Internal energy is an important concept in thermodynamics. Matter consists of atoms and molecules, and these are made up of particles having kinetic and potential energies. The internal energy of a system is tentatively defined as the sum of the kinetic energies of all of its constituent particles, plus the sum of all the potential energies of interaction among these particles.
Note that internal energy does not include potential energy arising from the interaction between the system and its surroundings. If the system is a glass of water, placing it on a high shelf increases the gravitational potential energy arising from the interaction between the glass and the earth. But this has no effect on the interaction between the molecules of the water, and so the internal energy of the water does not change.
During a change of state of the system the internal energy may change from an initial value U1 to a final value U2. The change in internal energy is denoted as ΔU = U2 − U1.
Heat transfer is energy transfer, so when a quantity of heat Q is added to a system and it does no work during the process, the internal energy increases by an amount equal to Q; that is, ΔU = Q. When a system does work W by expanding against its surroundings and no heat is added during the process, energy leaves the system and the internal energy decreases. That is, when W is positive, ΔU is negative, and vise versa. So ΔU = −W. When both heat transfer and work occur, the total change in internal energy is:
(Eq1)
U2 − U1 = ΔU = Q − W
This can be arranged to the form:
(Eq2)
Q = ΔU + W
The message of Eq2 is that in general when heat Q is added to a system, some of this added energy remains within the system, changing its internal energy by an amount ΔU; the remainder leaves the system again as the system does work W against its surroundings. Because W and Q may be positive, negative, or zero, ΔU can be positive, negative, or zero for different processes.
Eq1 or Eq2 is the first law of thermodynamics when taking into consideration only the internal energy. It is a generalization of the principle of conservation of energy to include energy transfer through heat as well as mechanical work. In every situation in which it seems that the total energy in all known forms is not conserved, it has been possible to identify a new form of energy such that the total energy, including the new form, is conserved. There is energy associated with electric fields, with magnetic fields, and, according to the theory of relativity, even with mass itself.
At the beginning of this discussion internal energy was tentatively defined in terms of microscopic kinetic and potential energies. This has drawbacks, however. Actually calculating internal energy in this way for any real system would be hopelessly complicated. Furthermore, this definition isn't an operational one because it doesn't describe how to determine internal energy from physical quantities that can be measured directly.
The change in internal energy ΔU during any change of a system as the quantity given by Eq1, ΔU = Q − W. This is an operational definition, because Q and W can be measured. It does not define U itself, only ΔU. This is not a shortcoming, because the internal energy of a system can be defined to have a specified value in some reference state, and then Eq1 can be used to define the internal energy in any other state.
While Q and W depend on the path from one state to another, ΔU = Q − W is independent of the path. The change in internal energy of a system during any thermodynamic process depends only on the initial and final states, not on the path leading from one to another. Internal energy depends only on the state of a system. In changes of state, the change in internal energy is path independent.
There's nothing wrong with thinking of internal energy as microscopic mechanical energy, but in the interest of precise operational definitions, internal energy, like heat, can and must be defined in a way that is independent of the detailed microscopic structure of the material.
For a cycle the total internal energy change must be zero. Then:
U2 = U1
and
Q = W
Another special case occurs in an isolated system, one that does no work on its surroundings and has no heat flow to or from its surroundings. For any process taking place in an isolated system:
W = Q = 0
and therefore:
U2 − U1 = ΔU = 0
In other words, the internal energy of an isolated system if constant.