## I apologize if my previous response was not helpful. Let me provide a step-by-step explanation of how to calculate the cryoscopic constant and use it to determine the value of Ka for the weak acid HX.

1. Calculate the cryoscopic constant of water (Kf):

The cryoscopic constant (Kf) represents the degree of the freezing point depression caused by a solute in a solution. For water, the cryoscopic constant is approximately 1.86 Â°C/m. This means that for every 1 molal (1 mol/kg) concentration of solute, the freezing point of water is depressed by 1.86 Â°C.

2. Calculate the freezing point depression (delta T):

Given that the freezing point depression is 0.5 Â°C, you can use the formula:

delta T = i * Kf * m

where i is the van't Hoff factor, Kf is the cryoscopic constant of water, and m is the molality of the solution.

3. Determine the van't Hoff factor (i):

The van't Hoff factor represents the number of particles into which a solute dissociates in a solvent. For the weak acid HX, it partially dissociates into H+ (hydrogen ion) and X- (conjugate base). Therefore, the van't Hoff factor (i) for HX is 2 (1 for H+ and 1 for X-).

4. Solve for the molality (m):

The molality (m) of the solution is known to be 0.250 mol/kg.

5. Solve for the concentration of HX:

To determine the concentrations of H+, X-, and HX, you can assume that the concentration of H+ and X- are equal, denoted as Y. The concentration of HX can be calculated by subtracting Y from the initial concentration of HX, which is 0.250 mol/kg.

6. Express the sum of concentrations:

The sum of HX, H+, and X- should equal the effective molality of the solution, which was determined in step 2 (m(effective)).

7. Solve for Y:

Using the equation from step 6, solve for Y by substituting the known values and solving for Y.

8. Calculate Ka:

With the value of Y determined, you can calculate Ka using the equation:

Ka = [X-][H+] / [HX]

Substitute the value of Y into the equation and solve for Ka.

I hope this explanation helps you understand the steps to calculate the cryoscopic constant and determine the value of Ka for the weak acid HX. Please let me know if you need further assistance or if you have any other questions.