Question

$1$ mole of an ideal gas in a cylindrical container have the $P-V$ diagram as shown in figure. If $V_2=4V_1$ , then the ratio of temperatures $\frac{T_1}{T_2}$ will be

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{3}{2}$
• D. $\frac{3}{4}$

Answer 1

Answer: (A)

Ideal gas equation, $PV=nRT$
or $T=\frac{PV}{nR}...(i)$
According to question,
$P{V}^{1/2}=$ constant $(A)$
multiplying both side by $\sqrt{V}$
or $PV=A\sqrt{V}...(ii)$
From eqns. $(i)$ and $(ii)$
$\therefore T=\frac{A\sqrt{V}}{nR}$
$\Rightarrow T\propto \sqrt{V}$
Now $\frac{T_1}{T_2}=\sqrt{\frac{V_1}{V_2}}$
$=\sqrt{\frac{V_1}{4V_1}}=\frac{1}{2}\left[\because V_2=4V_1\right]$
$\therefore \frac{T_1}{T_2}=\frac{1}{2}$