⇒Kinetic Theory of Gases

Correct Option is (B)

According to Charles's Law, the volume of an ideal gas is directly proportional to its absolute temperature when pressure is held constant. i.e., $\mathrm{V}\propto \mathrm{T}$

This is depicted by graph $\mathrm{Q}$.

Correct Option is (A)

Density $\propto \mathrm{P}/\mathrm{T}$

So, $\frac{{\mathrm{d}}_2}{{\mathrm{d}}_1}=\frac{{\mathrm{P}}_2}{{\mathrm{T}}_2}\times \frac{{\mathrm{T}}_1}{{\mathrm{P}}_1}$

$\therefore$ The new density is:

$\begin{array}{rl}& {\mathrm{d}}_2={\rho }_0\times \frac{3{\mathrm{P}}_0}{2{\mathrm{T}}_0}\times \frac{{\mathrm{T}}_0}{{\mathrm{P}}_0}\\ & {\mathrm{d}}_2=\frac{3}{2}{\rho }_0\end{array}$

Correct Option is (A)

The units of PV can be calculated as follows:

The unit of pressure is $\mathrm{Kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$

The unit of volume is ${\mathrm{m}}^3$

$\therefore$ The unit of PV is

Unit $={\mathrm{Kgm}}^{-1}{\mathrm{s}}^{-2}{\mathrm{m}}^3={\mathrm{Kgm}}^2{\mathrm{s}}^{-2}$

This is a unit of energy.

Correct Option is (D)

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

For the first gas, we have,

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

For the second gas, we have,

$\begin{array}{rl}& (2\mathrm{P})\left(\frac{\mathrm{V}}{4}\right)={\mathrm{N}}_2{\mathrm{k}}_{\mathrm{B}}(2\mathrm{T})\\ & \mathrm{PV}=4{\mathrm{N}}_2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\text{.... (ii)}\end{array}$

From equations (i) and (ii)

${\mathrm{N}}_1=4{\mathrm{N}}_2\Rightarrow \frac{{\mathrm{N}}_1}{{\mathrm{N}}_2}=4$

Correct Option is (D)

$\begin{array}{rl}& P=\frac{F}{A}\\ \therefore P& \propto F\text{..... (i)}\end{array}$

From Ideal Gas Equation,

$\mathrm{P}=\frac{\mathrm{nRT}}{\mathrm{V}}$

When V remains constant,

$\mathrm{P}\propto \mathrm{T}$ ..... (ii)

Comparing (i) and (ii)

$\mathrm{F}\propto {\mathrm{T}}^{\mathrm{x}}$

$\therefore \mathrm{x}=1$