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.
$P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$
$\Rightarrow \left(\frac{P_2}{P_1}\right)={\left(\frac{V_1}{V_2}\right)}^{\gamma }$ $\Rightarrow V_2=V_1{\left[\frac{P_1}{P_2}\right]}^{\frac{1}{\gamma }}=V_1{C}^{1/\gamma }(C>1)$
$\Rightarrow V_2^{\prime }=V_1^{\prime }{\left[\frac{P_1}{P_2}\right]}^{\frac{1}{{\gamma }^{\prime }}}=V_1{C}^{1/{\gamma }^{\prime }}$
$\because \gamma >{\gamma }^{\prime }$
$\because$ [$\gamma \to$ Monotonic ${\gamma }^{\prime }\to$ Polyatomic]
$\Rightarrow V_2^{\prime }>V_2$