Molar specific heat is the amount of heat required to raise the temperature of one mole of any material by \(1\text{K}\) (or \(1{}^ \circ {\rm{C}}\)). If the molar specific heat is measured at constant volume, it is called molar specific heat at constant volume denoted by \(C_\rm{v}\). And if it is measured at constant pressure, it is called molar specific heat at constant pressure denoted by \(C_\rm{p}\). Here we discuss the relationship between molar specific heat at constant volume and molar specific heat at constant pressure. First let's talk about the specific heat at constant volume for an ideal gas along with the principle of equipartition of energy.
What is the relationship between specific heat at constant volume and equipartition principle?
The process at constant volume is isovolumetric process, and we are talking about the specific heat of an ideal gas at constant volume. There are no interactions between molecules in an ideal gas and all energy the molecules possess is the total kinetic energy of the molecules. The kinetic theory of an ideal gas shows that the kinetic energy depends only on the absolute temperature and there are no other forms of energy in an ideal gas other than the kinetic energy, so the total kinetic energy is the internal energy of the system of an ideal gas. Therefore, the internal energy of a system of an ideal gas depends only on the temperature. The concentration or the number of moles of an ideal gas does not change, so we always keep that constant here.
If a small quantity of heat \(dq\) is added to a system of an ideal gas at constant volume, the added heat increases the internal energy of the system. And since there are no other forms energy other than kinetic energy, the added heat is equal to the change in total kinetic energy of all molecules. You know the added heat is \(dq = n{C_\text{v}}dT\) and from the kinetic theory of an ideal gas, the change in total kinetic energy is \(dK = \frac{3}{2}nRdT\). Note that we didn't write the subscript \(t\) to represent translational kinetic energy here but the kinetic energy is always translational in ideal gas. Now,
\[\begin{align*} dq &= dK\\ {\rm{or,}}\quad n{C_{\rm{v}}}dT &= \frac{3}{2}nRdT\\ {\rm{or,}}\quad {C_{\rm{v}}} &= \frac{3}{2}R \tag{1} \label{1} \end{align*}\]
The above result for the value of molar specific heat at constant volume is found to agree with the experimental results for monoatomic gases only not for diatomic or polyatomic gases. That's because the molecules of monoatomic gases are more like point particles and nearly behave like an ideal gas but that's not true for diatomic and polyatomic gases.
For mono-atomic gases a molecule (molecule of single atom) has three components of velocity; it means it can move randomly in any direction. But for diatomic gases a molecule can also rotate about two axes perpendicular to each other and each perpendicular to its own axis joining the two atoms. The Figure 1 shows the possible axes of rotation for a diatomic gas molecule. The moment of inertia about the axis joining the centres of the atoms is very small and the kinetic energy associated with this rotation is negligible. Therefore, we can neglect the rotation of diatomic molecule about its own axis joining the centres of two atoms. In other words there is no energy distribution to this motion. The number of velocity components either translational or angular needed to describe the motion of a molecule is called the number of degrees of freedom. In other words the number of energy distributions for every possible motion of a molecule is called the number of degrees of freedom. And therefore a diatomic molecule has five total possible motions and five degrees of freedom. The greater amount of specific heat for diatomic or polyatomic molecules is due to the greater number of energy distributions per molecule or greater number of degrees of freedom.
The principle of equipartition of energy states that there is an average energy equal to \(\frac{1}{2}kT\) per degree of freedom of a molecule in a system in thermodynamic equilibrium. It means one degree of freedom has the average energy of \(\frac{1}{2}kT\) and therefore a molecule with three degrees of freedom has the energy of \(\frac{3}{2}kT\) and so on. Note that \(k\) is the Boltzmann constant. Thus, the average kinetic energy associated with a diatomic molecule is \(\frac{5}{2}kT\). If there are \(N\) number of molecules in the system, total kinetic energy of all molecules is \(\frac{5}{2}NkT=\frac{5}{2}nN_AkT\) where \(N_A\) is the Avogradro's number and \(n\) is the number of moles. Note that \(R = N_Ak\) and so the total kinetic energy of the system is
\(\frac{5}{2}nRT\). And if you do the calculation to find the molar specific heat at constant volume as above for a system of diatomic gas with change in kinetic energy \(dK = \frac{5}{2}nRdT\) due to the added heat \(dq = nC_\text{V}dT\), you'll find:
\[{C_{\rm{V}}} = \frac{5}{2}R \tag{2} \label{2}\]
You might think that the added heat is used to increase only the kinetic energy of the system of diatomic gas; you considered diatomic gas as an ideal gas without considering potential energy of interaction between molecules. The complication can be neglected if you think the diatomic gas molecule as point particle but still has two atoms and shows rotational motion. This approaches the ideal behaviour of diatomic gas but not completely ideal as the rotation of the molecule is not possible in ideal nature. Still the above result in Eq. \eqref{2} is worth mentioning which agrees approximately with the experimental results of molar specific heat at constant volume of a system of diatomic gas.
What is the relationship between molar specific heat at constant volume(\(C_\text{V}\)) and molar specific heat at constant pressure(\(C_\rm{p}\)) of a system of ideal gas?
Both \(C_\rm{V}\) and \(C_\rm{p}\) are molar specific heats but one specific heat is measured at constant volume and the other is measured at constant pressure. The thermodynamic process carried out at constant volume is isovolumetric process and that carried out at constant pressure is isobaric process. The first law of thermodynamics tells us that when a particular quantity of heat is added to a system, no work is done by the system if the volume of the system remains constant. A small quantity of heat \(dQ\) is added to a system at constant volume and according to the first law of thermodynamics the added heat is completely used to increase the internal energy of the system of ideal gas and
\[dQ = dU = n{C_{\rm{V}}}dT \tag{3} \label{3}\]
where \(dU\) is the increase in internal energy. In the process carried out at constant pressure or isobaric process the system does work and the first law of thermodynamics tells us that some amount of added heat is used to increase the internal energy and the remaining amount is used in doing work by the system, so, the system undergoes expansion keeping the pressure constant. The heat \(dQ\) added in isobaric process for the change in temperature \(dT\) is \(dQ = nC_\text{p}dT\), and therefore,
\[\begin{align*} dQ &= dU + dW\\ {\rm{or,}}\quad dQ &= dU + pdV \\ {\rm{or,}}\quad n{C_{\rm{p}}}dT &= dU + pdV \tag{4} \label{4} \end{align*}\]
The important thing for an ideal gas is that the internal energy of a system of an ideal gas depends only on the temperature of the system. And therefore for the same temperature change \(dT\) for both isovolumetric and isobaric processes, the change in internal energy is the same. So, we can replace \(dU\) in Eq. \eqref{4} by \(n{C_\rm{V}}dT\) from Eq. \eqref{3}. Also from the equation of ideal gas we have \(pdV = nRdT\). Thus we can rewrite Eq. \eqref{4} and,
\[\begin{align*} {C_{\rm{p}}}dT &= n{C_{\rm{V}}}dT + nRdT\\ {C_{\rm{p}}} &= {C_{\rm{V}}} + R \tag{5} \label{5}. \end{align*}\]
The above equation Eq. \eqref{5} shows that the molar specific heat at constant pressure is greater than the molar specific heat at constant volume. In short \(C_\rm{p}\) is greater than \(C_\rm{V}\) as it should be. You can understand why \(C_\rm{p}\) is greater than \(C_\rm{V}\) as some quantity of heat is used to do work in isobaric process (constant pressure process). Note that heat and temperature are entirely different things.
Another useful quantity is the ratio of molar specific heat at constant pressure to the molar specific heat at constant volume denoted by \(\gamma\), that is, \(\gamma = {C_\rm{p}}/{C_\rm{V}}\). For monoatomic ideal gas, \({C_\rm{V}} = 3R/2\) and using Eq. \eqref{5} we can find
\[\gamma = \frac{{{C_{\rm{p}}}}}{{{C_{\rm{V}}}}} = \frac{{{C_{\rm{V}}} + R}}{{{C_{\rm{V}}}}} = \frac{{5R}}{{3R}} = 1.67\]
In the same way you can also find the value of \(\gamma\) for diatomic gases using \({C_\rm{V}} = 5R/2\) and you'll obtain the result to be \(\gamma = 1.40\). The values of \(\gamma\) obtained for monoatomic or diatomic or polyatomic gases are approximately in agreement with the experimental values.
