1. What is the volume occupied by$\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{mol}$of an ideal gas at standard conditions- that is$\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{atm}\mathbf{}(\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{01}\mathbf{\times }{\mathbf{10}}^5\mathbf{Pa})$and$\mathbf{273}\mathbf{K}\mathbf{?}$
2. Show that the number of molecules per cubic centimetre (the Loschmidt number) at standard conditions is$\mathbf{2}\mathbf{.}\mathbf{69}\mathbf{\times }{\mathbf{10}}^{1\mathbf{9}}$.

If the internal energy of the gas is a function of temperature only, then the gas is called as an ideal gas. The ideal gas obeys the ideal gas law, which relates the state parameters of the gas as-

$\mathbf{PV}\mathbf{=}\mathbf{nRT}$

1. The number of moles of gas$n=1\text{\hspace{0.17em}mol}$
2. The standardpressure$P=1.00\text{\hspace{0.17em}atm}=1.01\times {10}^5\text{\hspace{0.17em}Pa}$
3. The standard temperature$T=273\text{\hspace{0.17em}K}$
4. The ideal gas constant$R=8.31\text{J}/\text{mol}\cdot \text{K}$

$\begin{array}{l}PV=nRT\\ V=\frac{nRT}{P}\end{array}$

For the given values of state parameters, the equation becomes-

$V=\frac{1\text{\hspace{0.17em}mol}\times 8.31\text{J}/\text{mol}\cdot \text{K}\times 273\text{\hspace{0.17em}K}}{1.01\times {10}^5\text{\hspace{0.17em}Pa}}$

$\begin{array}{c}V=2.24\times {10}^{-2}{\text{m}}^3\\ =22.4\text{\hspace{0.17em}L}\end{array}$

The volume occupied by 1 mole of the gas is$22.4\text{\hspace{0.17em}L}$

b.

The number of molecules in 1 mole of the gas is.$6.02\times {10}^{23}$

The number of molecules in$22.4\text{\hspace{0.17em}L}$ is$6.02\times {10}^{23}$, so the number of molecules in $1{\text{\hspace{0.17em}cm}}^{\text{3}}$of gas is-

$\begin{array}{c}N=\frac{6.02\times {10}^{23}}{22.4\text{\hspace{0.17em}L}}\times \frac{1{\text{\hspace{0.17em}m}}^3}{{10}^6{\text{cm}}^3}\\ =\frac{6.02\times {10}^{23}}{2.24\times {10}^{-2}{\text{m}}^3}\times \frac{1{\text{\hspace{0.17em}m}}^3}{{10}^6{\text{cm}}^3}\\ =2.69\times {10}^{19}\end{array}$

Hence, number of molecules in$1{\text{\hspace{0.17em}cm}}^{\text{3}}$ of gas$N=2.69\times {10}^{19}$.

The volume occupied by the one mole of the gas at standard temperature and pressure is $22.4\text{\hspace{0.17em}L}$and the number of molecules in one cubic centimeter of the gas is.$2.69\times {10}^{19}$