1. Use kinetic theory relation

For a monoatomic ideal gas, the average kinetic energy per molecule is

$\frac{3}{2}kT = 4\times 10^{-14}\ \text{erg}$

So,

$kT = \frac{2}{3}\times 4\times 10^{-14} = \frac{8}{3}\times 10^{-14}\ \text{erg}$

2. Use ideal gas equation in molecular form

$PV = NkT$

Hence,

$N = \frac{PV}{kT}$

3. Convert pressure into cgs units

Given pressure = $2$ cm of mercury.

Pressure due to a mercury column:

$P = h\rho g$

with

• $h = 2\ \text{cm}$
• $\rho = 13.6\ \text{g/cm}^3$
• $g = 980\ \text{cm/s}^2$

Therefore,

$P = 2\times 13.6\times 980$

$P = 26656\ \text{dyne/cm}^2$

4. Volume in cgs units

Given

$V = 4\ \text{cm}^3$

So,

$PV = 26656\times 4 = 106624\ \text{erg}$

(since $1\ \text{dyne/cm}^2 \times 1\ \text{cm}^3 = 1\ \text{dyne}\cdot\text{cm} = 1\ \text{erg}$)

5. Calculate number of molecules

$N = \frac{106624}{\left(\frac{8}{3}\times 10^{-14}\right)}$

$N = 106624\times \frac{3}{8}\times 10^{14}$

$N = 39984\times 10^{14}$

$N \approx 3.9984\times 10^{18}$

$N \approx 4.0\times 10^{18}$

6. Match with options

This corresponds to:

Option D: $4.0\times 10^{18}$