One mole of an ideal gas is taken along the process in which $P V^{x} =$ constant. The graph shown represents the variation of molar heat capacity of such a gas with respect to ' $x$ ' . The values of $c^{'}, x^{'}$ and degree of freedom, respectively, are given by (dotted line in the diagram is asymptote to the graph on both sides. The horizontal solid line is not part of the graph and is an asymptote to the graph) R is the universal gas constant.

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$\frac{7}{2} R, \frac{5}{2}, 5$

$\frac{5}{2} R, \frac{5}{3}, 3$

$\frac{7}{2} R, \frac{7}{5}, 5$

$\frac{5}{2} R, \frac{7}{5}, 3$

answer is B.

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Detailed Solution

At $X = \infty, C = \frac{3}{2} R$
From $P V^{x} =$ constant
$\Rightarrow P^{1 / x} V =$ another constant
So at $X = \infty, V =$ constant
Hence $C = C_{v} = \frac{5}{2} R$
And then $C_{p} = C_{v} + R = \frac{7}{2} R$
At $X = 0, P =$ constant and $C = C'$
At $X = X', C = 0,$ so the process is adiabatic , hence $X' = \frac{C_{p}}{C_{v}} = \frac{7}{5}$
Degree of freedom, f : $C_{v} = \frac{F R}{2} = 3 R$