By way of elucidation, let us consider the case of a gas?' subject to variations such that any two of the three variables pres sure, temperature and volume, will suffice to define its condition at any given moment; and let us assume that the only forms of energy to be consideed are heat-energy and the energy of elastic compression. Then the fore going equation takes the form A # p- as, where T is the absolute temperature, 0 is the entropy, v is the volume of the gas, and p is the pressure to which it is subject, per unit of its bounding surface. (See THERMODY NAMICS). The first term on the right is then the quantity of heat-energy absorbed, and the second is the quantity of compression-energy absorbed. (The negative sign is affixed to the last term because we are considering the energy added to the system, and the internal energy due to compression increases when v decreases).
If the body or system undergoes any kind of a cyclic change, such that its final state is in all respects identical with its initial state, then the algebraic sum of all the changes of E (the internal or intrinsic energy), summed up for the entire cycle, will be zero ; for if this were not the case, then by causing the body to pass around the cycle repeatedly, in one certain di rection, we could obtain an indefinite supply of energy from it; and this would violate the principle of the conservation of energy. (See below). Suppose, now, that in the special case we are considering, the body undergoes the following cycle : (1) With its temperature constantly equal to T1 it passes from the state in which 0=-01 to the state in which 0=0,; (2) with # constantly equal to 0, it passes from the state in which T=T3 to the state in which T= T3; (3) with T constantly equal to T. it passes from the state in which 0=0,
to the state in which 0-=-0,; and (4) with 0 constantly equal to 01 lit returns to its initial state, so that T changes from T, to T. In each stage the heat absorbed will be ob tained by integrating along the path that is followed. Thus in the first stage T1 units of heat will be absorbed. In the second stage there will be no heat absorbed, because 0 does not change. In the third stage T. (01-0,) units of heat will be absorbed; and in the final stage no heat will be absorbed. But as T, the absolute temperature, is essen tially positive, it follows that 2;(01 02) = #,) is negative, if T1(02 01) is posi tive. Hence if heat enters the body during the first stage of the cycle, heat is rejected by the body during the third stage. The amount of heat that is absorbed in the course of the whole cycle is T,(0, 01) T2(02 --01) T,) ("3-03), and in view of the principle of the conservation of energy, this must have been converted into some other form of energy. But it is not represented by any increase in the internal energy of the body, because a com plete cycle has been described, and the body has returned to its original state. Hence it has been transformed into mechanical energy. The only heat that has entered the body (in the positive direction) is the quantity taken in during the first stage of the cycle. Hence the efficiency of the conversion of heat-energy into mechanical-energy is