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Apparent contradiction between specific heat ( i.e internal energy, in this case) and average kinetic energy (i.e temp). For ex- if i take two gases [He] and [Xe] then xenon have more specific heat capacity than helium, So in other words, at equal volume, pressure and no.of moles, [Xe] has more "internal energy" than [He], and acc.to Avogadro's law or PV=nRT they should have same temperature too ! But how can both have same temperature, pressure and volume when [Xe] clearly has more internal energy ?

Or is it possible for two gases to have different internal energy but same pressure and temperature, if no.of moles/molecules and Volume is equal ? If so then why ? I can't seem to find any reason or intuition for this !

Any help is appreciated !

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The equation of state does not tell everything about a thermodynamic system. Moreover, the specific heat is not related to the value of the internal energy but to the variation of internal energy when the temperature changes.

A very simple example (even simpler than the case of interacting gases like Xenon and Helium) may help to understand the previous points.

Let's consider two equal volumes containing the same number of moles of two perfect gases at the same temperature. Gas A is made by monoatomic molecules, while gas B is made by di-atomic molecules. The equation of state is the same for both $PV=nRT$, therefore the pressure is the same. However specific heat and internal energy are not the same, being $U_A=\frac32 nRT$ and $U_B=\frac52 nRT$.

The reason for that has to do with the different roles of the energy of a single molecule in the internal energy and pressure. In this very simple case where all the energy is kinetic, the full energy (sum of the translational and rotational contributions) enters the thermodynamic internal energy. Only the translational degrees of freedom enter in the case of pressure, and this explains the equality of pressure.

In the case of interacting systems, the situation is even more complex but the main idea remains the same: the knowledge of the equation of state alone is not sufficient to reconstruct internal energy and specific heat. This is a basic fact of the description of any thermodynamic system.

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  • $\begingroup$ Only the translational degrees of freedom enter in the case of pressure, and this explains the equality of pressure. 🤯🤯 This is what i was missing ! dots are connected now. Thanks for your time ! $\endgroup$
    – Abhay
    Dec 20 '20 at 10:12
  • $\begingroup$ And also i forgot that xe is a monoatomic gas too while asking, but you still got it correct ! $\endgroup$
    – Abhay
    Dec 20 '20 at 10:16
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(a) There's something odd about your data. The website I consulted gave $c_v$, the molar heat capacity at constant volume as 12.4717 $\text J\ \text{mol}^{-1}\ \text {K}^{-1}$ for both helium and neon (I couldn't find Xenon). Both of these were measured at 25 °C, though for inert gases there is very little change of $c_v$ with temperature.

This value of $c_v$ agrees to five sig figs with the theoretical value of $\frac32 R$.

So at equal temperatures the same amount (number of moles) of the gases have the same internal energy, given by $$U=\tfrac32 nRT.$$ This works for all monatomic gases at lowish densities (so they behave as ideal gases).

(b) However for diatomic gases, such as oxygen, nitrogen, hydrogen, a good approximation for the molar heat capacity at constant volume is $c_v = \frac52 RT$, so $$U=\tfrac52 nRT.$$

The reason is that the molecules of these gases have kinetic energy of rotation as well as translation (moving about!). We say that diatomic molecules at ordinary temperatures have 5 "degrees of freedom", 3 translational and 2 rotational. Kelvin temperature is proportional to the average kinetic energy per degree of freedom, so a mole of diatomic molecules has 5/3 times the internal energy of a mole of monatomic molecules at the same temperature!

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A very simplified explanation of temperature is that in thermal equilibrium, energy is distributed among excitation modes in a statistically even manner (that is, random fluctuations will result in particular modes transiently having more energy, but on average the distribution is even). So substances with more excitation modes will have more internal energy for the same temperature. Monatomic molecules have just translational excitation (that is, kinetic energy from the molecule as a whole moving). Diatomic molecules also have vibrational energy within the molecule.

There are also excitation modes of electron orbitals, but those aren't a significant factor at room temperature.

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