Which of the following options are correct regarding the molar heat capacity $(C)$ for the gas whose $P V$ graph is shown below.

C = \infty

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C = 0

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C = R

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C = 2 R

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Solution

The correct option is B $C = 0$
We know, internal energy $(U)$
$U = \frac{n f R T}{2} = \frac{P V f}{2}$ .... $(1)$ $[P V = n R T]$
Also, molar heat capacity $(C)$
$C = \frac{\partial U}{\partial T} = \frac{\partial}{\partial T} (\frac{P V f}{2})$ ... $(2)$ (from $(1)$ )
As, $P V =$ constant [given]
$\Rightarrow \frac{P V f}{2} =$ constant..... $(3)$
[Degree of freedom $(f)$ of a gas is constant]
From $(2)$ and $(3)$ ,
$C = 0$

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P V

For polytropic process $P V^{n} = c o n s t a n t, C_{m}$ (molar heat capacity) of an ideal gas is given by

P V^{1 / 4} = constant

{P V}^{n} =

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p V^{2}

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