Question

The mean free path of molecules of a certain gas at STP is 1500d, where d is the diameter of the gas molecules. While maintaining the standard pressure, the mean free path of the molecules at 373K is approximately

• A. 1500d
• B. 750d
• C. 2049d
• D. 1098d

Answer 1

The Correct Option is C

The mean free path \( \lambda \) of molecules is given by the formula: \[ \lambda = \frac{RT}{\sqrt{2} \pi d^2 N_A P}, \] where \( R \) is the gas constant, \( T \) is the temperature, \( d \) is the diameter of gas molecules, \( N_A \) is Avogadro's number, and \( P \) is the pressure. At constant pressure, the mean free path is directly proportional to temperature \( T \): \[ \lambda \propto T. \] Given: \[ \lambda_\text{STP} = 1500d \quad \text{at} \quad T = 273 \, \text{K}. \] For \( T = 373 \, \text{K} \), the new mean free path \( \lambda \) is: \[ \frac{\lambda}{\lambda_\text{STP}} = \frac{T}{T_\text{STP}}. \] Substituting the values: \[ \frac{\lambda}{1500d} = \frac{373}{273}. \] Simplify: \[ \lambda = 1500d \cdot \frac{373}{273} \approx 2049d. \] Thus, the mean free path of the molecules at \( 373 \, \text{K} \) is approximately \( \boxed{2049d} \).