Before You Begin

Question #44057

Difficulty: 8/10 | 🟢 Free

The Question

A $0.10 \text{ mol kg}^{-1}$ aqueous solution of a weak monoprotic acid \ce{HA (aq)} freezes at $-0.2232^\circ\text{C}$ at $1 \text{ atm}$ . The cryoscopic constant of water is $K_f = 1.86 \text{ K kg mol}^{-1}$ . Assume the solution behaves ideally with respect to the solvent (osmotic coefficient $\phi = 1$ ), such that the freezing point depression is strictly proportional to the total molality of all solute particles. However, the acid dissociation equilibrium is non-ideal. If the mean ionic activity coefficient of the ions \ce{H+ (aq)} and \ce{A- (aq)} in this solution is $\gamma_{\pm} = 0.80$ and the activity coefficient of the unionized acid is $\gamma_{HA} = 1.0$ , what is the true thermodynamic acid dissociation constant ( $K_a^\circ$ ) of \ce{HA} at the freezing point?

Options

A.
$3.20 \times 10^{-3}$
B.
$5.00 \times 10^{-3}$
C.
$4.00 \times 10^{-3}$
D.
$7.81 \times 10^{-3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Numerical_Precision

Improve calculation stamina and avoid rounding errors in high-stakes competitive examinations.

Concepts Behind This Question

Standard State Conditions|Raoult’s Law Ideal vs Non-Ideal|Van’t Hoff Factor (i) Application

Why This Question Is Tricky

Common Mistakes

This problem tests the critical distinction between the apparent equilibrium constant ( $K_m$ ) derived from colligative properties and the true thermodynamic equilibrium constant ( $K_a^\circ$ ).

Students frequently fall into the trap of assuming ideal solute behavior ( $\gamma = 1$ ) and stopping at $K_m = 5.00 \times 10^{-3}$ .

The 'Hidden Variable' is the mean ionic activity coefficient ( $\gamma_{\pm}$ ), which must be squared to correct the activities of both the cation and anion in the numerator of the equilibrium expression.

Distractors include ignoring the activity coefficient entirely (Option B), failing to square it (Option C), or incorrectly dividing by it (Option D).

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Thermodynamics - Class 11 Part 1
Concept	Standard State Conditions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free