In thermodynamics the total energy of a system consists of kinetic energy of motion of the system as a whole, potential energy of the system as a whole due to external force fields, and energy contained within the system known as internal energy,
$$
E = E_{\mathrm{k}} + E_{\mathrm{p}} + U \, . \tag{1}
$$
That is the way it is said in many books, lecture notes and online source. For instance, the first paragraph of Wikipedia article on internal energy says:
In thermodynamics, the internal energy is one of the two cardinal
state functions of the state variables of a thermodynamic system. It
refers to energy contained within the system, and excludes kinetic
energy of motion of the system as a whole, and the potential energy of
the system as a whole due to external force fields. It keeps account
of the gains and losses of energy of the system.
Well, if a system is just one macroscopic body, I have no problems with these definition. But what if we have more than one body in our system?
Consider, for instance, an isolated system composed of two bodies of mass $m_{1}$ and $m_{2}$. According to classical mechanics, between two masses there always always exists a gravitational force
$$
F = \frac{G m_{1} m_{2}}{r_{12}^2} \, ,
$$
and there exist a gravitational potential energy
$$
V = – \frac{G m_{1} m_{2}}{r_{12}} \, ,
$$
due to this interaction.
Now, this potential energy $V$ is not a part of $E_{\mathrm{p}}$ in (1), since it is not due to external, but rather internal force field.
Clearly, it is also not a part of $E_{\mathrm{k}}$.
Where is it then? Is it included into internal energy $U$ or it is just an additional term for the total energy that has nothing to do with internal energy $U$?
In other words, the increase in which quantity (according to the 1st law of thermodynamics) is equal to the heat supplied to the system plus the work done on it: internal energy counted with or without this potential energy $V$ due to gravitational interaction?
Best Answer
In principle, the gravitational potential energy should be included into total internal energy, but in practice, most often it is not. I know of two reasons.
because for systems that are discussed in thermodynamics, it is believed that gravitational energy is negligible compared to electromagnetic potential energy of the constituting particles;
because it is difficult to include $1/r^2$ forces such as electromagnetic or gravitational force to calculations based on standard statistical physics in a unique and convincing way.