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](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Book%3A_Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/07%3A_State_Functions_and_The_First_Law/7.13%3A_Heat_Capacities_for_Gases-_Cv_Cp) 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;


$$ C\_p(T) = C\_V(T) + R$$




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For some real diatomic gases, check the chart at [Wikipedia](https://en.wikipedia.org/wiki/Molar_heat_capacity). 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.


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