⇒Kinetic Theory of Gases

Correct Option is (D)

According to the ideal gas equation $\mathrm{PV}={\mathrm{Nk}}_{\mathrm{B}}\mathrm{T}$

For jar $\mathrm{P}$, we have

$\mathrm{PV}={\mathrm{N}}_1{\mathrm{k}}_{\mathrm{B}}\mathrm{T}$ ..... (i)

For jar Q, we have,

(2P) $\left(\frac{V}{4}\right)={\mathrm{N}}_2{\mathrm{k}}_{\mathrm{B}}(2\mathrm{T})$

$\Rightarrow \mathrm{PV}=4{\mathrm{N}}_2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}$ ..... (ii)

From equations (i) and (ii)

$\begin{array}{rl}& N_1=4N_2\Rightarrow \frac{N_1}{N_2}=4\\ \therefore & N_1:N_2=4:1\end{array}$

Correct Option is (A)

For isothermal process: ${\mathrm{P}}_1{\mathrm{V}}_1={\mathrm{P}}_2{\mathrm{V}}_2$

$\therefore {\mathrm{P}}_2=\frac{{\mathrm{P}}_1{\mathrm{V}}_1}{{\mathrm{V}}_2}=\frac{{\mathrm{P}}_1}{2}(\because \frac{{\mathrm{v}}_1}{{\mathrm{v}}_2}=\frac{1}{2})$

For adiabatic process : ${\mathrm{P}}_2{\mathrm{V}}_2^{\gamma }={\mathrm{P}}_3{\mathrm{V}}_3^{\gamma }$

$\begin{array}{rl}\therefore P_3& =P_2{\left(\frac{V_2}{V_3}\right)}^{\gamma }=P_2{\left(\frac{2V_1}{V_3}\right)}^{\gamma }=\frac{P_1}{2}{\left(\frac{1}{8}\right)}^{5/3}[\because \frac{V_1}{V_3}=\frac{1}{16}]\\ & =\frac{P_1}{2}\cdot \frac{1}{32}=\frac{P_1}{64}\end{array}$