An aqueous solution containing 2.1 g of a protein in 200 mL of solution is found to be isotonic with a 5 10⁻⁴ M aqueous solution of BaCl ₂ at the same temperature. Assuming complete dissociation of BaCl ₂ , the molar mass of the protein is x 10³ g mol ⁻¹ . The value of x is ________.

Step-by-step solution

For isotonic solutions at the same temperature, their effective molarities (osmotic concentrations) must be equal: _ protein = _ BaCl ₂ C_ protein = i C_ BaCl ₂ For the protein solution: C_ protein = moles of protein Volume in L = 2.1 / M 0.2 = 10.5 M For the strong electrolyte BaCl ₂ , assuming complete dissociation: BaCl ₂ Ba ²⁺ + 2 Cl ⁻ The van't Hoff factor i = 3 . Effective concentration of BaCl ₂ : i C_ BaCl ₂ = 3 5 10⁻⁴ = 15 10⁻⁴ M Equating the two concentrations: 10.5 M = 15 10⁻⁴ M = 10.5 15 10⁻⁴ = 0.7 10⁴