with temperature. A larger magnitude of this term indicates a stronger temperature dependence. If $E_a > 0$, a higher value of $E_a$ leads to a larger positive value of $\frac{E_a}{RT^2}$, meaning $k$ increases more rapidly with $T$. Thus, stronger temperature dependence corresponds to higher activation energy. If $E_a < 0$, a more negative value of $E_a$ (e.g., -100 kJ/mol compared to -50 kJ/mol) leads to a more negative value of $\frac{E_a}{RT^2}$, meaning $k$ decreases more rapidly with $T$. In this case, stronger temperature dependence corresponds to a lower (more negative) activation energy. Since the statement implies a general relationship without specifying the sign of $E_a$, and it holds true only for $E_a > 0$, this statement is not universally correct if $E_a$ can be negative. However, in the context of typical chemical kinetics problems at this level, activation energy is often assumed to be positive unless specified. If we assume $E_a \ge 0$, then statement (A) is correct.

(B) If a reaction has zero activation energy, its rate is independent of temperature. If $E_a = 0$, the Arrhenius equation becomes: $k = A e^{-0/RT} = A e^0 = A$ The rate constant $k$ is equal to the pre-exponential factor $A$, which is independent of temperature. Therefore, the rate of the reaction is independent of temperature. This statement is correct.

(C) The stronger the temperature dependence of the rate constant, the smaller is the activation energy. This is the converse of statement (A). As analyzed above, if $E_a > 0$, stronger temperature dependence implies higher $E_a$. If $E_a < 0$, stronger temperature dependence implies more negative (smaller) $E_a$. Since it does not hold for $E_a > 0$, this statement is not universally correct. If we assume $E_a \ge 0$, then this statement is incorrect.

(D) If there is no correlation between the temperature and the rate constant then it means that the reaction has negative activation energy. "No correlation" means that the rate constant $k$ does not change with temperature $T$. This implies $\frac{dk}{dT} = 0$. From the relation $\frac{dk}{dT} = \frac{E_a}{RT^2} k$, for $\frac{dk}{dT}$ to be zero (assuming $k \neq 0$ and $T \neq 0$), we must have $E_a = 0$. If $E_a < 0$, the rate constant $k$ does change with temperature (it decreases as $T$ increases), meaning there is a correlation. Thus, this statement is incorrect.

Revisiting (A) and (C) assuming $E_a \ge 0$: If $E_a \ge 0$, then $\frac{E_a}{RT^2} \ge 0$. A larger value of $E_a$ leads to a larger value of $\frac{E_a}{RT^2}$, meaning a stronger temperature dependence (rate constant increases more significantly with temperature). Thus, statement (A) is correct under this common assumption. Statement (C) is incorrect.

Based on this analysis, statements (A) and (B) are correct. The question asks for the number of correct statements.

Number of correct statements = 2.