# Is it possible for two gases to have different internal energy but equal pressure and temperature?

(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!

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 role of the energy of a single molecular has in the case of internal energy and of the pressure. In this very simple case where all the energy is kinetic, the full energy (sum of the translational and rotational contributions) enters in 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.