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At high temperatures, the specific heat at constant volume $\text{C}_{v}$ has three degrees of freedom from rotation, two from translation, and two from vibration.

That means $\text{C}_{v}=\frac{7}{2}\text{R}$ by the Equipartition Theorem.

However, I recall the Mayer formula, which states $\text{C}_{p}=\text{C}_{v}+\text{R}$.

The ratio of specific heats for a diatomic molecule is usually $\gamma=\text{C}_{p}/\text{C}_{v}=7/5$.

What is then the specific heat at constant pressure? Normally this value is $7/5$ for diatomic molecules?

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"At high temperatures, the specific heat at constant volume $C_v$ has three degrees of freedom from rotation, two from translation, and two from vibration." I can't understand this line. $C_v$ is a physical quantity not a dynamical system. So how can it have a degrees of freedom?? You can say the degrees of freedom of an atom or molecule is something but it is wrong if you say the degrees of freedom of some physical quantity(like temperature, specific heat etc.) is something. Degrees of freedom is the number of independent coordinates necessary for specifying the position and configuration in space of a dynamical system.

Now to answer your question, we know that the energy per mole of the system is $\frac{1}{2} fRT$. where $f$= degrees of freedom the gas.
$\therefore$ molar heat capacity, $C_v=(\frac{dE}{dT})_v=\frac{d}{dT}(\frac{1}{2}fRT)_v=\frac{1}{2}fR$

Now, $C_p=C_v+R=\frac{1}{2}fR+R=R(1+ \frac{f}{2})$

$\therefore$ $\gamma=1+ \frac{2}{f}$

Now for a diaatomic gas:- enter image description here

A diaatomic gas has three translation(along x,y,z asis) and two rotational(about y and z axis) degrees of freedom. i.e. total degrees of freedom is $5$.

Hence $C_v=\frac{1}{2}fR=\frac{5}{2}R$ and $C_p=R(1+ \frac{f}{2})=R(1+ \frac{5}{2})=\frac{7}{2}R$

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A diatomic molecule will have 7 degrees of freedom at high temperatures. However, the ratio of specific heats that you cited is for diatomic molecules around room temperatures, which have 5 degrees of freedom.

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In particular, when the thermal energy $k_B T$ is smaller than the spacing between the quantum energy levels, the contribution of the vibrational and rotational degrees of freedom will fall. At room temperature, the contribution of the vibrational degree of freedom of a diatomic molecule is often 0, and so $C_v$ will be $R/2$ lower than expected.

Furthermore, since a rotation about the bond between the two atoms in a diatomic molecule is not really a rotation, there are actually only 6 degrees of freedom for a diatomic molecule at high temperatures: 3 translational, 2 rotational, and 1 vibrational. When you take away the vibrational degree of freedom at lower temperatures, only 5 remain, and you get $C_v = 5/2R$ and $C_p = C_v + R = 7/2R$.

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Molecules are quite different from the monatomic gases like helium and argon. With monatomic gases, thermal energy comprises only translational motions. Translational motions are ordinary, whole-body movements in 3D space whereby particles move about and exchange energy in collisions—like rubber balls in a vigorously shaken container (see animation here [19]). These simple movements in the three dimensions of space mean individual atoms have three translational degrees of freedom. A degree of freedom is any form of energy in which heat transferred into an object can be stored. This can be in translational kinetic energy, rotational kinetic energy, or other forms such as potential energy in vibrational modes. Only three translational degrees of freedom (corresponding to the three independent directions in space) are available for any individual atom, whether it is free, as a monatomic molecule, or bound into a polyatomic molecule.

As to rotation about an atom's axis (again, whether the atom is bound or free), its energy of rotation is proportional to the moment of inertia for the atom, which is extremely small compared to moments of inertia of collections of atoms. This is because almost all of the mass of a single atom is concentrated in its nucleus, which has a radius too small to give a significant moment of inertia. In contrast, the spacing of quantum energy levels for a rotating object is inversely proportional to its moment of inertia, and so this spacing becomes very large for objects with very small moments of inertia. For these reasons, the contribution from rotation of atoms on their axes is essentially zero in monatomic gases, because the energy spacing of the associated quantum levels is too large for significant thermal energy to be stored in rotation of systems with such small moments of inertia. For similar reasons, axial rotation around bonds joining atoms in diatomic gases (or along the linear axis in a linear molecule of any length) can also be neglected as a possible "degree of freedom" as well, since such rotation is similar to rotation of monatomic atoms, and so occurs about an axis with a moment of inertia too small to be able to store significant heat energy

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