So I was going through this answer which states:
In a fluid, the internal energy is the sum of the internal energy (in turn, the sum of the kinetic and potential energy) of each molecule
And I've seen many people state something similar. But this ends up confusing me. For simplicity, let's consider a system: a thermal gas in equilibrium inside a box and use Euler's homogeneous function theorem to get:
$$ U = TS - PV $$
where $U$ is the potential energy, $T$ is the temperature, $S$ is the entropy, $P$ is the pressure and $V$ is the volume. Now, let us focus on the first term of the RHS:
$$ TS = -k_B T \sum_i p_i \ln p_i $$ where $k_B$ is Boltzmann's constant and $p_i$ is the probability. Since $p_i = e^{-E_i/k_B T}$ ($E_i$ is the i'th energy state) then:
$$ TS = \sum_i p_i E_i $$
This is just your average energy of the volume considered. Now, let us think of the the term $PV$. The pressure term only affects your center of mass. The energy associated with it thus also be related to the center of mass. $K_{CM}$ is the translational kinetic energy associated with an ideal gas:
$$ \frac{3}{2} K_{CM} = PV $$
Thus internal energy has excess translational energy excluded from it? I think I must be confusing something. Feel free to point out flaws in my argument.
