Question

Two vessels \( A \) and \( B \) are of the same size and are at the same temperature. Vessel \( A \) contains \( 1 \, \text{g} \) of hydrogen and vessel \( B \) contains \( 1 \, \text{g} \) of oxygen. \( P_A \) and \( P_B \) are the pressures of the gases in \( A \) and \( B \) respectively. Then \( \frac{P_A}{P_B} \) is:

• A. \( 8 \)
• B. \( 16 \)
• C. \( 32 \)
• D. \( 4 \)

Hint:

At constant temperature and volume, the pressure of a gas is directly proportional to its number of moles. For comparisons, use the relationship \( P \propto n \).

Answer 1

The Correct Option is B

The ideal gas equation is PV = nRT, where P is pressure, V is volume, T is temperature, and n is the number of moles. This implies that pressure is directly proportional to the number of moles: \( P \propto n \). Step 1: Calculate the number of moles for each gas. For hydrogen (\( H_2 \)), the molar mass is \( M_{H_2} = 2 \, \text{g/mol} \). With a mass of 1 g, the number of moles is \( n_A = \frac{1}{2} = 0.5 \, \text{mol} \). For oxygen (\( O_2 \)), the molar mass is \( M_{O_2} = 32 \, \text{g/mol} \). With a mass of 1 g, the number of moles is \( n_B = \frac{1}{32} = 0.03125 \, \text{mol} \). Step 2: Compute the ratio of pressures. Given \( P \propto n \), the ratio of pressures \( \frac{P_A}{P_B} \) is equal to the ratio of moles \( \frac{n_A}{n_B} \). \( \frac{P_A}{P_B} = \frac{n_A}{n_B} = \frac{0.5}{0.03125} \). Simplifying this ratio gives: \( \frac{P_A}{P_B} = 16 \). Final Answer: \( \boxed{16} \)