Question

Vacancy diffusion coefficient: $D = D_0 \exp(-Q/RT)$

• A. Dimensionally correct
• B. Requires $D_0/L$ to be correct
• C. Requires Q dimensionless
• D. All formulas fail dimensionally

Hint:

Arguments of transcendental functions (like $\exp$, $\ln$, $\sin$, $\cos$) must always be dimensionless.
This rule is extremely helpful for verifying physical formulas quickly.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks to evaluate the dimensional consistency of the Arrhenius equation for the vacancy diffusion coefficient:
\[ D = D_0 \exp\left(-\frac{Q}{RT}\right) \]

Step 2: Key Formula or Approach:
For any equation containing an exponential function of the form $y = e^x$, the exponent $x$ must be a dimensionless quantity.
If the exponent is dimensionless, then the dimension of the left-hand side must match the dimension of the pre-exponential factor:
\[ [D] = [D_0] \]

Step 3: Detailed Explanation:

• Let us analyze the dimensions of the terms inside the exponent, $-\frac{Q}{RT}$:
- $Q$ is the activation energy for diffusion, typically in Joules per mole ($[\text{J/mol}]$).
- $R$ is the universal gas constant, in Joules per mole-Kelvin ($[\text{J/(mol} \cdot \text{K)}]$).
- $T$ is the absolute temperature, in Kelvin ($[\text{K}]$).

• Evaluating the dimensions of the denominator $RT$:
\[ [RT] = [\text{J/(mol} \cdot \text{K)}] \times [\text{K}] = [\text{J/mol}] \]

• Evaluating the dimensions of the ratio $\frac{Q}{RT}$:
\[ \left[\frac{Q}{RT}\right] = \frac{[\text{J/mol}]}{[\text{J/mol}]} = [1] \quad (\text{dimensionless}) \]

• Because the exponent is dimensionless, the term $\exp(-Q/RT)$ is a dimensionless number.

• Both the diffusion coefficient $D$ and the pre-exponential factor $D_0$ have identical units of area per unit time ($[\text{m}^2/\text{s}]$).

• Therefore, the equation is fully dimensionally correct without requiring any modifications.

Step 4: Final Answer:
The equation is dimensionally correct.