### Question 2.17

NCERT Class XI Physics

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 ?

### Physical Concept and Assumptions

Assumption 1: hydrogen is an ideal gas so we can use 22.4 liters as the molar volume.

Assumption 2: each hydrogen molecule is a rigid sphere of diameter 1 Å angstorm.

Forces of attraction between two hydrogen atoms are small so they spread as a gas at STP [273.15 K and 100 kPa]. As a result there is a lot of empty space between two atoms of the gas.

What we have to do: We have to compare the sum of volumes of all hydrogen atoms [i.e., if they stay together as a solid] to the volume of the container 22.4 L.

### Video Explanation

Please watch this youtube video for a quick explanation of the solution:

### Solution in Detail

$\displaystyle 1 \text{ liter} = 10^{-3} \:m^3$

$\displaystyle \therefore \text{molar volume} = 22.4 \times 10^{-3}\:m^3$

Vol. of 1 atom $\displaystyle = \frac 16 \pi D^3 = \frac 16 \pi (10^{-10})^3$

Vol. of 1 mol atom $\displaystyle = 6 \times 10^{23} \times \frac 16 \pi (10^{-10})^3$

$ \begin{aligned} \frac{\text{molar vol.}}{\text{atomic vol}} &= \frac{22.4 \times 10^{-3}}{6 \times 10^{23} \times \displaystyle \frac 16 \pi (10^{-10})^3}\\ \\ &\approx 10^{4} \text{(see video)} \:\underline{Ans} \end{aligned} $

Why is this ratio so large? Ans: because force of attraction between two hydrogen atoms is small so they spread out as a gas.

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