Question

Statement 1: Equal volumes of monoatomic and polyatomic gases are adiabatically compressed separately to equal compression ratio $\left(\frac{P_{2}}{P_{1}}\right)$. Then monoatomic gas will have greater final volume.
Statement 2: Among ideal gases, a molecules of a monoatomic gas has the smallest number of degrees of freedom.

• A. Statement-1 is true, Statement-2 is false
• B. Statement-1 is false, Statement-2 is true
• C. Statement-1 is true, Statement-2 is true and Statement-2 is a correct explanation for Statement-1
• D. Statement-1 is true, Statement-2 is true but Statement-2 is not a correct explanation for Statement-1

Answer 1

Answer: (B)

Statement-1 is false, Statement-2 is true.
$P_{1} V_{1}^{\gamma}=P_{2} V_{2}^{\gamma}$
$\Rightarrow \left(\frac{P_{2}}{P_{1}}\right)=\left(\frac{V_{1}}{V_{2}}\right)^{\gamma}$ $\Rightarrow V_{2}=V_{1}\left[\frac{P_{1}}{P_{2}}\right]^{\frac{1}{\gamma}}=V_{1} C^{1 / \gamma}(C > 1)$
$\Rightarrow V_{2}'=V_{1}'\left[\frac{P_{1}}{P_{2}}\right]^{\frac{1}{\gamma^{\prime}}}=V_{1} C^{1 / \gamma'}$
$\because \gamma>\gamma'$
$\because$ [$\gamma \rightarrow$ Monotonic $\gamma' \rightarrow$ Polyatomic]
$\Rightarrow V_{2}'>\,V_{2}$