Question

P-V diagram of a diatomic gas is a straight line passing through the origin. What is the molar heat capacity on the gas in the process?
A. $R$
B. $2R$
C. $3.33\,K$
D. $3R$

Answer 1

Hint: Molar heat capacity is defined as the amount of heat that is required to rise the temperature of one mole of a substance by a one kelvin. It is given in the question that P-V diagram of a diatomic gas is a straight line passing through the origin, we can say that the pressure and the volume are directly proportional to each other. Here, we will use the formula used for calculating the molar heat capacity on the gas which is given below.

Formula used:
The formula used for calculating the molar heat capacity on the gas is given below
$C = \dfrac{R}{{\gamma - 1}} + \dfrac{R}{{1 - x}}$
Here, $C$ is the molar specific heat, $\gamma $ is the adiabatic exponent, $R$ is the universal gas constant and $x$ is the polytropic index.

Complete step by step answer:
As given in the question, P-V diagram of a diatomic gas is a straight line passing through the origin, therefore, we can say that the pressure and the volume are directly proportional to each other and is given below
$P \propto V$
$ \Rightarrow \,P = kV$
Here, $k$ is the constant of proportionality.
$ \Rightarrow \,\dfrac{P}{V} = k$
$ \Rightarrow \,P{V^{ - 1}} = k$
Comparing it with $P{V^x} = k$, we get
$x = - 1$
Now, the formula used for calculating the molar heat capacity on the gas is given below
$$C = \dfrac{R}{{\gamma - 1}} + \dfrac{R}{{1 - x}}$$
Here,$\gamma $ is the adiabatic exponent and its value is equal to $1.4$ for air. Also, the value of $x$ is $ - 1$ in this process. Therefore, the above equation will become
$C = \dfrac{R}{{1.4 - 1}} + \dfrac{R}{{1 - \left( { - 1} \right)}}$
$ \Rightarrow \,C = \dfrac{R}{{0.4}} + \dfrac{R}{{1 + 1}}$
$ \Rightarrow \,C = \dfrac{R}{{0.4}} + \dfrac{R}{2}$
$ \Rightarrow \,C = \dfrac{{10R}}{4} + \dfrac{R}{2}$
$ \Rightarrow \,C = \dfrac{{5R}}{2} + \dfrac{R}{2}$
$ \Rightarrow \,C = \dfrac{{6R}}{2}$
$ \therefore \,C = 3R$
Therefore, the molar heat capacity on the gas in the process is $3R$.

Hence, option D is the correct option.

Note:The above process is the polytropic process that is defined as a thermodynamic process that obeys the relation given below
$P{V^n} = C$
Here, $P$ is the pressure, $V$ is the volume, $n$ is the polytropic index and $C$ is the constant.Here, in the above solution, we have used $n$ instead of $x$ and $k$ instead of $C$.