Internal energy

Definition

When the system can transition from a reference point of extensive variables $X^{*}$ to another state $X$ is achievable, for an arbitral temperature $T$ , either the transition $(T; X) \overset{a}{\to} (T^{*}; X^{*})$ or the reverse transition $(T^{*}; X^{*}) \overset{a}{\to} (T; X)$ should be achievable via an adiabatic operation $a$ .

In this context, the internal energy $U$ is defined as follows:

If the transition $(T; X) \overset{a}{\to} (T^{*}; X^{*})$ is possible, then the internal energy $U$ is given by

U (T; X) = W_{a} ((T; X) \to (T^{*}; X^{*})),

where $W_{a}$ denotes the work performed by the surrounding on the system during the adiabatic operation $a$ .

If the reverse transition $(T^{*}; X^{*}) \overset{a}{\to} (T; X)$ is possible, then $U$ is defined as:

U (T; X) = - W_{a} ((T^{*}; X^{*}) \to (T; X)) .

Important note

Note

The internal energy $U$ is a thermodynamic state function and a form of energy that corresponds to the same fundamental quantity used across other domains of physics.

1. Sign convention

There is some arbitrariness in the sign convention for work and heat exchanged between a system and its surrounding in thermodynamics textbooks. Here the convention designates work done by the system on the surrounding is designated as positive.

2. Reference state

By definition, there is an inherent quantitative indeterminacy in internal energy due to the choice of reference point. In other words, it is a degree of freedom to add or subtract a constant offset to the whole without affecting its physical meaning. Such indeterminacy is common in quantities like potential in electromagnetics, where only differences in values are physically significant.

Reference

Tasaki Chapter 4, P63