Question

Find the molar heat capacity of an ideal gas with adiabatic exponent \['\gamma '\] for the polytropic process \[P{V^n} = {\rm{constant}}\].

Answer 1

Hint: Molar heat capacity of an ideal gas is the amount of heat required to raise the temperature of the gas by \[1{\rm{ K}}\] at constant pressure. We will use the concept of the ideal gas equation and the first law of thermodynamics to deduce the final expression for the molar heat capacity of the given ideal gas.

Complete step by step answer:
Using the concept of the first law of thermodynamics, we can write:
\[{C_P} = \dfrac{{PdV}}{{ndT}} + {C_V}\]……(1)
Here \[{C_P}\] is molar heat capacity, \[{C_V}\] is the specific heat of the ideal gas at constant volume and P is the pressure, T is the temperature, n is the number of moles of the given ideal gas and V is the volume.
We know that for the relationship between temperature and volume in a polytropic process is given as:
\[T{V^{n - 1}} = {\rm{constant}}\]
On differentiating the above equation with respect to T, we get:
\[\dfrac{{dV}}{{dT}} = - \dfrac{V}{{T\left( {n - 1} \right)}}\]
Substitute \[\left[ { - \dfrac{V}{{T\left( {n - 1} \right)}}} \right]\] for \[\dfrac{{dV}}{{dT}}\] in equation (1).
\[\begin{array}{l}
{C_P} = \dfrac{P}{n}\left[ { - \dfrac{V}{{T\left( {n - 1} \right)}}} \right] + {C_V}\\
{C_P} = {C_V} - \dfrac{{PV}}{{nT\left( {n - 1} \right)}}
\end{array}\]……(2)
Let us write the equation of an ideal gas.
\[PV = nRT\]
Here, P is pressure, V is volume, T is temperature and n is the number of moles of the given ideal gas.
On substituting \[nRT\] for \[PV\] in equation (2).
\[\begin{array}{l}
{C_P} = {C_V} - \dfrac{{nRT}}{{nT\left( {n - 1} \right)}}\\= {C_V} - \dfrac{R}{{\left( {n - 1} \right)}}
\end{array}\]……(3)
We know that the value of specific heat at constant volume of an ideal gas is given as below:
\[{C_V} = \dfrac{R}{{\gamma - 1}}\]
Substitute \[\dfrac{R}{{\gamma - 1}}\] for \[{C_V}\] in equation (3).
\[\begin{array}{l}
{C_P} = \dfrac{R}{{\gamma - 1}} - \dfrac{R}{{\left( {n - 1} \right)}}\\
{C_P} = R\left[ {\dfrac{{n - \gamma }}{{\left( {n - 1} \right)\left( {\gamma - 1} \right)}}} \right]
\end{array}\]

Note:At very higher temperature and lower pressure, a gas starts behaving like an ideal gas. Also, if the forces of interaction of molecules are negligible then at very high temperature and lower pressure, we can consider a gas as an ideal gas.