Question

The mean free path of the molecule of a certain gas at 300 K is $2.6\times {{10}^{-5}}m$ . The collision diameter of the molecule is 0.26 nm. Calculate
(a) Pressure of the gas, and
(b) Number of molecules per unit volume of the gas.
A. (a) $1.281\times {{10}^{23}}{{m}^{-3}}$ (b) $5.306\times {{10}^{2}}Pa$
B. (a) $1.281\times {{10}^{22}}{{m}^{-3}}$ (b) $5.306\times {{10}^{3}}Pa$
C. (a) $12.81\times {{10}^{23}}{{m}^{-3}}$ (b) $53.06\times {{10}^{2}}Pa$
D. (a) $2.56\times {{10}^{23}}{{m}^{-3}}$ (b) $10.612\times {{10}^{2}}Pa$

Answer 1

Hint: The formula used to calculate the pressure of the gas is as follows.
\[P=\dfrac{KT}{\sqrt{2}\pi {{\sigma }^{2}}\lambda }\]
Where P = Pressure of the gas
K = $\dfrac{R}{{{N}_{A}}}$ , R= Gas constant, ${{N}_{A}}$ = Avogadro number
T = Temperature of the gas
$\sigma $ = Diameter of the gas molecule
$\lambda $ = Mean free path

Complete step-by-step answer: - In the question it is asked to calculate the pressure of the gas and number of molecules per unit volume of the gas by using the data given in the question.
a) Initially we have to calculate the pressure of the gas by using the below formula.
\[P=\dfrac{KT}{\sqrt{2}\pi {{\sigma }^{2}}\lambda }\]
Where P = Pressure of the gas
K = $\dfrac{R}{{{N}_{A}}}$ , R= Gas constant = 8.314 , ${{N}_{A}}$ = Avogadro number = $6.023\times {{10}^{23}}$
T = Temperature of the gas = 300 K
$\sigma $ = Diameter of the gas molecule = 0.26 nm = 0.26 $\times {{10}^{-10}}m$
$\lambda $ = Mean free path = $2.6\times {{10}^{-5}}m$
- Substitute the above values in the above formula to get the pressure of the gas.
\[ P=\dfrac{KT}{\sqrt{2}\pi {{\sigma }^{2}}\lambda } \\
=\dfrac{8.314\times 300}{\sqrt{2}\times 3.14\times 0.2\times {{10}^{-10}}\times 2.6\times {{10}^{-5}}} \\
 =\dfrac{2494.5}{4700.86} \\
 P=5.30\times {{10}^{2}} \\
\]
- The pressure of the gas at 300 K is $P=5.30\times {{10}^{2}}$.
b) Now we have to calculate the Number of molecules per unit volume of the gas by using the formula below.
P = cRT
P = Pressure of the gas = $5.30\times {{10}^{2}}$
c = Concentration of the gas
R = Gas constant = 8.314
T = Temperature of the gas = 300 K
- Substitute all the known values in the above formula to calculate the concentration of the gas.
P = cRT
\[\Rightarrow c=\dfrac{5.30\times {{10}^{2}}}{8.314\times 300} \\
\therefore c=2.12\times {{10}^{-3}} \\
\]
- From concentration we can calculate the number of molecules by using the below formula.
Number of molecules = (Concentration of the gas) (Avogadro Number)
Number of molecules of the gas = $2.12\times {{10}^{-2}}\times 6.023\times {{10}^{23}}=1.28\times {{10}^{21}}molecules/volume$

Note: To calculate the number of molecules of the gas first we have to find the pressure of the gas. By using pressure we have to calculate the concentration of the gas later using concentration of the gas we can calculate the number of molecules of the gas.