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.
 A: 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.
A: 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. 
A: 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.
