Given, density of gas at point $A, {\rho }_A={\rho }_0$

General equation for an ideal gas

$pV=nRT$

$\Rightarrow pV=\frac{m}{M}RT$  [$m\to$mass, $M\to$ molecular mass]

$\Rightarrow p=\frac{m}{V}·\frac{R}{M}·T$

$\Rightarrow p=\rho \left(\frac{R}{M}\right)T ...\left(\mathrm{i}\right)$    [where density $\rho =\frac{m}{V}$]

Applying Equation. (i) at point $A$ and $B$, respectively.

$p={\rho }_A\left(\frac{R}{M}\right)T_0 ...\left(\mathrm{ii}\right)$

$2p={\rho }_B\left(\frac{R}{M}\right)3T_0 ...\left(\mathrm{iii}\right)$

From Equations. (ii) and (iii), we get

$\frac{1}{2}=\frac{{\rho }_A}{{\rho }_B}·\frac{1}{3}$

$\Rightarrow {\rho }_B=\frac{2}{3}{\rho }_A=\frac{2}{3}{\rho }_0$