In this cycle of operations heat first entered, and then left the gas, and with this entrance or rejection of heat, the condition of the gas with respect to its power of doing work changed. We investigate this flow of heat, and the concomitant change of properties of the substance, with regard to which the flow took place, by forming the concept called entropy. We make the convention that when heat enters a substance the entropy of the latter increases, and when heat leaves it its entropy decreases. We call the quantity of heat entering or leaving a substance Q, and the temperature of the substance T. Then Q/T is proportional to the change of entropy of the substance when the quantity of heat, Q, enters or leaves it.

Now it is a fact of our experience that heat can only flow, of itself, from a hotter to a colder body. Consider two such bodies forming an isolated system, the temperature of the hotter one being T2°, and that of the colder one T1°. Let Q units of heat flow from the body at T2° to that at T1° no work being done.

Then the loss of entropy of the hotter body is Q/T2°, and the gain of entropy of the colder body is Q/T1°. The nett change of entropy of the system is Q/T1° − Q/T2°. Since T2° is greater than T1°, Q/T2° is less than Q/T1°. Therefore the expression Q/T1° − Q/T2° is positive, that is, the entropy of the system, as a whole, has increased. When heat flows from a hotter to a colder body the nett entropy of the two bodies, therefore, increases.

But we can also cause heat to flow from a colder to a hotter body by effecting a compensatory energy-transformation. Such a compensation would not occur by itself in any system capable of effecting an energy-transformation, if it is to be effected some external agency must act on the transforming system. We can suppose it to happen in a perfectly reversible imaginary mechanism. Suppose a Carnot engine works in the positive direction, taking heat from a reservoir at temperature T2°, and giving up part of this heat to a refrigerator at T1°, and doing a certain amount of work W. Suppose that this work is stored up, so to speak, say by raising a heavy weight, which can then fall and actuate the same Carnot engine in the opposite (negative) direction. The engine then exactly reverses its former series of operations. The work it did is reconverted into heat, and as much of this heat flows from the refrigerator into the source, that is, from a colder to a hotter body, in the negative operations, as flowed from the source to the refrigerator in the positive operations. In this primary energy-transformation, combined with a compensatory energy-transformation, there is no change of entropy. The mechanism is an ideal one—the limit to an irreversible mechanism.

But—and now we appeal to experience and cease to work with ideal mechanisms—the actual engine which we can design and work is one in which there will be friction, in which some parts will conduct heat imperfectly, and other parts will insulate heat imperfectly. Let the friction generate q units of heat, and let the quantity of heat which is “wasted” by imperfect conduction and insulation be q1. This heat will flow into the refrigerator, or will be radiated or conducted to the surrounding medium, which we suppose to be at the same temperature as the refrigerator. If, then, we divide this total quantity of heat by the temperature T1°, we get q + q1/T1° = S1 as the quantity of entropy which is generated as the result of the imperfections of the engine, in addition to the quantity of entropy, S, which would be generated if the engine were a perfect one. Both S and S1 are positive.

Also in the working of the engine in the negative direction a certain quantity of entropy, S1, is generated for reasons similar to those mentioned above.

The entropy generated when the engine works in the positive direction is therefore S + S1, and when it works negatively the quantity generated is also S1. The entropy destroyed when the engine works negatively is S. The total change of entropy is therefore 2S1 + S − S, that is, 2S1. In an actual energy-transformation combined with a compensatory energy-transformation there is therefore an increase of entropy.

We can generalise these statements so that they will apply not only to a heat-engine but to all mechanisms which effect energy-transformations. In all such transformations entropy is generated. Therefore the Entropy of the Universe tends to a maximum.

AVAILABLE AND UNAVAILABLE ENERGY

Consider the Carnot engine as a perfect mechanism. It takes heat-energy from a source at a temperature T2°, and it gives up heat to a refrigerator at a temperature T1°, T2° being greater than T1°. In the adiabatic expansion 1→2 the gas continues to expand until its temperature becomes equal to that of the refrigerator. It cannot, then, expand and do work any longer, and thus the proportion of the heat, Q2, received from the source, which can be converted into work, depends on the difference of temperature T2° − T1°. The greater is this difference the greater will be the proportion of the heat-energy received which can be converted into work. If the engine were a perfect one, and if the gas were also a perfect one (that is a gas which would continue to expand according to the equation for the adiabatic expansion of gases), and if the refrigerator were absolutely cold, then all the heat energy received from the source could be converted into work.