Question

An ideal gas undergoes a quasi-static, reversible process in which its molar heat capacity $C$ remains constant. If during this process the relation of pressure $P$ and volume $V$ is given by $P{V^n}\, = {\text{constant}}$, then $n$ is given by (Here ${C_P}$ and ${C_V}$ are molar specific heat at constant pressure and constant volume , respectively)
A. $n = \dfrac{{{C_P}}}{{{C_V}}}$
B. $n = \dfrac{{C - {C_P}}}{{C - {C_V}}}$
C. $n = \dfrac{{{C_P} - C}}{{C - {C_V}}}$
D. $n = \dfrac{{C - {C_V}}}{{C - {C_P}}}$

Answer 1

Any thermodynamics process that obeys the relation $P{V^n}\, = c$ is called a polytropic process.
Here, $V$ is the volume, $P$ is the pressure $c$ is a constant and $n$ is the polytropic index.
In a polytropic process, molar specific heat capacity is given as,
$C = {C_V} + \dfrac{R}{{1 - n}}$
Where $R$ is the universal gas constant.
According to Mayer’s formula, ${C_P} - {C_V} = R$
Where ${C_P}$ is the molar specific heat capacity of an ideal gas at constant pressure, ${C_V}$ is its molar specific heat capacity at constant volume and $R$ is the universal gas constant.

Complete step by step answer:
Any thermodynamics process that obeys the relation $P{V^n}\, = c$ is called a polytropic process.
Here, $V$ is the volume, $P$ is the pressure $c$ is a constant and $n$ is the polytropic index.
This equation can describe multiple expansion and compression.
The name polytropic is given to a general process where
$P{V^n}\, = {\text{constant}}$
Depending upon the value of the polytropic index equation can represent isothermal process, isobaric process, adiabatic process etc.
In a polytropic process, molar specific heat capacity is given as,
$C = {C_V} + \dfrac{R}{{1 - n}} \\\ C - {C_V} = \dfrac{R}{{1 - n}} \\\$
We need to find $n$ . Solving for $n$ we get
$1 - n = \dfrac{R}{{C - {C_V}}}$

$$n = 1 - \dfrac{R}{{C - {C_V}}} \\\ = \dfrac{{C - {C_V} - R}}{{C - {C_V}}} \\\$$

The specific heat capacity of a substance is defined as the heat supplied per unit mass per unit rise in temperature.
According to Mayer’s formula, ${C_P} - {C_V} = R$
Where ${C_P}$ is the molar specific heat capacity of an ideal gas at constant pressure, ${C_V}$ is its molar specific heat capacity at constant volume and $R$ is the universal gas constant.
So, we can substitute $R = {C_P} - {C_V}$
On substituting we get,

$$n = \dfrac{{C - {C_V} - {C_P} + {C_V}}}{{C - {C_V}}} \\\ = \dfrac{{C - {C_P}}}{{C - {C_V}}} \\\$$

So, the correct answer is option B.

Note:
Here the equation given is the polytropic process $P{V^n}\, = {\text{constant}}$. It should not be confused with adiabatic process which looks similar given as $P{V^\gamma }\, = {\text{constant}}$. The term polytropic is used in a general sense and adiabatic process is a polytropic process when polytropic index is $\gamma$ which is given as $\gamma = \dfrac{{{C_P}}}{{{C_V}}}$