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Question: For polytropic process \(pV^n\)= constant, \({C_m}\) (molar heat capacity) of an ideal gas is given ...

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

Explanation

Solution

We have to know that a polytropic process is a thermodynamic process which follows the relationship: pVn=Cp{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
pVn=Cp{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 Cp{C_p} to the heat capacity at constant volume Cv{C_v} is the ratio of heat capacity given by the term γ\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,
pV(1−γ)K+γ=Cp{V^{\left( {1 - \gamma } \right)K + \gamma }} = C
Here C represents constant
K=δqδwK = \dfrac{{\delta q}}{{\delta w}}
γ=CpCv\gamma = \dfrac{{{C_p}}}{{{C_v}}}
With the coefficient of polytropic n=(1−γ)K+γn = \left( {1 - \gamma } \right)K + \gamma .
We could derive the equation of molar heat capacity for an ideal gas as,
dV=dq+dWdV = dq + dW
nCV,m⋅dT=nCm⋅dT−p⋅dVn{C_{V,m}} \cdot dT = n{C_m} \cdot dT - p \cdot dV
Cm=CV,m+p⋅dVn⋅dT{C_m} = {C_{V,m}} + \dfrac{{p \cdot dV}}{{n \cdot dT}} → (1)
Here,
pVn=kp{V^n} = k and pV=nRTpV = nRT
Therefore, kV1−n=nRTk{V^{1 - n}} = nRT
k(1−n)v−n⋅dV=nRdTk\left( {1 - n} \right){v^{ - n}} \cdot dV = nRdT
dVdT=nRk(1−n)V−n\dfrac{{dV}}{{dT}} = \dfrac{{nR}}{{k\left( {1 - n} \right){V^{ - n}}}} → (2)
From equation (1) and (2) as,
Cm=CV,m+R(1−n){C_m} = {C_{V,m}} + \dfrac{R}{{\left( {1 - n} \right)}}
Cm{C_m} (molar heat capacity) of an ideal gas is given by Cm=CV,m+R(1−n){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=1n = 1 , pV=CpV = C this effect is equivalent to an isothermal process under the consideration of ideal gas law, because then pV=nRTpV = nRT . If the value of n=γn = \gamma the process is equivalent to an adiabatic and reversible where there is no transfer of heat under the consideration of ideal gas law.