Question

An evacuated glass vessel weighs 40.0 g when empty, 135.0 g when filled with a liquid of density 0.95 g mL\(^{-1}\) and 40.5 g when filled with an ideal gas at 0.82 atm at 250 K. The molar mass of the gas in g mol\(^{-1}\) is: (Given: R = 0.082 L atm K\(^{-1}\) mol\(^{-1}\))

• A. 35
• B. 50
• C. 75
• D. 125

Hint:

Always keep an eye on the units! Since \(R\) is given in L atm, ensure your volume is converted from mL to Liters ($100\text{ mL} = 0.1\text{ L}$) before plugging it into the equation.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We first need to determine the volume of the vessel using the mass and density of the liquid. Once the volume is known, we can use the Ideal Gas Equation to find the molar mass of the gas.

Step 2: Key Formula or Approach:
1. Volume of vessel (\(V\)) = \(\frac{\text{Mass of liquid}}{\text{Density of liquid}}\)
2. Ideal Gas Law: \(PV = nRT = \frac{m}{M}RT\), where \(M\) is the molar mass.

Step 3: Detailed Explanation:
1. Find the volume of the vessel: \[ \text{Mass of liquid} = 135.0\text{ g} - 40.0\text{ g} = 95.0\text{ g} \] \[ V = \frac{\text{Mass}}{\text{Density}} = \frac{95.0\text{ g}}{0.95\text{ g/mL}} = 100\text{ mL} = 0.1\text{ L} \] 2. Find the mass of the gas: \[ m = 40.5\text{ g} - 40.0\text{ g} = 0.5\text{ g} \] 3. Calculate Molar Mass (\(M\)): Rearranging \(PV = \frac{m}{M}RT\): \[ M = \frac{mRT}{PV} \] Substitute the values: \(m = 0.5\text{ g}\), \(R = 0.082\), \(T = 250\text{ K}\), \(P = 0.82\text{ atm}\), \(V = 0.1\text{ L}\). \[ M = \frac{0.5 \times 0.082 \times 250}{0.82 \times 0.1} \] Notice that \(\frac{0.082}{0.82} = 0.1\): \[ M = \frac{0.5 \times 0.1 \times 250}{0.1} = 0.5 \times 250 = 125\text{ g/mol} \]

Step 4: Final Answer
The molar mass of the gas is 125 g mol\(^{-1}\).