Question

The unit of frequency factor in Arrhenius equation is:

• A. same as that of rate constant
• B. same as that of activation energy
• C. Dimensionless
• D. inverse of that of rate constant

Hint:

Dimensional analysis rule: In any transcendental function (such as \(\exp(x)\), \(\ln(x)\), or \(\sin(x)\)), the argument \(x\) and the evaluated output must be dimensionless. Therefore, in \(k = A \cdot \exp(-E_a/RT)\), the exponential term has no units, which means the frequency factor \(A\) must share the exact same units as the rate constant \(k\).

Answer 1

The Correct Option is A

Concept: The Arrhenius equation describes the dependence of the chemical reaction rate constant ($k$) on the absolute temperature ($T$): \[ k = A \cdot \exp\left( -\frac{E_a}{R \cdot T} \right) \] Where:
• \( k \) represents the reaction rate constant.
• \( A \) represents the pre-exponential factor, also known as the frequency factor.
• \( E_a \) represents the activation energy (\(\text{J}/\text{mol}\)).
• \( R \) represents the universal gas constant (\(\text{J}/(\text{mol}\cdot\text{K})\)).
• \( T \) represents the absolute temperature (\(\text{K}\)).

Step 1: Analyzing the units of the exponential term.
Let us examine the units of the term inside the exponent, $\frac{E_a}{R \cdot T}$: \[ \text{Units of } \left( \frac{E_a}{R \cdot T} \right) = \frac{\text{J}/\text{mol}}{\left(\frac{\text{J}}{\text{mol}\cdot\text{K}}\right) \cdot \text{K}} = \frac{\text{J}/\text{mol}}{\text{J}/\text{mol}} = 1 \quad \text{(Dimensionless)} \] Because the exponent $\frac{E_a}{R \cdot T}$ is a dimensionless quantity, the entire exponential term, $\exp\left( -\frac{E_a}{R \cdot T} \right)$, is also completely dimensionless.

Step 2: Equating units across the equation.
Let us analyze the dimensional units of both sides of the Arrhenius equation: \[ [\text{Units of } k] = [\text{Units of } A] \cdot \left[\text{Units of } \exp\left(-\frac{E_a}{RT}\right)\right] \] Since the exponential term is dimensionless, this simplifies directly to: \[ [\text{Units of } A] = [\text{Units of } k] \] Therefore, the dimensional unit of the frequency factor ($A$) is completely identical to the unit of the reaction rate constant ($k$). The specific units depend on the overall order of the chemical reaction (for example, $\text{s}^{-1}$ for a first-order reaction, or $\text{L}/(\text{mol}\cdot\text{s})$ for a second-order reaction).