One mole of an ideal gas at S.T.P. occupies 22.4 L. What is the ratio of molar volume to the atomic volume of a mole of hydrogen ? Take the radius of hydrogen molecule to be 1Å.

The correct option is 10⁴.

To find the ratio of the molar volume to the atomic volume of a mole of hydrogen, we can calculate both volumes step-by-step.

Step 1: Calculate the Molar Volume of Hydrogen Gas
We are given that one mole of an ideal gas at S.T.P. (Standard Temperature and Pressure) occupies a volume of 22.4 L. Let us convert this molar volume ($V_m$) into cubic meters (${\text{m}}^3$):
$V_m=22.4\text{L}=22.4\times {10}^{-3}{\text{m}}^3$

Step 2: Calculate the Volume of a Single Hydrogen Molecule
Let us treat the hydrogen molecule as a sphere. The radius of the hydrogen molecule is given as:
$r=1\text{Å}={10}^{-10}\text{m}$
The volume of a single sphere (one hydrogen molecule) is:
$V_1=\frac{4}{3}\pi r^3$
Substituting the value of the radius:
$V_1=\frac{4}{3}\times 3.14\times {\left({10}^{-10}\text{m}\right)}^3$
$V_1\approx 4.19\times {10}^{-30}{\text{m}}^3$

Step 3: Calculate the Atomic Volume of a Mole of Hydrogen
One mole of hydrogen contains Avogadro's number of molecules:
$N_A\approx 6.023\times {10}^{23}{\text{mol}}^{-1}$
The total molecular (or atomic) volume ($V_a$) of one mole of hydrogen is the volume occupied by all these molecules:
$V_a=N_A\times V_1$
$V_a=(6.023\times {10}^{23})\times (4.19\times {10}^{-30}{\text{m}}^3)$
$V_a\approx 2.52\times {10}^{-6}{\text{m}}^3$

Step 4: Find the Ratio of Molar Volume to Atomic Volume
Now, we calculate the ratio of the molar volume ($V_m$) to the atomic volume ($V_a$):
$\frac{V_m}{V_a}=\frac{22.4\times {10}^{-3}{\text{m}}^3}{2.52\times {10}^{-6}{\text{m}}^3}$
$\frac{V_m}{V_a}\approx 8.89\times {10}^3\approx {10}^4$
Therefore, the ratio is approximately of the order of ${10}^4$.