Specific heat of diatomic gases and equipartion energy

[Dario SLC]

My doubt it is simply if have other reason to don't use this principle for the specific heat of diatomic gases.

Homework Equations

$$U=NkT=nRT$$
$$u_n=\frac{U}{n}=RT\text{ molar energy}$$
$$u_N=\frac{U}{N}=kT\text{ average energy}$$
$$Z=\sum{e^{-\omega_i/kT}}\text{ with $\omega_i$ particular energy}$$ partition function

The Attempt at a Solution

I believe (better said, in the Sears book, Thermodynamics) that the equipartition energy is not valid because, for the energies relative to the vibrational motion and rotational motion, it has quantized.

If was valid the principle of equipartition energy, this say:
$$U_t=\frac3{2}kT$$
when $U_t$ is relative at the translation motion, and the factor three is for the degree of freedom, ie:
$$\frac{1}{2}kT+\frac{1}{2}kT+\frac{1}{2}kT$$ for each degree of freedom.

In the vibrational motion, like in the solid if we are model like oscillator and it have the quadratic coordinate for the velocity (kinetic energy) and quadratic coordinate for the position (potential energy), only (since classical point of view), therefore the energy of vibration is $\frac{1}{2}kT+\frac{1}{2}kT=kT$, but it is not the real when $T\Longrightarrow0$ (then solved using quantized energy).

Here my question, ONLY because the energy of vibration is quantized, the principle of partition energy I can't use it?

(Similar to rotational motion)

Thanks!

 

[ehild]

My doubt it is simply if have other reason to don't use this principle for the specific heat of diatomic gases.

Homework Equations

$$U=NkT=nRT$$
$$u_n=\frac{U}{n}=RT\text{ molar energy}$$
$$u_N=\frac{U}{N}=kT\text{ average energy}$$
$$Z=\sum{e^{-\omega_i/kT}}\text{ with $\omega_i$ particular energy}$$ partition function

The Attempt at a Solution

I believe (better said, in the Sears book, Thermodynamics) that the equipartition energy is not valid because, for the energies relative to the vibrational motion and rotational motion, it has quantized.

If was valid the principle of equipartition energy, this say:
$$U_t=\frac3{2}kT$$
when $U_t$ is relative at the translation motion, and the factor three is for the degree of freedom, ie:
$$\frac{1}{2}kT+\frac{1}{2}kT+\frac{1}{2}kT$$ for each degree of freedom.

In the vibrational motion, like in the solid if we are model like oscillator and it have the quadratic coordinate for the velocity (kinetic energy) and quadratic coordinate for the position (potential energy), only (since classical point of view), therefore the energy of vibration is $\frac{1}{2}kT+\frac{1}{2}kT=kT$, but it is not the real when $T\Longrightarrow0$ (then solved using quantized energy).

Here my question, ONLY because the energy of vibration is quantized, the principle of partition energy I can't use it?

(Similar to rotational motion)

Thanks!

Both the rotational and vibrational energy of molecules are quantized. The Equipartion Principle can only be used at high enough temperatures when kT>>ΔE, the energy difference between the subsequent energy levels.
Usually, the rotational energy levels are much closer than the vibrational levels, so as the rotational levels are excited at near-room temperatures, but most of the molecule stay at their ground level of vibration. You can count with the rotational degrees of freedom, two for a two-atomic molecule, but the vibrational degrees of freedom are frozen, and come in at elevated temperatures.
If you have a system with close vibrational levels, (liquid water, for example) you should count with the vibrational degrees of freedom. Water molecules are connected with various weak hydrogen bonds, performing low-energy vibrational modes. That makes the water have high specific heat at room temperature.