Question

During an experiment, an ideal gas is found by obey a condition $\frac{P^2}{\rho }=\text{constant}\left[\rho \text{is density of the gas}\right]$. The gas is initially at temperature $T$, pressure $P$ and density $\rho$. If the gas expands such that density changes to $\frac{\rho }{2}$, choose the correct statements.

• A. The pressure of the gas changes to $\sqrt{2}P$
• B. The temperature of the gas changes to $\sqrt{2}T$
• C. The graph of the above process on the $P-T$ diagram is parabola.
• D. The graph of the above process on the $P-T$ diagram is hyperbola.

Answer 1

Answer: (A), (D)

The correct options are
A The pressure of the gas changes to $\sqrt{2}P$
D The graph of the above process on the $P-T$ diagram is hyperbola.

Given $\frac{P^2}{\rho }=$ Constant

As density changes from $\rho \to \frac{\rho }{2}$, let pressure change from $P$ to $P^{\prime }$. So,
$\frac{P^2}{\rho }=\frac{P^{\prime 2}}{\rho /2}$
$\Rightarrow P^{\prime }=\sqrt{2}P$

We know that $\rho \propto \frac{1}{V}$ and $V=\frac{nRT}{P}$,
$\Rightarrow \frac{P^2}{\rho }\propto P^{2}V\propto PT$=Constant
$\therefore PT=\text{constant}........\left(1\right)$
Since $P\propto \frac{1}{T}$, $PT$ - diagram will be a rectangular hyperbola.

Let Temperature be $T^{\prime }$ when pressure is $P^{\prime }$,
From $\left(1\right)$
$\Rightarrow PT=P^{{}^{\prime }}{T}^{{}^{\prime }}\Rightarrow T^{{}^{\prime }}=\frac{PT}{\sqrt{2}P}$
$\Rightarrow T^{{}^{\prime }}=\frac{T}{\sqrt{2}}$