# 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.${C_{p,m}} + \dfrac{R}{{\left( {n - 1} \right)}}$

Answer

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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.

**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

$K = \dfrac{{\delta q}}{{\delta w}}$

$\gamma = \dfrac{{{C_p}}}{{{C_v}}}$

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.

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