# Are Cp and Cv functions of temperature?

I have read that Cp and Cv are independent of temperature for the case of a perfect gas whereas for an ideal gas it will vary with temperature. The variation in Cp and Cv with temperature is attributed to the activation of the additional DOFs at high temperature. Although over the normal range of temperature it can be considered almost independent of temperature.

My question is

1. Do Cp and Cv vary at the same rate with respect to temperature at high temperatures (I have read somewhere that the variation of Cp and Cv are different but it was long back ago and I can't find the reference).
2. If Cp and Cv vary with temperature at different rates, then the Meyer's relation will be applicable only to certain limited cases like at low temperatures for ideal gas.

Update :

1. For monoatomic gases there are no rotational DOFs. I don't know about vibrational modes. So I think the Meyer's relation will hold true at all temperatures for monoatomic gas.

2. Link regarding the convergence of Cp and Cv value at absolute zero temperature.

If we take for 1 mol of an ideal gas the definition of enthalpy:

$$H = U + pV$$

then differentiation according to T leads to:

$$\frac{\mathrm{d}H}{\mathrm{d}T} = \frac{\mathrm{d}U}{\mathrm{d}T} + \frac{\mathrm{d}(pV)}{\mathrm{d}T}$$

As $$\frac{\mathrm{d}H}{\mathrm{d}T} = C_p$$, $$\frac{\mathrm{d}U}{\mathrm{d}T} = C_V$$, $$pV=RT$$,

it directly leads for ideal gases to;

$$C_p(T) = C_V(T) + R$$

For some real diatomic gases, check the chart at Wikipedia. There is easily seen that:

• Rotational modes for light molecules like $$\ce{H2}$$ are at room temperature still discriminated by quantization.
• Heavy molecules of halogens have their vibrations already active, with various discrimination degree.
• I still couldn't get my head around the answer to first question. In the case of ideal gas does Cp and Cv vary at different rate with respect to temperature? I have edited the question with a new link to a page. It says that the Cp and Cv converge to 0 at absolute zero temperature. So clearly when Cp and Cv have different value at same temperature and if they change at same rate with respect to temperature ,then there is no way they could converge at 0 at absolute zero temperature (assuming extrapolation was done linear). Commented Sep 17, 2022 at 9:58
• Yes, heat capacities do converge to 0 for T->0, but not for ideal gas, that is always gaseous. Remember that ideal gas model fails miserably to approximate gas behaviours near 0 K. It is unusable extrapolation. Commented Sep 17, 2022 at 10:03
• What gas behavior near zero K? An ideal gas is simply a mathematical figment not a gas. Heat capacity is a discontinuous function at absolute zero IF it is actually zero that implies if 0 K is ever reached it will be permanent, No Exit! The behavior near absolute zero could be related to possible quantization of molecular motion. Commented Sep 18, 2022 at 3:09
• @jimchmst No reality models of science are reality. We speak here in context of ideal gas model of real gas behaviour, not about real gases. The heat capacity of real solid matter near T=0 K is proportional to T^3, as there is no translation nor rotation, and vibration modes of solid lattice are heavily quantized. Commented Sep 18, 2022 at 7:36