Question

Concerning the chemical potentials of components in a binary system at constant pressure, the correct statement(s) is/are

• A. For single-phase equilibrium at a given temperature, chemical potentials of the components change with alloy composition.
• B. For two-phase equilibrium at a given temperature, chemical potential of any component in both phases is same.
• C. For two-phase equilibrium at a given temperature, chemical potentials of the components change with alloy composition.
• D. For single-phase equilibrium of a given composition, chemical potentials of the components do not change with temperature.

Hint:

Remember these key rules for chemical potential (\(\mu\)) in phase diagrams (at constant T & P): - {In a single-phase region:} \(\mu\) changes as you change the alloy composition. - {In a two-phase region:} \(\mu\) of a component is constant across the entire region (for a given temperature), because the compositions of the individual phases are fixed.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Chemical potential (\(\mu\)) is a form of potential energy that can be absorbed or released during a chemical reaction or phase transition. It is the partial molar Gibbs free energy. A key principle of thermodynamic equilibrium is that the chemical potential of any given component must be uniform throughout all phases in which it is present.
Step 2: Detailed Analysis of Each Statement:
- (A) For single-phase equilibrium at a given temperature, chemical potentials of the components change with alloy composition.
In a single-phase solid solution (e.g., \(\alpha\)-phase), the chemical potential of a component (say, component A) is given by \(\mu_A = \mu_A^\circ + RT\ln(a_A)\), where \(a_A\) is the activity of A. The activity is related to the mole fraction (composition), \(a_A = \gamma_A X_A\). As the alloy composition (\(X_A\)) changes, the activity (\(a_A\)) changes, and therefore the chemical potential (\(\mu_A\)) must also change. This statement is correct.
- (B) For two-phase equilibrium at a given temperature, chemical potential of any component in both phases is same.
This is the fundamental condition for phase equilibrium. If two phases, \(\alpha\) and \(\beta\), are in equilibrium, then for any component 'i' present in both phases, its chemical potential must be equal in both phases: \(\mu_i^\alpha = \mu_i^\beta\). If this were not true, there would be a net flux of component 'i' from the phase with higher chemical potential to the one with lower chemical potential, meaning the system would not be in equilibrium. This statement is correct.
- (C) For two-phase equilibrium at a given temperature, chemical potentials of the components change with alloy composition.
In a two-phase region of a binary phase diagram at a constant temperature, the compositions of the two individual phases in equilibrium are fixed (given by the ends of the tie-line). Since the composition of each phase is fixed, the chemical potential of a component within that phase is also fixed. Changing the overall alloy composition (moving along the tie-line) only changes the relative amounts (lever rule) of the two phases, not their compositions or the chemical potentials of the components within them. Therefore, this statement is incorrect.
- (D) For single-phase equilibrium of a given composition, chemical potentials of the components do not change with temperature.
The chemical potential is the partial molar Gibbs free energy (\(\mu_i = (\partial G / \partial n_i)_{T,P,n_j}\)). The Gibbs free energy is temperature-dependent (\(dG = -SdT + VdP\)). Similarly, the chemical potential is also temperature-dependent. The relationship is \((\partial \mu_i / \partial T)_{P, comp} = -\bar{S}_i\), where \(\bar{S}_i\) is the partial molar entropy. Since entropy is generally not zero, the chemical potential changes with temperature. This statement is incorrect.
Step 3: Why This is Correct:
Statements (A) and (B) describe fundamental principles of the thermodynamics of solutions and phase equilibria. Statement (A) reflects how chemical potential depends on concentration, and statement (B) states the core condition for phase equilibrium.