(A fellow noder queried my claim - (see: Heat Energy) - that there were two types of energy; saying, with justice, that at an atomic level, at least in the case of an ideal gas, there was only one: the kinetic energy of the particle's motion.)

How Many Types of Energy Are There?

People like Heisenberg* and Schrödinger* held that energy could not be observed only changes in the energy of something. (Sporus seems to recall.) Against this background the question should be changed to:
In how many ways may a system change its energy?

Two Is the Answer:

There are precisely two ways a system may change its energy:

Discussion:

(Use, as the example system, a cylinder, sealed at one end and with a sliding piston fitted in the other, open end. Trapped between the blind end and the piston is an ideal gas - visualised as mass points ricocheting around randomly, with no forces acting between them.)

Case 1.Pushing in the piston changes the size of the system (defining the system to be the volume of ideal gas). If this compression is adiabatic - the cylinder is insulated so that heat cannot escape - the system increases its energy. This illustrates case I. Work has been done on the system.

(If the cylinder is not insulated the system does not change its energy*: it simply looses all added energy to the surroundings. The compression raises its temperature - the moving piston bats the particles up to higher speeds; temperature measures the particle's kinetic energy - so it looses heat, "warming" its environs. This is not a problem, perhaps, because, in this case, the system changes its entropy so that case I. and case II. cancel each other out.)

It might be objected that different systems exist which do not change their size when work is done on them, an electrical system for example. An electrical analogy for the gas compression is forcing some charge into a capacitor (a device for storing electrical charge - electrons) with a voltage. Defining the original volume of electrons in the capacitor to be the system, it is seen that forcing more electrons in squeezes this volume down into a smaller compass. It changes is size.

An objection, in turn, to this argument is that it is theory. For all anyone knows electrons, for example, could be fictions; dreamt up to facilitate thought but having no reality. This seems to amount to the objection that nothing has been measured:

The process would have caused the capacitor to swell slightly. This could be measured. It seems reasonable to infer from the measured swelling that something inside has shrunk, changed its size; just as one could infer the same from the slight swelling of the cylinder in the ideal gas example.

Case 2.
Touching the cylinder with something at a high temperature increases the energy of the system by heating it. In this case the system's entropy increases; illustrating case II.

Entropy:

This argument gives the imagination a handle on entropy. As pressure (the pressure of the gas) is to temperature and as work energy is to heat energy so size is to entropy.

*Since it appears easy to observe that the system has changed its size and since no change of energy has occurred the remarks about Heisenberg and Schrödinger may be misleading.