# Why does the specific heat of a real gas depends on temperature but not for the ideal gas?

The specific heat of a real gas, unlike an ideal gas, depends on temperature. How can we physically understand this? Thanks.

• It depends on how one defines an ideal gas. We engineers include temperature dependence of specific heat in our definition of ideal gases. Physicists, on the other hand do not. Engineers regard ideal gases as the limiting behavior of real gases at low specific volume. Sep 25, 2017 at 13:21
• A physicist with a good knowledge of thermodynamics should know that the thermodynamic ideal gas definition does not require that the specific heat capacity is constant. Thus engineers and physicists agree if the latter have done their homework. Nov 29, 2018 at 22:15
• Based on the answers so far, there seems to be disagreement about if the question is about non-ideal gases (i.e., those with interactions between particles) or about the "freezing out" of vibrational and rotational degrees of freedom in gases of non-interacting molecules. Nov 30, 2018 at 15:54

## 3 Answers

The heat capacity (specific heat times the mass of the gas) is defined to be how much the internal energy of the gas changes due to changes in temperature, which can be done either at constant pressure $$C_P=\left.\frac{\partial U}{\partial T}\right)_P$$ or at constant volume $$C_V = \left.\frac{\partial U}{\partial T}\right)_V.$$ Notice that both $$C_P$$ and $$C_V$$ will be constant if the internal energy $$U$$ is a linear function of the temperature. This is of course the case for the ideal case, for which $$U_{\mathrm{ideal}} = \frac{3}{2}N k_B T,$$ where $$N$$ is the number of particles (I've assumed monatomic here, but the linear dependence in $$T$$ is true of diatomic and polyatomic ideal gases as well). Microscopically, this form of the internal energy results from the fact that all of an ideal gas's energy is kinetic. Real gases, however, also have internal energy due to the potential energy of the interactions between particles. So, the total internal energy of a real gas is $$U_{\mathrm{real}} = U_{\mathrm{ideal}} + U_{\mathrm{pot}}.$$ In textbooks, the potential part is often called the "excess" internal energy. The way it depends on temperature is different for every gas, since it depends on the details of their interactions. In general, though, it will not depend linearly on $$T$$. Then, the heat capacity (either $$C_V$$ or $$C_P$$) also has two parts: $$C = \frac{\partial U}{\partial T} = \frac{3}{2}Nk_B + \frac{\partial U_{\mathrm{pot}}}{\partial T}$$ for a montaomic gas.

• In some cases $U_{\mathrm{pot}}$ is not a function of T, and C is identical to that for ideal gases. May 8, 2019 at 10:38
• Clausius said that temperature is the result of translational movement of of gas particles. He made this distinction from rotational and vibrational movement which can contribute to energy but not temperature. So even in an an ideal gas, shouldn't the heat capacity equation above also include a term to account for this? [like dU/d(vibration)+ dU/d(rotation) ] May 20, 2021 at 12:44
• @aquagremlin Yes, that's right. In a gas of molecules, one might prefer to think of the rotational and/or vibrational degrees of freedom as part of the "ideal" heat capacity. I don't have to think about molecules very much in my own work, so I am biased toward grouping rotations and vibrations in with the "potential" term. May 20, 2021 at 17:14

Ideal gas in real sense is not a physical reality. We treat it as monatomic gas. In monatomic gases only translational degree of freedom is effective,which is three. At any high temperature rotational and vibrational degree of freedom are not effective. Thus its specific heat is independent of temperature. On the other hand real gases may be monatomic,diatomic or in general polyatomic. In polyatomic gases vibrational degree of freedom becomes effective at higher temperature. For example in case of diatomic gases two vibrational degree of freedom becomes effective at higher temperature and their molar sp heat becomes 7/2 R from 5/2R.

The tempurature dependence of heat capacitance was one of the historical failures of classical physics, which predicts constant heat capacitance (i.e. $$\frac{\partial}{\partial T} \frac{n}{2}k_\text{B}T$$). The temperature dependence of solids and gases was not explained until the advent of quantum mechancis, in which the vibrational states are quantized.

An ideal gas is a classical gas and will have constant heat capacitance.