Question

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

Answer 1

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.

Complete step by step answer:
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
$K={\displaystyle \frac{\delta q}{\delta w}}$
$\gamma ={\displaystyle \frac{C_p}{C_v}}$
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,
$dV=dq+dW$
$n{C}_{V,m}\cdot dT=n{C}_m\cdot dT-p\cdot dV$
$C_m=C_{V,m}+{\displaystyle \frac{p\cdot dV}{n\cdot dT}}$ → (1)
Here,
$p{V}^n=k$ and $pV=nRT$
Therefore, $k{V}^{1-n}=nRT$
$k\left(1-n\right)v^{-n}\cdot dV=nRdT$
${\displaystyle \frac{dV}{dT}}={\displaystyle \frac{nR}{k\left(1-n\right)V^{-n}}}$ → (2)
From equation (1) and (2) as,
$C_m=C_{V,m}+{\displaystyle \frac{R}{\left(1-n\right)}}$
$C_m$ (molar heat capacity) of an ideal gas is given by $C_m=C_{V,m}+{\displaystyle \frac{R}{\left(1-n\right)}}$.
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.