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} \quad ...(i) $
According to question,
$ PV^{1/2} = $ constant $ (A) $
multiplying both side by $ \sqrt{V} $
or $ PV = A\sqrt{V} \quad ...(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} $