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 Å). Why is this ratio so large ?

Answer 1

Given, the molar volume of an ideal gas at standard temperature and pressure is 22.4 L and the size of a hydrogen molecule is about 1  A o .

Let r be the radius of the hydrogen molecule.

The radius of the hydrogen molecules is,

r= 1  A o 2 =0.5  A o

The volume of one hydrogen molecule is,

V= 4 3 π r 3 = 4 3 π ( 0.5  A o × 10 −10  m 1  A o ) 3 =0.52× 10 −30   m 3

One mole of hydrogen contains 6.023× 10 23 molecule.

So, the total atomic volume of one mole of hydrogen molecule is,

V atomic =V×n =( 5.236× 10 −31   m 3 )( 6.023× 10 23 ) =3.15× 10 −7   m 3 ≅3× 10 −7   m 3

The molar volume of an ideal gas is,

V molar =22.4 l× 1000  cm 3 1 l × 10 −6   m 3 1  cm 3 =22.4× 10 −3   m 3

Hence the ratio of the molar volume to the atomic volume of a mole of hydrogen is,

ratio= V molar V atomic

Substitute the given value in above expression.

ratio= 22.4× 10 −3   m 3 3× 10 −7   m 3 =7.5× 10 4 ≈ 10 4

The ratio of the volumes comes out to be very large because the inter-atomic separation in hydrogen gas is much larger than the size of a hydrogen atom.