Question

A 100 L cylinder containing H2 exerted a pressure of 4 atm at 300 K. It was accidentally opened and some H2 was escaped. When it was closed, it exerted a pressure of 3 atm at 300 K. The number of moles of H2 remaining in the cylinder is equal to

• A. \( \frac{1}{2} R \)
• B. \( R \)
• C. \( \frac{1}{R} \)
• D. \( 2R \)

Hint:

Remember, when dealing with ideal gases, the number of moles is proportional to pressure when volume and temperature are constant.

Answer 1

The Correct Option is C

Given the ideal gas law: \[ PV = nRT \] Where: - \( P \) is pressure - \( V \) is volume - \( n \) is the number of moles - \( R \) is the gas constant - \( T \) is temperature At initial conditions: \[ P_1 = 4 \, \text{atm}, \, V = 100 \, L, \, T = 300 \, K \] Using the ideal gas law: \[ P_1 V = n_1 R T \] So, the number of moles initially in the cylinder is: \[ n_1 = \frac{P_1 V}{RT} \] At final conditions: \[ P_2 = 3 \, \text{atm}, \, V = 100 \, L, \, T = 300 \, K \] Using the ideal gas law again: \[ P_2 V = n_2 R T \] Thus, the number of moles remaining in the cylinder is: \[ n_2 = \frac{P_2 V}{RT} \] Since the volume and temperature remain constant, the ratio of the final to initial moles is: \[ \frac{n_2}{n_1} = \frac{P_2}{P_1} = \frac{3}{4} \] Thus, the number of moles remaining in the cylinder is \( \boxed{\frac{1}{R}} \).

Answer 2

Step 1: Understand the initial condition using the ideal gas law.
Given:
- Initial pressure, \( P_1 = 4 \, \text{atm} \)
- Volume, \( V = 100 \, \text{L} \)
- Temperature, \( T = 300 \, \text{K} \)
- Gas constant, \( R \) (value not given explicitly, but will be kept symbolic)

Using ideal gas law:
\[ P_1 V = n_1 R T \implies n_1 = \frac{P_1 V}{R T} = \frac{4 \times 100}{R \times 300} = \frac{400}{300R} = \frac{4}{3R}. \]

Step 2: Understand the condition after some gas escaped.
Final pressure, \( P_2 = 3 \, \text{atm} \)
Volume and temperature remain constant.
Number of moles remaining:
\[ P_2 V = n_2 R T \implies n_2 = \frac{P_2 V}{R T} = \frac{3 \times 100}{R \times 300} = \frac{300}{300R} = \frac{1}{R}. \]

Step 3: Conclusion.
The number of moles of \(\text{H}_2\) remaining in the cylinder is \(\boxed{\frac{1}{R}}\).