Question

Design an approach to estimate the activity coefficient of a component in a non-ideal liquid mixture:

• A. Use the ideal gas law
• B. Apply the Gibbs-Duhem equation with experimental data
• C. Assume it equals mole fraction
• D. Use the Second Law only

Hint:

Use the Gibbs–Duhem equation when estimating activity coefficients in non-ideal mixtures — especially when supported by experimental data.

Answer 1

The Correct Option is B

In a non-ideal liquid mixture, the activity coefficient $\gamma_i$ represents the deviation of the behavior of a real solution from an ideal one. It provides a correction factor to the chemical potential or fugacity of a component.
The most accurate way to estimate activity coefficients is by using experimental data in conjunction with the Gibbs–Duhem equation, which relates the changes in activity coefficients of all components in a mixture.
The differential form of the Gibbs–Duhem equation is:
\[ \sum x_i d\ln \gamma_i = 0 \]
This equation ensures thermodynamic consistency and allows us to estimate the activity coefficients across a range of compositions if experimental data (such as vapor–liquid equilibrium data) are available.
Let’s review the other options:
- (1) Use the ideal gas law: This assumes ideal behavior and doesn't account for real solution interactions — inappropriate for non-ideal mixtures.
- (3) Assume it equals mole fraction: This is valid only for ideal mixtures, where $\gamma_i = 1$, not for non-ideal ones.
- (4) Use the Second Law only: While entropy concepts underpin thermodynamic consistency, this approach alone doesn't provide a practical means to calculate activity coefficients.
Therefore, the best and most scientifically accurate approach is to apply the Gibbs–Duhem equation with experimental data.