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Physics Question on Dimensional analysis and its applications

One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen ? (Take the size of hydrogen molecule to be about 1 A˚\text\AA ). Why is this ratio so large ?

Answer

Radius of hydrogen atom, r = 0.5 A˚\text\AA = 0.5 × 10−1010^{-10} m
Volume of hydrogen atom = 43πr3\frac{4}{3}\pi r^3
= 43×227×(0.5×10−10)3\frac{4}{3}\times \frac{22}{7}\times(0.5 \times 10^{-10})^3
= 0.524×10−30m30.524 \times 10^{-30}\text m^3
Now, 1 mole of hydrogen contains 6.023×10236.023\times 10^{23} hydrogen atoms.
∴ Volume of 1 mole of hydrogen atoms, Va\text V_a = 6.023×10236.023\times 10^{23} × 0.524×10−300.524 \times 10^{-30}
= 3.16×10−7m33.16 \times 10^{-7}\text m^3
Molar volume of 1 mole of hydrogen atoms at STP,
Vm\text V_m = 22.4 L = 22.4×10−3m322.4\times 10^{-3}\text m^3
∴VmVn\frac{\text V_m}{\text V_n} = 22.4×10−33.16×10−7\frac{22.4\times 10^{-3}}{3.16\times 10^{-7}} = 7.08×1047.08\times 10^4

Hence, the molar volume is 7.08×1047.08\times 10^4 times higher than the atomic volume. For this reason, the inter-atomic separation in hydrogen gas is much larger than the size of a hydrogen atom.