If the chemical potential of a univalent cation M+ is denoted μ+ and that of a univalent anion X− is denoted μ−, the total molar Gibbs energy of the ions in the electrically neutral solution is the sum of these partial molar quantities. The molar Gibbs energy of an ideal solution is
| Gmideal = μ+ideal + μ−ideal | (1) |
However, for a real solution of M+ and X− of the same molality,
| Gm = μ+ + μ− = μ+ideal + μ−ideal + RT ln γ+ + RT ln γ− = Gmideal + RT ln γ+γ− | (2) |
All the deviations from ideality are contained in the last term.
There is no experimental way of separating the product γ+γ− into contributions from the cations and the anions. The best we can do experimentally is to assign responsibility for the nonideality equally to both kinds of ion. Therefore, for a 1,1-electrolyte, we introduce the mean activity coefficient as the geometric mean of the individual coefficients:
| γ± = (γ+γ−)1/2 | (3) |
and express the individual chemical potentials of the ions as
| μ+ = μ+ideal + RT ln γ± | μ− = μ−ideal + RT ln γ± | (4) |
The sum of these two chemical potentials is the same as before, eqn 2, but now the nonideality is shared equally.
We can generalize this approach to the case of a compound MpXq that dissolves to give a solution of p cations and q anions from each formula unit. The molar Gibbs energy of the ions is the sum of their partial molar Gibbs energies:
| Gm = pμ+ + qμ− = Gmideal + pRT ln γ+ + qRT ln γ− | (5) |
If we introduce the mean activity coefficient
| γ± = (γ+pγ−q)1/s s = p + q | (6) |
and write the chemical potential of each ion as
| μi = μiideal + RT ln γ± | (7) |
we get the same expression as in eqn 2 for Gm when we write
| G = pμ+ + qμ− | (8) |
However, both types of ion now share equal responsibility for the nonideality.
Multiple Choice Questions
1. Why is it necessary to introduce a "mean activity coefficient" (γ±) rather than measuring individual ion activity coefficients directly?
- A) Individual ions cannot exist in a solid state matrix.
- B) Solutions must remain electrically neutral, making it experimentally impossible to isolate and measure single-ion nonideality.
- C) Cations always have a greater impact on Gibbs free energy deviations than anions.
- D) Ideal solutions require both ions to behave completely identically under all conditions.
View Answer & Explanation
Correct Answer: B
Explanation: As noted in the text, there is no experimental technique available to isolate or separate the ionic interactions of single cations from single anions due to the strict requirement of overall electrical neutrality in solutions. Therefore, responsibility for nonideality is assigned equally via a geometric mean.
2. For a generalized electrolyte solution containing a compound with the formula M3X2, what is the value of the exponent s used in calculating the mean activity coefficient γ±?
- A) s = 1
- B) s = 5
- C) s = 6
- D) s = 1.5
View Answer & Explanation
Correct Answer: B
Explanation: According to Equation (6), the scaling variable s is defined as the total stoichiometric number of ions produced per formula unit, s = p + q. For M3X2, p = 3 and q = 2, so s = 3 + 2 = 5.
3. In a real electrolyte solution, which specific mathematical relationship defines how individual chemical potentials (μi) share the nonideality evenly using the mean activity coefficient?
- A) μi = μiideal · RT ln γ±
- B) μi = μiideal / (RT ln γ±)
- C) μi = μiideal + RT ln γ±
- D) μi = Gmideal + pRT ln γ+
View Answer & Explanation
Correct Answer: C
Explanation: Equation (7) explicitly establishes that by substituting the mean activity coefficient (γ±) into the equation for each individual ionic species (μi = μiideal + RT ln γ±), both ions share equal accountability for the non-ideal behavior while maintaining the mathematically correct total Gibbs energy.
Reference: Atkins' Physical Chemistry