INFINITY

What is really meant when the mathematician uses the concept of infinity in his operations? Suppose that we take a line of finite length and divide it into halves, and then divide each half into halves, and so on ad infinitum. We make cuts in the line, and these cuts have no magnitude, so that the sum of the lengths into which we divide the line is equal to the length of the undivided line. We can divide the line into as many parts as we choose, that is, into an “infinite” number of parts.

Suppose that we are making a thing which is to match another thing, and suppose that we can make the thing as great as we choose. If, then, no matter how great we make the thing, it is still too small, the thing that we are trying to match is infinitely great.

Substitute “small” for “great,” and this is also a definition of the infinitely small.

Clearly the idea of infinity does not reside in the results of an operation, but in its tendency. It inheres in our intuition of striving towards something, but not in the results of our striving.

FUNCTIONALITY

If we pour some mercury into a U-tube closed at one end, the air in this end will be contained in a closed vessel under pressure. We can increase the pressure by pouring more mercury into the open end of the tube. We can measure the volume of the air by measuring the length of the tube which it occupies. We can measure the pressure on this air by measuring the difference of length of the mercury in the two limbs of the tube. By taking all necessary precautions we shall find that for each value which the pressure attains there is a corresponding value of the volume of the air.

We thus find the pressure values, p₁, p₂, p₃, p₄, p₅, etc., and the corresponding volumes, v₁, v₂, v₃, v₄, v₅, etc., and we may then plot these values so as to make a graph.

In this figure the values represented along the horizontal axis are pressure-values, and those represented along the vertical axis are volume-values. We have so made the experiment that we can make the pressure-values whatever we choose—let us call them the values of the independent variable or argument. For each value of the pressure, or argument, there is a corresponding value of the volume, which depends on the pressure—let us call these values of the volume values of the dependent variable or function.

We can make arbitrary values of the pressure, but whenever we do this the corresponding values of the volume are fixed. We say, then, that the volume is a function of the pressure. In general, when we choose one value of an independent variable, or argument, there can be only one, or a small number, of values of the dependent variable, or function. If there are two or more values of the function for one value of the argument each of these is necessarily determined by the value which we choose to assign to the argument. There is a strict functionality between the two series of variables. In the experiment we have chosen this functionality is expressed by the equation pv = k(1+at), where p is the pressure, v the volume, k and a constants, and t is the temperature at which the experiment is carried out. In a number of experiments like that which we have mentioned, k, a, and t are the same throughout, and this is why we call them constants. We give p any value we like, and then v can be calculated from the equation.

RATE OF VARIATION

THE NOTION OF THE LIMIT

But the reasoning would be faulty. The line ff₁ only touches the curve, it does not coincide with an element of the curve. Also at the point b₁ the pressure has a certain definite value, and there is no change. At the corresponding point b₁₁ the volume also has a certain definite value, and there is no change. There can therefore be no rate of variation. The value of the tangent does not give us a measure of the rate of variation: it gives us the limit to the rate of variation, when the pressure is changing in the immediate vicinity of b₁.

We must stick to the notion of a pressure change in the immediate vicinity of b₁. What do we mean by “immediate vicinity”? We mean that we are thinking of a range of pressure-values in which the particular pressure-value b₁ is contained, but not as an end-point. We mean also that we choose a definite standard of approximation to the value b₁, so that any pressure-value within our interval differs from b₁ by less than this standard of approximation. It means further that, no matter how small is the number representing this standard of approximation, any pressure-value within the interval will differ from b₁ by less than this number. This is what we really mean when we say that the interval we are thinking about is an “infinitely small one.”

Now corresponding to this interval of pressure-values in the immediate vicinity of b₁, there will be an interval of volume-values in the immediate vicinity of b₁₁, and, as before, any one of these volume-values will differ from b₁₁ by less than any number representing a standard of approximation to b₁₁. We then find the point on the curve corresponding to both b₁ and b₁₁, that is b, and we draw the line ff₁, and find the tangent of the angle which this line makes with op. The value of this tangent is the limit of the rate of variation of the volume of the gas when the pressure undergoes a change in the immediate vicinity of b₁.

“Rate of variation” is a function of the argument “pressure.” This function has the limit l for a value of its argument b₁, when, as the argument varies in the immediate vicinity of b₁, the value of the function approximates to l within any standard whatever of approximation.36

(Another example.)

Let us consider the case of a stone failing from a state of rest. Observations will show that when the stone has fallen for one second it has traversed a space of 16 feet; at the end of two seconds it has fallen through 64 feet; and at the end of three seconds the space traversed is 144 feet. From these and similar data we can deduce the velocity of motion of the stone as it passes any point in its path.

The velocity is the space traversed in a certain time s/t. If we take any easily observable space (say five feet) on either side of the point chosen, and then determine the times when the stone was at the extremities of this interval, and divide the interval of space by the interval of time, we shall obtain the average velocity of motion of the stone over this fraction of the whole path chosen. But the velocity did not vary in a constant manner during this interval (as we see by considering the spaces traversed during the first three seconds of the fall). Therefore our average velocity does not accurately represent the velocity of the stone as it passes the point at the middle of the path chosen.

We therefore reduce the length of the path more and more so as to make the average velocity approximate closer and closer to the velocity near the middle portion of the path. In this way we find the ratio ds/dt, where ds is a very small interval of path containing the point chosen, but not as an end-point, and dt is a very small interval of time. Perhaps this average velocity may be near enough for our purposes, but perhaps it may not. The interval of path ds is still a finite interval, and dt is still a finite time, and so long as these values are finite ones the velocity deduced from them remains a mean one. All that we can say is that it approximates to the velocity, as the arbitrary point was passed, within a certain standard of approximation.

Obviously the smaller the interval ds, the closer will be this approximation. Suppose, then, that we diminish ds till it “becomes zero.” It might appear now that when ds coincides with the point chosen we shall obtain the velocity of the stone at this point. But if there is no interval of path, and no interval of time, there can be no velocity, which is an interval of path divided by an interval of time; and if the stone is “at the point,” it does not move at all. We must stick to the idea of intervals of space and time, and yet we must think of these intervals as being so small that no error whatever is involved in regarding the mean velocity deduced from them as the “true velocity.” We therefore think of the point as being placed in an interval of path, but not at an end-point of this interval. We think of the velocity as a mean one, but we must have a standard of approximation, so that we may be able to say that the mean velocity approximates to the “actual” or limiting velocity of the stone as it passes the point, within this standard of approximation. The smaller we make the interval, the closer will the mean velocity approximate to the limiting velocity.

We therefore think of the stone as moving in the immediate vicinity of the point in the sense already discussed. We say that the “immediate vicinity” is an interval such that any point in it, p₁, approximates to the arbitrary point p which we are considering within any standard of approximation: that is, no point in the interval is further away from p than a certain number expressing the standard of approximation, and this can be any number, however small. We say the same thing about the interval of time. That is to say, we make the intervals as small as we like: they can be smaller than any interval which will cause an error in our deduced velocity, no matter how small this error may be.

The limit of the velocity of a stone falling past a point in its path is, therefore, that velocity towards which the mean velocities approximate within any standard of approximation, when we regard the interval of space as being the immediate vicinity of the point, and the interval of time as being the time in the immediate vicinity of the moment when the stone passes the point. The limit of the velocity is not ds/dt but ds/dt, dt and ds being, not finite intervals of time and space, but “differentials.” We determine this limit by the methods of the differential calculus.

FREQUENCY DISTRIBUTIONS AND PROBABILITY

Let the reader keep a note of the number of trumps held by himself and partner in a large number of games of whist (the cards being cut for trump). In 200 hands he may get such results as the following:

No. of trumps in his own and partner’s hands—0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.

No. of times this hand was held—0, 0, 0, 1, 9, 29, 53, 52, 35, 14, 6, 1, 0, 0.

He should note also the number of times that trumps were spades, clubs, diamonds, and hearts: he will get some such results as the following: spades, 46; clubs, 53; diamonds, 51; hearts, 50.

The numbers in the lower line of the first series form a “frequency distribution,” for they tell us the frequency of occurrence of the hands indicated in the numbers above them. “No. of trumps” is the independent variable, and “no. of times these nos. of trumps were held” is the dependent variable.

A frequency distribution represents the way in which the results of a series of experiments differ from the mean result. A particular result is expected from the operation of one, or a few, main causes. But a number of other relatively unimportant causes lead to the deviation of a number of results from this mean or characteristic one. Yet since one, or a few, main causes are predominant, the majority of the results of the experiment will approximate closely to the mean; and a relatively small proportion will deviate to variable distances on either side of the mean. If a pack of cards were shuffled so that all the suits were thoroughly mixed among each other, then we should expect the trumps to be as equally divided as possible between the four players. But a number of causes lead to irregularities in this desired uniform distribution, and so the results of a large number of deals deviate from the mean result. It is possible, by an application of the theory of probability, to calculate ideal, or theoretical frequency distributions, basing our reasoning on the considerations suggested above. We then find that the observed and calculated frequency distributions may be very much alike.

In biological investigation, far more than in physical investigation, we deal with mean results. It is, however, just as important that the mean should be considered as the individual divergences from the mean. We want to know the mean results, and the way and the extent in which the individual results diverge from the mean.

There is a mean or “ideal” result, but we must think of a great number of small independent causes which cause the actually obtained results to diverge from this mean. If these small un-co-ordinated causes are just as likely to cause the results to be less than the mean, as greater than the mean, we shall obtain a frequency distribution resembling the one given above, in that the variations from the mean are equal on both sides of the mean. But if the general tendency of the small un-co-ordinated causes is to cause the results, on the whole, to tend to be greater than the mean, then the frequency distribution will be “one-sided,” that is, if we represent it by a curve the latter will be an asymmetrical one. Curves which are asymmetrical are those most frequently obtained in biological, statistical investigations.

MATTER

Our generalised notion of matter is that it is the physical substance underlying phenomena. Immediately, or intuitively, we attain the notion of matter because of our perceptions of touch, and our perception of muscular exertion. The distance sense-receptors, visual, auditory, and olfactory, would not give us this intuition of matter.

Material things are extended, that is, they have form, and they exclude each other, so that they cannot occupy the same place. They appear to us to be aggregates of different nature: they may be solid and homogeneous, like a piece of metal; or solid and porous, like a piece of pumice-stone; or loose and granular, like sand; or viscous or liquid, like pitch or water. They may have colour. They are opaque, or transparent in various degrees. They may have odour. Material things, as they are perceived by the distance sense-receptors, appear to have qualities.

Material things are aggregates of molecules. The aggregates may possess essential form, like that of a crystal, or an organism. The form of the aggregate may be essential and homogeneous, so that it consists of molecules, all of which are of the same kind, like a crystal. It may be heterogeneous and essential, like the body of the organism, when it consists of molecules which are not all of the same kind. The aggregates may have accidental form, like that of a river valley, or a delta, or a mountain, and the form in these, and similar cases, is not a part of the essential nature of the aggregate.

The molecules are selections (in the mathematical sense) of some of about eighty different kinds of atoms. A molecule is a small number of atoms arranged together in a definite way, and its nature depends, not only on the kinds of atoms of which it is composed, but also on the arrangement of these atoms. Two or more different arrangements of the same atoms are, in general, different molecules.

MASS

When matter is perceived by the tactile and muscular sense organs, we have the intuition of mass. It is heavy, and the degree of heaviness is proportional to the quantity of matter in the body which we feel, that is, to its mass. Heaviness is synonymous with weight, but weight does not depend alone on the quantity of matter in the body. If the latter were removed to an infinite distance from the earth or other cosmic bodies, its weight would disappear, but its mass would remain. We could still touch and move it, and we should still find that different degrees of muscular exertion would be necessary when bodies of different masses had to be moved.

INERTIA

If the body were in motion, we should find that muscular exertion is necessary in order that it might be brought to rest; and if it were at rest, we should find that muscular exertion was necessary in order that it might be moved. The body, matter in general, possesses inertia, and this is its most fundamental attribute. Mass we can only conceive in terms of inertia. If two bodies were at rest, and if the same degree of muscular exertion conferred on each the same initial velocity of motion, their masses would be equal. If the same degree of muscular exertion conferred different velocities on different bodies, their masses would be different, and would vary directly with the initial velocities conferred.

FORCE

The feeling which we experience when we move a body from a state of rest, or stop a body which is moving, is what we call force. If on climbing a stair in the dark we think there is one step more than there is, and so have the queer, familiar, feeling of treading on nothing, we have the intuition of energy; but when we tread on the steps, and so raise our body, we have the intuition of force. Force is that which accelerates the velocity of a mass. If the latter is at rest, we consider it to have zero velocity. If it is moving, and we stop it, there is still acceleration, but this is negative.

Matter, that is, the substantia physica, is clearly to be conceived only in terms of energy. It is, to our direct intuitions, resistance, or inertia, that which requires energy in order that it may be made to undergo change. Our static idea of physical solidity, or massiveness, disappears on ultimate analysis. Molecules are made up of atoms, and the atoms are assumed to have all the characters of matter: we could not see them, of course, even if we possessed all the magnifying power that we wished, for they would be too small to reflect light. Modern physical theory is compelled to regard atoms as complex, and imagines them as being composed of moving electrons. The electron is immaterial—it is the unit-charge of electricity. It is said to possess mass, but mass is now understood to mean inertia. So long as the electron is moving, it sets up a field of energy round it, and this field—the electro-magnetic one—extends in all directions. Periodic disturbances in it constitute radiation, and this radiation travels with the velocity of light. It is because of the existence of this field that we are obliged to postulate the existence of an ether of space. Unfamiliar to us until the discovery of Hertzian waves and “wireless” telegraphy, this electro-magnetic radiation in space is now accessible to our direct intuitions. We can initiate it by setting electrons in motion, that is, by expending energy (producing the sparking in the transmitters of the wireless telegraphy apparatus); and we can stop it, if it is in existence, by absorbing the energy (in the receivers of the wireless telegraphy apparatus). This is essentially what we understand by the inertia of gross matter. We set a body in motion by expending energy on it (the explosion of the powder in a cartridge, which converts potential chemical energy into the kinetic energy of the moving projectile); and we can stop a body which is in motion by absorbing this energy of motion (by causing the projectile to strike against a target, when the kinetic energy of its motion becomes the kinetic energy of the heat of the arrested body).

Inertia is therefore the same thing whether it be the inertia of visible, material bodies, or the inertia of invisible, material molecules, or the inertia of the immaterial, non-tangible ether. It is the condition that energy-changes must occur if anything accessible to our observation is to change its state of rest or motion.

ENERGY

Energy is therefore indefinable. It is an elemental aspect of our experience.

Nature to us is an aggregate of particles in motion. We have to speak of massive particles, whether we call these visible material bodies, or molecules, or atoms, or electrons, in order that we may describe nature. We must employ the fiction of a substantia physica. We only know the substance or matter in terms of energy; it is really the latter that is known to us. It is the poverty of our language, or rather it is the legacy of a materialistic age, that compels us to speak of particles that move, rather than of motions as entities in themselves.

Considering, then, the idea of particles in motion as a fiction necessary for clear description, we can study energy. There is only one kind, or form, of energy which presents itself to our aided or unaided intuitions, that is kinetic energy. Bodies that move possess this energy represented by their motion: they can be made to do work, that is, their energy can be transformed into other forms of energy. All things are in motion. A gas consists of molecules incessantly moving with high velocity, and colliding and rebounding from each other. The energy of a gas is the sum of one-half of the masses of all the molecules, multiplied by the squares of the velocities of all the molecules, that is, S 1/2mv². This is also the kinetic energy of a projectile, or of a planet revolving round the sun. Kinetic energy is that of the uniform, unchanging motion of some entity possessing mass, but we must extend our notion of mass so as to include immaterial, imponderable entities such as electrons.

This energy cannot be destroyed or created—the law of conservation of energy. This is a principle or mode of our thought. We are unable scientifically or philosophically to think of an entity ceasing to be. Dreams and phantoms show us entities which are real while they last, but which cease to exist. If we do attempt to think of entities that appear from, or disappear into, nothing, we surrender the notion of reality. The more we think of it the more clearly we shall see that the things which we call real are the things which are conserved.

Yet energy, to our immediate intuitions, seems to disappear. A flying bullet strikes against a target and becomes flattened out into a motionless piece of lead. A red-hot piece of iron cools down to the temperature of its surroundings. A golf-ball driven up the side of a hill comes to rest in the grass. A current of electricity passing through water is used up, that is, electricity of a higher potential is required to force the current through water than to force it through thick copper wire. In all these cases we might think that energy is lost, but we cannot believe this. The kinetic energy of the flying bullet becomes transformed into the increase of the kinetic energy of the molecules of the metal of which the bullet was composed; for the latter becomes greatly heated when its flight is arrested and this increased heat ought to be equal to the kinetic energy of the bullet in flight. The red-hot piece of iron cools, and the kinetic energy of its molecules becomes less and less, but this does not cease to exist, for the energy is simply transferred by radiation and conduction to the surrounding bodies, the temperature of which it raises. The golf-ball driven up the hill comes to rest and loses its kinetic energy. Some of this has been transferred to the air through which it passes, the latter being heated very slightly; some of it is expended by friction with the grass over which the ball rolls before coming to rest, and this energy is traceable in heat-effects, or in mechanical effects, but the rest of it apparently ceases to exist. But this would be contradictory to the principle of conservation, and so we say that the lost kinetic energy has become potential. The current of electricity may heat the water through which it passes, and some of the energy which seems to disappear is so to be traced, but the greater fraction is apparently lost. A quantity of free hydrogen and oxygen is, however, generated, and we say that the kinetic energy of the moving electrons has become transformed into the potential chemical energy of the gaseous mixture.

POTENTIAL ENERGY

Therefore, if energy disappears or appears, we do not say that it is destroyed or is created: we invent potential energies, into which we suppose that the energies in question have become transformed, in order that we may still think of them as being subject to an a priori principle of conservation. Although a particle of radium continually generates heat, we do not therefore think of the first principle of energetics as being invalidated, for we suppose that the energy which thus appears was really potential in the atoms of radium. But it was contrary to all our former experience of atoms that they should contain any other energy than that of their own motion, and so the further assumption was made that the atom, at least the atom of the radio-active substance, is really complex, and not simple, as chemical theory demands. It is made up of smaller particles, and possesses a definite structure. In certain circumstances the atom may disintegrate, and the energy which held together its particles, whether these were simpler corpuscles or electrons, is given off as the heat which the radio-active substance apparently generates. The potential energy of the chemical atom is therefore a hypothesis which has been devised in order to preserve the validity of the law of conservation, and the reality of this hypothesis is being tested by investigation. If we accept it as true, are the deductions made from it justified in our experience? That is the test which must be satisfied in all the hypotheses where potential energies are invented, and the potentials are only real if the test is satisfactory. The golf ball at rest at the top of the hill is a different entity from the golf ball at rest at the bottom of the hill: it is capable of developing energy, for a touch may cause it to roll down the hill, when most of the energy which was expended in order to drive it to the top of the hill will reappear in the form of the kinetic energy of motion of the ball. The atoms of hydrogen and oxygen which were dissociated by the energy of the electric current are different things from the atoms of hydrogen and oxygen which are combined together to form the molecules of water. Their state when the gases are in the elementary condition, or are “free,” is that of molecules moving rapidly and incessantly, rebounding from each other after colliding with each other: they possess energy of position—potential energy—because they are separate from each other. If they “combine,” as when a minute electric spark explodes the mixture of gases, they tractate together, and remain in proximity to each other, becoming molecules of water. The energy which became potential in the gaseous mixture, when the electric energy of the current seemed to disappear, now appears as the heat generated by the combustion, that is, as the greatly increased kinetic energy of the molecules of the gas (steam) which takes the place of the mixture of hydrogen and oxygen. Previous to the explosion this gas was a mixture of molecules of hydrogen and oxygen (2H₂+2O) at the ordinary temperature, but after the explosion it consists of a smaller number of molecules at a very much higher temperature.

What is “energy of position”? The golf ball at the bottom of the hill was at a distance of R feet from the centre of the earth, but at the top of the hill it is at a distance of R + 100 feet from the centre of the earth. In the first case it was free to fall R feet, but in the second case it is free to fall R + 100 feet. The atoms of the constituent molecules of water occupy the position H-O-H, the bonds (-) indicating that the atoms are very close together; but when the water is decomposed by an electric current, the atoms occupy the positions O-O + H-H + H-H, the (+) indicating that the atoms are relatively far apart from each other. Now the golf ball and the earth, or the atoms of hydrogen and oxygen, are physically the same material entities, whether they are close together or far apart, yet when the earth and the ball, or the atoms of oxygen and hydrogen, are separated from each other, their “properties” are different from what they are when they are close together. What is it that makes the difference? It is that which is between them. Is it, in the last case, “the potential energy of chemical affinity”? This dreadful phrase is actually used in a recent book on biology: “In the elements carbon and oxygen, so long as they remain separate, a certain amount of energy remains latent. When the carbon and oxygen atoms are allowed to come together and unite, this potential energy of chemical affinity is liberated as kinetic energy.” What is changed by the tractation and pellation (the terms suggested by Soddy in place of the anthropomorphic ones, “attraction” and “repulsion”)? It is the ether which has become changed in some way. Potential energy resides therefore in the ether of space.

ISOTHERMAL AND ADIABATIC CHANGES

Let us consider the changes which occur in a gas under the influence of changes in temperature and pressure, premising that the remarks which we have to make can be applied to bodies in the liquid and solid conditions, with some necessary modifications. A gas, then, consists of a very great number of particles, or molecules, in motion. These molecules move in straight lines at very high velocities, and if the envelope in which the gas is contained is a restricted one, the molecules collide with each other, and with the walls of the envelope; and, being assumed perfectly elastic, they rebound from each other, and from the walls of the vessel, with the same velocity which they had when they collided. The pressure of the gas (say that of steam at a temperature of 110°C., and a pressure of 120lbs. to the square inch in a steam boiler) is the sum of the impacts of the molecules on the walls of the containing vessel. When the temperature is high the molecules are moving at a higher mean velocity than when the temperature is lower, and their mean free path tends to become greater. The volume of a certain mass of gas, that is, the volume occupied by a certain very great number of molecules, is greater the higher is the temperature, provided the envelope is one capable of yielding. If we reduce the capacity of the envelope in which the gas is contained, the pressure will rise, for the intrinsic energy of the gas is still the same; but we have done work on it, and by the law of conservation this work, or at least the energy represented by it, must still exist. It is represented by the decreased length of free path of the molecules, and this means that the impacts on the walls of the vessel will be greater than they were. There is, therefore, a certain relation between the volume of a gas and its pressure, and this relation can be represented by an equation involving the temperature, the pressure, and the volume.

The diagram represents the pressure and the volume of a gas when these things change. There are two conditions, (1) when the heat developed by the compression is allowed to escape through the walls of the vessel to the outside, or when the heat lost in the expansion of the gas is compensated by the conduction of heat through the walls of the vessel from outside; and (2) when the heat developed is retained in the gas, as when the latter is contained in a vessel the walls of which do not conduct heat. The pressure of the gas is measured along the horizontal axis, and the volume is measured along the vertical axis, and a curve is drawn so that for any value of the pressure there is a corresponding value of the volume. Thus the values of the pressures p and p₁ in the diagram correspond to the value of the volume v. The curve relating the change of pressure with a corresponding change of volume is, in general, that called a rectangular hyperbola. But there are two kinds of such curves: (1) that which we obtain by plotting the corresponding values of pressure and volume, when the temperature of the gas remains constant throughout the series of changes, that is, when the rise of temperature which would occur when the gas is compressed is compensated by the conduction of this heat to the outside of the vessel containing the gas. Such a series of changes of pressure and volume is called an isothermal one. (2) When the heat developed by the compression of the gas is retained in the gas, as when the walls of the vessel in which these changes are effected are such as do not conduct heat: such a series of changes is called an adiabatic one. Adiabatic curves are steeper than are isothermal ones.

THE CARNOT ENGINE

This is an imaginary mechanism which performs a certain cycle of operations. It does not really exist, but the conception of its operation is of the greatest value in the consideration of energy-transformations, and it is for this reason that we discuss it here.

Consider a gas, or some other substance capable of expanding or contracting. It contains intrinsic energy, and it is capable of doing work. Thus, since a gas can expand indefinitely it can be made to do mechanical work. A mass of gas at a pressure p₁, and having a volume v₁, and at a temperature T°, can do work by expanding till its pressure is reduced to p, and its volume increased to v. If it expands adiabatically its temperature will fall to t°. Let us suppose that t° is the temperature of the surrounding medium: the gas cannot therefore cool further, and we can obtain no more work from it. If the gas is the substance which we wish to employ as the working substance in the Carnot engine, we must therefore bring it back to the condition represented by A. That is, we must raise its temperature to T°, we must reduce its volume to v₁, and we must increase its pressure to p₁.

Thus the steam of an engine is (say) at a temperature of 110°C., and a pressure of 120lbs. to the square inch. When it has passed through the cylinder and condenser it is water at a temperature of, say, 15°C., and it is at atmospheric pressure. We must, therefore, bring it back to its former condition by heating this water in the boiler till it is steam under the former conditions of temperature and pressure.

Therefore we must, in order to obtain a self-acting engine, cause the working substance, and the mechanism of the engine, to perform a series of cyclical operations.

The Carnot engine is a cylinder containing a gas called the working substance S, and this gas can be brought into thermal contact with a source of heat, or a refrigerator, that is, the gas can be heated or cooled by a mechanism outside itself. The walls of the cylinder are made of some substance which is a perfect non-conductor of heat, but the bottom of the cylinder is made of a substance which conducts heat perfectly. There is a piston in the cylinder which fits it closely, but which moves up and down without friction. At the bottom of the latter is a valve which can be turned so as to place the bottom of the cylinder, and therefore the gas, in thermal contact with a reservoir of heat (+), or a refrigerator (-). But when the valve is turned so that the non-conducting part O fills the bottom, the gas is perfectly insulated, and heat can neither enter nor leave it.

Such an engine is, of course, an imaginary one, since there can be no mechanism in which there is not a certain amount of friction between moving parts, and there are no substances which conduct or insulate heat perfectly. The engine is, in fact, the limit to a series of engines each of which is supposed to be more perfect than the last one. It is a fiction which is of considerable use in theoretical work.

THE CARNOT POSITIVE CYCLE

We have therefore a substance which can be heated by contact with a hot body, and which can then expand, doing mechanical work by raising a piston, and perhaps turning a flywheel, and on which work is then done so that it returns to its original condition. This is a cycle of operations. If we consider only the changes which occur in the working substance we can represent these changes by a diagram.

First operation, (1?2). We suppose that the valve is turned so that the non-conducting plug closes the cylinder. The piston is in the position II (Fig.31). Heat cannot then enter or leave the gas. But the latter already contains heat: it is at a temperature of T₂°, so that it can expand doing work. Let it expand, forcing up the piston. During this operation the pressure of the gas will fall from a point on the vertical axis opposite 1 to a point opposite 2, and its volume will increase from a point on the horizontal axis beneath 1 to a point beneath 2. It will cool because it has expanded, and no heat is allowed to enter it during this act of expansion. The expansion is therefore adiabatic; the temperature falls from T₂° to T₁°; and work is done by the gas.

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.

THE CARNOT NEGATIVE CYCLE

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.

REVERSIBILITY

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.

ENTROPY

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 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.

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 Q₁T₁°/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.

INERT MATTER

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.