Question

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 \(\text\AA\) ). Why is this ratio so large ?

Answer 1

Radius of hydrogen atom, r = 0.5 \(\text\AA\) = 0.5 × \(10^{-10}\) m
Volume of hydrogen atom = \(\frac{4}{3}\pi r^3\) 
= \(\frac{4}{3}\times \frac{22}{7}\times(0.5 \times 10^{-10})^3\) 
= \(0.524 \times 10^{-30}\text m^3\)
Now, 1 mole of hydrogen contains \(6.023\times 10^{23}\) hydrogen atoms.
∴ Volume of 1 mole of hydrogen atoms, \(\text V_a\) = \(6.023\times 10^{23}\) × \(0.524 \times 10^{-30}\) 
= \(3.16 \times 10^{-7}\text m^3\)
Molar volume of 1 mole of hydrogen atoms at STP, 
\(\text V_m\) = 22.4 L = \(22.4\times 10^{-3}\text m^3\)
∴\(\frac{\text V_m}{\text V_n}\) = \(\frac{22.4\times 10^{-3}}{3.16\times 10^{-7}}\) = \(7.08\times 10^4\)

Hence, the molar volume is \(7.08\times 10^4\) 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.