3 g of a weak monobasic acid HA (molar mass 50 g mol ⁻¹ ) is dissolved in 1 kg of water. The freezing point of the resulting solution is -0.135^ C and its density is 1.003 g mL ⁻¹ . If the acid dissociation constant K_a of the acid is x 10⁻³ , the value of x is ________. (Given: K_f of water is 1.8 K kg mol ⁻¹ )

Step-by-step solution

Moles of HA = 3 50 = 0.06 mol Molality m = 0.06 mol 1 kg = 0.06 mol kg ⁻¹ Mass of solution = Mass of solvent + Mass of solute = 1000 + 3 = 1003 g Volume of solution = Mass Density = 1003 1.003 = 1000 mL = 1 L Molarity C = 0.06 mol 1 L = 0.06 M Using the freezing point depression formula: T_f = i K_f m 0.135 = i 1.8 0.06 i = 0.135 0.108 = 1.25 For a weak monobasic acid, i = 1 + 1 + = 1.25 = 0.25 The acid dissociation constant K_a is: K_a = C ^2 1 - = 0.06 (0.25)^2 1 - 0.25 = 0.06 0.0625 0.75 = 0.005 = 5 10⁻³ Thus, t