Step 1 : Understanding the Question:
The goal of this problem is to determine the molar mass of an unknown non-volatile solute using osmotic pressure data. Osmotic pressure is a colligative property, which means its value is proportional to the molar concentration of the solute particles in the solution. By measuring the pressure required to stop the flow of solvent through a semipermeable membrane, we can work backwards to find the number of moles of solute present, and from there, its molar mass.
Step 2 : Key Formulas and approach:
We use the van't Hoff equation for osmotic pressure: $\pi = CRT$.
Where:
1. $\pi$ is the osmotic pressure (0.82 atm).
2. $C$ is the molarity, which is $n/V = (w/M) / V$.
3. $R$ is the gas constant (0.0821 L$\cdot$atm/K$\cdot$mol).
4. $T$ is the absolute temperature in Kelvin.
Rearranging for Molar Mass ($M$): $M = \frac{wRT}{\pi V}$.
Our approach involves converting units (mL to L and $^\circ$C to K) and substituting them into the formula.
Step 3 : Detailed Explanation:
First, convert temperature to Kelvin: $T = 27 + 273 = 300$ K.
Convert volume to Liters: $V = 500$ mL = $0.5$ L.
Note the mass of solute $w = 2$ g and $\pi = 0.82$ atm.
Use the gas constant $R \approx 0.082$ L$\cdot$atm/K$\cdot$mol for easier calculation.
Substitute values into $M = \frac{2 \times 0.082 \times 300}{0.82 \times 0.5}$.
Simplify the fraction: notice that $0.082/0.82 = 0.1$.
The equation becomes: $M = \frac{2 \times 0.1 \times 300}{0.5}$.
Further simplify: $M = \frac{60}{0.5} = 120$.
The molar mass is $120$ g/mol, matching option (B).
Step 4 : Final Answer:
The molar mass of the solute is 120 g mol$^{-1}$.