# For polytropic process $pV^n$= constant, ${C_m}$ (molar heat capacity) of an ideal gas is given by:A.${C_{v,m}} + \dfrac{R}{{\left( {n - 1} \right)}}$B.${C_{v,m}} + \dfrac{R}{{\left( {1 - n} \right)}}$C.${C_{v,m}} + R$D.${C_{p,m}} + \dfrac{R}{{\left( {n - 1} \right)}}$

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Hint:We have to know that a polytropic process is a thermodynamic process which follows the relationship: $p{V^n} = C$ . Here p represents pressure, V represents volume and n is the polytropic index, and C represents constant. The equation of the polyprotic process could explain multiple expansion and processes of compression that includes transfer of heat.

We could say that the polytropic process where the relation of pressure-volume is written as
$p{V^n} = C$
The exponent n that contain any value which ranges from minus infinity to plus infinity based on the process.
At constant pressure ${C_p}$ to the heat capacity at constant volume ${C_v}$ is the ratio of heat capacity given by the term $\gamma$ .
For an ideal gas in a closed system going through a slow process with minute kinetic energy changes and potential energy the process is known as polytropic in such way that,
$p{V^{\left( {1 - \gamma } \right)K + \gamma }} = C$
Here C represents constant
$K = \dfrac{{\delta q}}{{\delta w}}$
$\gamma = \dfrac{{{C_p}}}{{{C_v}}}$
With the coefficient of polytropic $n = \left( {1 - \gamma } \right)K + \gamma$ .
We could derive the equation of molar heat capacity for an ideal gas as,
$dV = dq + dW$
$n{C_{V,m}} \cdot dT = n{C_m} \cdot dT - p \cdot dV$
${C_m} = {C_{V,m}} + \dfrac{{p \cdot dV}}{{n \cdot dT}}$ → (1)
Here,
$p{V^n} = k$ and $pV = nRT$
Therefore, $k{V^{1 - n}} = nRT$
$k\left( {1 - n} \right){v^{ - n}} \cdot dV = nRdT$
$\dfrac{{dV}}{{dT}} = \dfrac{{nR}}{{k\left( {1 - n} \right){V^{ - n}}}}$ → (2)
From equation (1) and (2) as,
${C_m} = {C_{V,m}} + \dfrac{R}{{\left( {1 - n} \right)}}$
${C_m}$ (molar heat capacity) of an ideal gas is given by ${C_m} = {C_{V,m}} + \dfrac{R}{{\left( {1 - n} \right)}}$.
Therefore, the option (B) is correct.

Note:
We have to know that for particular values of the polytropic index, the process would be synonymous with other some of the common processes. If the value of $n = 1$ , $pV = C$ this effect is equivalent to an isothermal process under the consideration of ideal gas law, because then $pV = nRT$ . If the value of $n = \gamma$ the process is equivalent to an adiabatic and reversible where there is no transfer of heat under the consideration of ideal gas law.