Question

The number of gas molecules per unit volume in a vessel is \(2\times10^{25}\text{ m}^{-3}\) and the surface area of each gas molecule is \(12.5\sqrt{2}\times10^{-20}\text{ m}^2\). If all the gas molecules are in motion, then the mean free path of the gas molecules (in \AA) is:

• A. \(500\)
• B. \(1000\)
• C. \(2000\)
• D. \(4000\)

Hint:

Remember that for moving gas molecules the factor \(\sqrt2\) appears in the denominator of the mean free path formula.

Answer 1

The Correct Option is C

Concept: Mean free path is \[ \lambda=\frac{1}{\sqrt{2}\,n\,\pi d^2} \] The given surface area of one molecule is \[ 4\pi r^2=\pi d^2 \] Hence, \[ \pi d^2=12.5\sqrt2\times10^{-20} \]

Step 1: Substitute in the mean free path formula. \[ \lambda = \frac{1} {\sqrt2(2\times10^{25}) (12.5\sqrt2\times10^{-20})} \] \[ = \frac{1} {50\times10^{5}} \] \[ = 2\times10^{-7}\text{ m} \]

Step 2: Convert into Angstrom units. \[ 1\AA=10^{-10}\text{ m} \] \[ \lambda = \frac{2\times10^{-7}} {10^{-10}} \] \[ =2\times10^3\AA \] \[ =2000\AA \]

Step 3: Final answer. \[ \boxed{2000\AA} \]