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# 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 and Why is this Ratio So Large - CBSE (Science) Class 11 - Physics

ConceptDimensional Analysis and Its Applications

#### 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 1Å). Why is this ratio so large?

#### Solution 1

Volume of one mole of ideal gas, V= 22.4 litre = 22.4 x 10-3 m3

Radius of hydrogen molecule = 1A/2 = 0.5 A = 0.5 x 10-10 m

Volume of hydrogen molecule = 4/3 πr3

=4/3 x 22/7 (0.5 x 10-10)3 m3

= 0.5238 x 10-30 m3

One mole contains 6.023 x 1023 molecules.

Volume of one mole of hydrogen, VH

= 0.5238 x 10-30 x 6.023 x 1023 m3 = 3.1548 x 10-7 m3

Now Vg/VH=22.4 x 10-3/3.1548 x 10-7 =7.1 x 104

The ratio is very large. This is because the interatomic separation in the gas is very large compared to the size of a hydrogen molecule.

#### Solution 2

Radius of hydrogen atom, r = 0.5 Å = 0.5 × 10–10 m

Volume of hydrogen atom = 4/3pir^3

= 4/3xx22/7xx(0.5xx10^(-10))^3

= 0.524 xx 10^(-30) m^3

Now, 1 mole of hydrogen contains 6.023 × 1023 hydrogen atoms.

∴ Volume of 1 mole of hydrogen atoms, Va = 6.023 × 1023 × 0.524 × 10–30

= 3.16 × 10–7 m3

Molar volume of 1 mole of hydrogen atoms at STP

Vm = 22.4 L = 22.4 × 10–3 m3

:.V_m/V_a = (22.4xx10^(-3))/(3.16 xx10^(-7)) = 7.08 xx 10^4

Hence, the molar volume is 7.08 × 104 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

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#### APPEARS IN

NCERT Solution for Physics Textbook for Class 11 (2018 to Current)
Chapter 2: Units and Measurements
Q: 17 | Page no. 36

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Solution 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 and Why is this Ratio So Large Concept: Dimensional Analysis and Its Applications.
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