Question

The Arrhenius expression for diffusion coefficient (D) is:

• A. \( D = D_0 e^{(-Q/RT)} \)
• B. \( D = D_0 e^{(Q/RT)} \)
• C. \( D = D_0 T e^{(-Q/RT)} \)
• D. \( D = D_0 + Q/RT \)

Hint:

An Arrhenius-type relationship is common for many thermally activated processes in metallurgy, including vacancy formation, recrystallization, and chemical reactions.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the standard Arrhenius mathematical expression that models how the solid-state diffusion coefficient \( D \) varies with absolute temperature \( T \).

Step 2: Key Formula or Approach:
The diffusion of atoms requires them to jump over potential energy barriers. The rate of these jumps is thermally activated, following the Arrhenius rate equation.

Step 3: Detailed Explanation:

• Equation Parameters: The standard equation is:
\[ D = D_0 e^{-\frac{Q}{RT}} \]
where:
\( D \) is the diffusion coefficient (or diffusivity) in \( \text{m}^2/\text{s} \),
\( D_0 \) is the temperature-independent pre-exponential factor,
\( Q \) is the activation energy for diffusion (in J/mol or eV/atom),
\( R \) is the gas constant, and
\( T \) is the absolute temperature in Kelvin.

• Physical Meaning: The exponential term \( e^{-Q/RT} \) represents the fraction of atom vibrations that possess energy equal to or greater than the activation energy barrier \( Q \).

• Plotting: Taking the natural logarithm of both sides yields:
\[ \ln D = \ln D_0 - \frac{Q}{R}\left(\frac{1}{T}\right) \]
A plot of \( \ln D \) vs \( 1/T \) yields a straight line with a slope of \( -Q/R \).


Step 4: Final Answer:
Thus, the correct Arrhenius expression is \( D = D_0 e^{(-Q/RT)} \), which matches Option (A).