Second operation, (2→3). The piston is now at the position I, that is, at the upper end of its stroke, and we must bring it back again to the lower end of the cylinder. The valve is turned so that the bottom of the cylinder is placed in thermal communication with the refrigerator (−), and the piston is pushed in to the position II. The gas is therefore compressed until its volume decreases from a point beneath 2 to a point beneath 3. As it is being compressed, heat is generated and its temperature would rise, but as this heat is generated it flows into the refrigerator, so that the temperature of the gas remains the same during the operation. The contraction is therefore an isothermal one; the temperature remains at T1°; and work is done on the gas from outside.
Third operation, (3→4). But the piston is not at the lower end of its stroke yet. We turn the valve so that the bottom of the cylinder is closed by the non-conducting plug O, and then push in the piston until it reaches the position III. The gas is still further compressed, and this compression generates heat. But the heat cannot escape, so that the temperature of the gas rises until it reaches T2°. The contraction is therefore an adiabatic one. Work is done on the gas.
Fourth operation, (4→1). The piston is now at the lower end of its stroke. We turn the valve so that the bottom of the cylinder is placed in communication with the source of heat (+). The gas expands from the point beneath 4 to the point beneath 1, raising the piston to the position II. This expansion of the gas would lower its temperature, but it is in communication with the source of heat, and so it does not cool, but draws heat from the source and remains at a constant temperature, T2°. The expansion is therefore an isothermal one. Work is done by the gas.
This completes the cycle. But the gas is heated, and when the piston is at position II, the valve is turned so as to close the cylinder by the non-conducting plug O. The heat already contained in the gas continues to expand, the latter doing more work, but this expansion causes the temperature to fall from T2° to T1°. This is the operation with which the cycle commenced.
Summarising the positive Carnot cycle, we see that the engine takes heat from a source (+) and gives up part of this to a refrigerator (−), (in an actual steam-engine heat is taken from the boiler and given up to the condenser water). If we measure the quantity of heat taken from the boiler in the steam which enters the cylinders we shall find that this quantity of heat is greater than the quantity which is given up to the condenser water. What becomes of the balance? It is converted into the mechanical work of the engine. The Carnot engine therefore takes a quantity of heat, Q2, from the source and gives up another quantity of heat, Q1, to the refrigerator. We find that Q2 is greater than Q1 and the balance, Q2 − Q1, is represented by the work done by the engine. Heat-energy falls from a state of high, to a state of low potential, and is partly transformed into mechanical work.
This is simply the positive cycle reversed. The reader should puzzle it out for himself if he is not already familiar with it. It consists of an adiabatic contraction 2→1, an isothermal contraction 1→4, an adiabatic expansion 4→3, and an isothermal expansion 3→2. A quantity of heat, Q1, is taken from the refrigerator at a temperature T1°, and another quantity, Q2, is given up to the source at a temperature T2°. But Q2 is greater than Q1, and the engine therefore gives up more heat than it receives, while, further, heat flows from a body at a low temperature to another body at a higher temperature. Where does the engine get this energy from? It gets it because work is done upon it by means of an outside agency, and all of this work is converted into heat.
The Carnot engine and cycle are therefore perfectly reversible. Not only can the engine turn heat into work, but it can turn work into heat. This perfect, quantitative reversibility is, however, a property of the imaginary mechanism only, and it does not exist in any actual engine.
Let us consider the cycle more closely. In the operation 4→1, which is an isothermal expansion, there is a flow of heat-energy from the source and a transformation of energy into work. The gas in the condition represented by the point 4 had a certain pressure and a certain volume. In the condition represented by the point 1, its pressure has decreased, its volume has increased, and its temperature is the same. Its physical condition has been changed, and to bring it back into its former condition something must be done to it. Let, then, the gas continue to expand without receiving any more heat, or parting with any: that is, let it undergo the adiabatic expansion 1→2 until its temperature falls to that of the refrigerator, T1°. We now compress the gas while keeping it at this temperature, that is, we cause it to undergo the isothermal contraction 2→3, during which operation it is giving up heat to the refrigerator, so that there is again a flow of heat-energy. We then compress it still further without allowing heat to escape from it, that is, we cause it to undergo the adiabatic contraction 3→4. During this operation the gas rises in temperature to T2°. It is now in the condition that it was when the cycle commenced.
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 QT 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 QT2°, and the gain of entropy of the colder body is QT1°. The nett change of entropy of the system is QT1° − QT2°. Since T2° is greater than T1°, QT2° is less than QT1°. Therefore the expression QT1° − QT2° 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 + q1T1° = 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.
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.
We cannot produce a refrigerator of absolute temperature 0°, and therefore only a certain proportion of the heat which is received by the engine can be transformed into mechanical work. But this work can be used to reverse the action of the engine, and thus the same fraction of the total heat-energy which was given to the refrigerator can be taken from it and given back to the source. The perfect engine is therefore reversible without loss of available energy.
Now consider still the engine as a mechanism which takes heat from a source and gives it to a refrigerator, but let it be an actual engine. Instead of giving up a certain fraction of the heat received to the refrigerator—a fraction equal to Q1 T1°T2°, it gives up rather more, because it is not a perfect mechanism, that is, it generates friction, etc. Some of the heat received thus ceases to be available for the performance of work; and passes into the refrigerator. The fraction of the heat-energy which passes into the refrigerator in the perfectly reversible engine was unavailable energy in the conditions in which the mechanism worked, or was imagined to work, but in the actual engine this fraction is increased. If we divide the increase of unavailable energy by the temperature of the refrigerator, the product is the increase of entropy generated in the actual engine over that generated in the ideal engine. Because of this reduction of available energy the actual engine is an irreversible mechanism.
This is the connection between unavailable energy and entropy. In all transformations some fraction of the transforming energy becomes heat, and this heat flows by conduction and radiation into the surrounding bodies. In general this heat simply raises the temperature of the medium into which it flows, and becomes unavailable for further transformations. With every transformation that occurs some part of the energy involved becomes unavailable. Therefore although the sum of the available and unavailable energy of the Universe remains constant, the fraction of unavailable energy tends continually to a maximum.
We can see now what is indicated by Bergson’s “inert matter.” It is not matter deprived of energy—such an expression has no meaning—it is energy which is unavailable for further transformations.
The matter in which we choose to say that this energy is inherent has become inert. Let us substitute for the Carnot engine the actual steam-engine of a ship, the condenser of which is cooled by the sea water which is taken in, and which is then heated and flows out again into the sea. The heat derived from the source, that is, from the furnace of the boiler where coal is burned to raise steam, thus passes out into the sea. Now the heat capacity of the sea is so great that the temperature of the water is not appreciably raised by this heat, which drains into it from the engine: even if it were appreciably raised, the heat would be conducted into the earth, or would be radiated out into space, and would then raise the temperature of the material bodies of the universe. But let all this heat remain in the sea. It then simply raises the temperature of the water by an exceedingly small amount, and the motions of the molecules become infinitesimally increased. But the heat becomes equally distributed by conduction and convection throughout the mass of the water in the sea, and as there are no differences in adjacent parts there are no means whereby the energy which thus passes into the sea can be again transformed.
A new order of things is the result of the processes we have indicated. The segregated, available heat-energy of material bodies has become transferred to the un-co-ordinated, diffuse, unavailable energies of the molecules which compose these bodies. The transformations which we can effect depend on the condition that the energy which we utilise is that of aggregates of molecules which are in a different physical condition, as regards this energy, from adjacent aggregates. But when this energy becomes equally distributed among the molecules of all the aggregates, the matter in which it inheres becomes inert. If we could, by a sorting process like that of Maxwell’s hypothetical demons, a process which does not expend the energy with which it deals, separate the molecules which were moving slowly from those which were moving more quickly, we could make this energy again available. But it must clearly be understood that our physics is the physics not of individual molecules, but of aggregates of molecules.