Popular Science Monthly/Volume 76/March 1910/The Second Law of Thermodynamics: Its Basis in Intuition and Common Sense

(1910)
The Second Law of Thermodynamics: Its Basis in Intuition and Common Sense by William Suddards Franklin

 THE SECOND LAW OF THERMODYNAMICS: ITS BASIS IN INTUITION AND COMMON SENSE[1]

By Professor W. S. FRANKLIN

LEHIGH UNIVERSITY

IT is the object of this article to give a simple account of that fundamental principle in physics which is known as the second law of thermodynamics. No generalization of modern physics is of greater importance, not even the principle of the conservation of energy, and no generalization of modern physics is based upon such deeply seated and such widely diffused human intuitions. It is the purpose of this article to give a sharp characterization to this widely diffused intuition.

Thermal Equilibrium

The most important single fact in connection with the study of the phenomena of heat is that a substance settles to a quiescent state in which there is no tendency to further change of any kind when it is left to itself and shielded from all outside disturbing influences. This quiescent state is called a state of thermal equilibrium. For example, the various objects in a closed room settle to thermal equilibrium; when a piece of red-hot iron is thrown into a pail of water, the mixture, at first turbulent, becomes more and more quiet and finally reaches a state of thermal equilibrium. A number of bodies which have settled to a common state of thermal equilibrium are said to have the same temperature. Thus a number of bodies left together in a closed room have the same temperature.

Atomics and Thermodynamics

In nearly every branch of physical science there are two more or less distinct modes of attack, namely, (a) a mode of attack in which the effort is made to develop conceptions of the physical processes of nature, and (b) a mode of attack in which the attempt is made to correlate phenomena on the basis of sensible things, things that can be seen and measured. In the theory of heat the first mode is represented by the application of the atomic theory to the study of heat phenomena, and the second mode is represented by what is called thermodynamics. In the first case one tries to imagine the nature of such processes as the melting of ice or the burning of coal, and in the second case one is content to measure the amount of heat absorbed or given off and to study the physical properties of the substance before and after the change has taken place.

The Atomic Theory of Heat.—The theory of heat properly includes the whole of chemistry, and every student of elementary chemistry is familiar with the use of the atomic theory in enabling one to form clear ideas of chemical processes. For example, the burning of hydrogen is thought of as the joining together of atoms of hydrogen and oxygen to form molecules of water vapor. The atomic theory is also of use in giving one clear ideas of the physical properties of substances. Thus, a gas is supposed to consist of a great number of particles in violent to-and-fro motion, and the gas exerts a pressure against the walls of the containing vessel because of the bombardment of the walls by the rapidly moving molecules of the gas. In addition to these two highly developed branches of the atomic theory (chemistry and the theory of gases), the atomic theory has been applied in a more or less vague but very useful way in the study of a great variety of heat phenomena as exemplified in the following quotation from Tyndall's "Heat a Mode of Motion."

When a hammer strikes a piece of lead the motion of the hammer appears to be entirely lost. Indeed in the early days it was supposed that what we now call the energy of the hammer was destroyed. But there is no loss. The motion of the massive hammer is transformed into molecular motion in the lead, and here our imagination must help us. In a solid body, although the force of cohesion holds the atoms together, the atoms are supposed, nevertheless, to vibrate within certain limits, and the greater the amount of mechanical action invested in the body by percussion, compression, or friction, the greater will be the rapidity and the wider the amplitude of the atomic oscillations.[2]

Thermodynamics.—To understand the essential features of the science of thermodynamics it is necessary to revert to the discussion of work and energy. Whenever a substance gives up energy which it has in store the substance always undergoes change. Thus, the fuel which supplies the energy to a steam engine and the food which supplies the energy to a horse undergo a chemical change; the steam which carries the energy of the fuel from a boiler to a steam engine cools off or undergoes a thermal change when it gives up its energy to the engine; a clock spring changes its shape as it gives up its energy in driving a clock; an elevated store of water changes its position as it gives its energy to a water wheel; the heavy flywheel of a steam engine does the work of the engine for a few moments after the steam is shut off and the fly-wheel changes its velocity as it gives up its energy.

In mechanics the theory of energy is discussed in connection with mechanical changes only, thermal and chemical changes being carefully ignored, and in taking up the study of thermal and chemical changes it is important to understand that our study is not to be concerned with thermal and chemical actions themselves, but with their results. The actions themselves are as a rule extremely complicated. Thus the details of behavior of the coal and air in a furnace are hopelessly complicated! The important practical thing,[3] however, is the amount of steam that can be produced by a pound of coal, and this depends upon (1) the condition of the water from which the steam is made, that is, whether the water is hot or cold to start with, (2) the condition of the air and of the coal which are to combine in the furnace, (3) the pressure and temperature of the steam which is to be produced, and (4) the condition of the flue gases as they enter the chimney. That is to say, the only things which it is necessary to consider are things which relate to quiescent substances. A quiescent substance may be said to be in a standing condition or state and the whole subject of heat (thermodynamics) may be said to refer to changes of state, that is, to changes from one quiescent condition to another quiescent condition without regard to the details of action which lead from one quiescent condition to the other.

Mechanical Energy and Heat Energy

In studying thermal and chemical changes we have to do with a new kind of energy. The gravitational energy of an elevated store of water can be wholly converted into mechanical work, the energy of two electrically charged bodies can be wholly converted into mechanical work (for example, by allowing the charged bodies to move towards each other), the kinetic energy of a moving car can be wholly converted into mechanical work, and so on. On the other hand, the energy of the hot steam which enters a steam engine from a boiler can not be wholly converted into mechanical work. Any store of energy which can be wholly converted into mechanical work may be called mechanical energy. The energy of the hot steam which enters a steam engine from a boiler is called heat energy.[4]

In the attempt to exclude all thermal changes from the purely mechanical discussion of energy one is confronted by the fact that friction (with its accompanying thermal changes) is always in evidence everywhere. In every actual case of motion the moving bodies are subject to friction and to collision, their energy is dissipated and they come to rest. This dissipation of energy is always accompanied by the generation of heat, and experience shows that the amount of heat generated is equivalent to the energy dissipated (first law of thermodynamics). It is important to understand that the term dissipation of energy refers to the conversion of mechanical energy into heat by friction or collision.[5] Thus energy is dissipated in the bearing of a rotating shaft, energy is dissipated when a hammer strikes a nail, and so on. The atomic theory enables one to form a clear idea of the dissipation of energy. Thus, the energy of the regular motion of a hammer is converted into energy of irregular molecular motion when the hammer strikes a nail.

It is worth while to give a statement of the first law of thermodynamics reduced to its simplest terms. A given substance is heated by the dissipation of work and brought back to its initial state by being cooled by contact with another (cooler) substance B. Then, if the loss of heat to surrounding bodies is carefully avoided, the thermal effect produced in substance B is exactly the same as would be produced in it if it had been heated directly by the dissipation of the original amount of work. Therefore a substance which is heated by the dissipation of work stores something which is equivalent to the work and which is called heat.

Gay Lussacs Law and the Air Thermometer.—When a number of closed vessels containing different gases at the same pressure are carried from a cool cellar, for example, to a warm room, they all suffer the same rise of temperature, and all of the gases show the same increase of pressure. That is to say, all gases follow the same law of increase of pressure with increase of temperature, the volumes of the containing vessels being constant. This fact was discovered by Gay Lussac and it is called Gay Lussac's Law. This law affords a convenient basis for the definition of temperature ratios, convenient because not dependent upon any particular gas. The ratio of two temperatures (provisionally defined) is the ratio of the pressures of a constant volume of gas at the respective temperatures. That is, if p and p′ are pressures of a constant volume of a gas at temperatures T and T′, respectively, then we have by definition

 ${\displaystyle T/T\prime =p/p\prime }$ (1)

The air thermometer is a device for measuring the ratio of two temperatures by observing the pressures of a constant volume of dry air at the respective temperatures.

Simple Knowledge Assumed

A great deal of simple every-day knowledge is always taken for granted in a treatise on thermodynamics. It is stated above that the important things in connection with the generation of steam in a boiler by the burning of coal are (a) the temperature of the feed water, (b) the temperature and pressure of the steam which is produced, (c) the character of the coal, (d) the temperature and composition of the air, and (e) the temperature and composition of the flue gases. In a certain sense this is true, but of course the fundamentally important thing is the knowledge that coal will burn and convert water into steam. Such fundamental knowledge is always taken for granted in the study of thermodynamics. The nature of fire is not an object of study in thermodynamics, but every one knows what fire is in a simple practical way; every one knows that an object becomes hot when it is placed upon a hot stove; and every one knows that steam will squirt out of a hole in a steam boiler under pressure. In the experience of the writer only one case has ever come to notice in which this kind of fundamental knowledge seemed to be lacking. A student was asked to define what is meant by the heat of combustion of coal, and he gave it correctly up to a certain point by saying that it was the number of thermal units generated by one pound of coal; it was, however, impossible to lead the young man by indirect suggestion to add the important qualifying phrase "when the coal is burned," and upon being asked explicitly how one gets heat out of coal, the young man actually replied, with some embarrassment, "Why, Professor, I don't know." Of course, he did know, but apparently he could only think that the study of thermodynamics must refer to unfamiliar and elegant things. No, thermodynamics refers to the things of the kitchen and to the things of the furnace, although the science of thermodynamics is so organized that it talks only of the things that go into and of the things which come out of those mysterious places where maids and furnace-men rule.

Limitations of Mechanics

The science of mechanics applies to the more or less ideal phenomena which are associated with the motion of rigid bodies either singly or in connected machines; with the regular motion of distortion of elastic bodies like the bending of a bow or the oscillation of a string; and with ideally simple motion of flow of liquids and gases like the smooth flow of water from an orifice in a tank. In every actual case of motion, however, we always encounter turbulence more or less marked, and the science of mechanics, which is the science of describing the phenomena of motion, fails utterly if we attempt to consider the minute details of the phenomena of motion which are involved in this turbulence. Let one, for example, watch the movement of the water at a point in a brook. There is indeed a fairly steady average velocity of the water at the point and a certain mean rhythmic variation, but superposed upon this average motion there is an erratic variation of velocity which is infinitely manifold, the details of which are beyond the scope of any descriptive science. A descriptive science like mechanics is concerned with how things progress as a phenomenon develops itself; how the structural parts of a bridge stretch and shorten as a car passes across the bridge; how the pressure and temperature of steam vary during the successive stages of admission, expansion and exhaust of a steam engine; how electro-motive force, current strength, electro-magnetic force and all of the changing variables play in the operation of a dynamo. But who could recite the story of the most minute details of these phenomena? It can not be done, and if it could be done, it would be of no avail, for these details can never be twice alike and the very essence of a science lies in the discovery of types which in their important features recur so that a knowledge of these types may serve as a basis for anticipation and design.

Fire is the most familiar example of a turbulent phenomenon, and its most striking characteristic is that its progress is not dependent upon any external driving cause; when once started it goes forward of itself and with a rush. Tyndall in referring to this matter says that to account for the propagation of fire was one of the philosophical difficulties of the eighteenth century. A spark was found sufficient to initiate a conflagration, and the effect seemed beyond all proportion greater than the cause; herein lay the philosophical difficulty. Indeed the simple idea of cause and effect is not applicable to physical phenomena involving turbulence. Every physical phenomenon involving turbulence is to some extent self-sustaining, and every such phenomenon has a certain impetuous quality which may carry it beyond anything that is commensurate with the original initiating cause. These remarkable characteristics of turbulence are now definitely formulated as the second law of thermodynamics.

The most important practical thing in connection with the turbulent aspect of any physical phenomenon is its general result or consequence, just as the important thing about the burning of a house is the loss. How utterly useless and uninteresting it would be, for example, to study the minutest details of a conflagration (assuming such study to be possible), recording the height and breadth and the irregular and evanescent distribution of temperature throughout each flicker of consuming flame, the story of each crackling sound and the extent and character of every yield and sway of timber and wall! The fact is that we are immersed in an illimitable sea of phenomena, every single detail of which is infinitely manifold, and no completely adequate science can ever be developed.

Physical science, then, aside from those branches which are dependent upon the atomic theory, consists of three branches, namely, (1) mechanics, including hydraulics, electricity and magnetism, light and sound; the science of those phenomena in which turbulence may for practical purposes be ignored; (2) statistical physics, the science of those phenomena in which turbulence introduces an appreciable and practically important erratic element. Such phenomena can be studied only by the statistical method, the record of individual cases and the study of averages. Meteorology is the best example of statistical physics, although every physical phenomenon has its statistical aspect; and (3) thermodynamics. Some of the features of thermodynamics have already been pointed out. It is the study of changes of state of substances. A most important aspect of thermodynamics remains, however, to be considered and the preliminary idea of this new aspect may be obtained by drawing a parallel. In every-day life we see the fire-insurance companies concerned with certain broad features of statistical physics in their examinations and records of fires, and we see them also concerned with a profit and loss account which is wholly abstracted from the details of the phenomena of conflagration. Thermodynamics is the profit and loss branch of physics as it were; and like the profit and loss branch of fire insurance, thermodynamics is completely abstracted from any consideration of the details of any physical phenomenon. Thermodynamics is concerned with the measurement and counting of that type of physical degeneration which accompanies turbulence just as fire insurance is concerned with the estimate and counting of what we might call, using a fine phrase, structural degeneration by fire.

Thermodynamic Degeneraton

Every one has a feeling of the irretrievable effects of disaster, the collapse of a bridge, the destruction of a house by fire, or the wreck of a ship; these things involve losses which indeed may be forgotten after reconstruction, but never balanced. The havoc wrought is essentially irreparable. It is desirable to use the word degeneration in a very narrow technical sense when we come to consider the second law of thermodynamics, and the way may be paved to a clear understanding of the later and accepted use of this word in physics by applying it now to designate that aspect of disaster which is irreparable. The burning of a building, for example, is a process of degeneration. It is very important, however, to avoid the carrying over of this idea of structural degeneration into thermodynamics, where a much more limited conception of degeneration arises. The term thermodynamic degeneration applies to the effects of turbulence which always plays a certain havoc in a system. Thus a certain degeneration is associated with the turbulence which is produced when a hot iron is dipped into water, a certain degeneration is associated with the escape of a compressed gas through an orifice, a certain degeneration is associated with the flow of heat from a region of high temperature to a region of low temperature, a certain degeneration is associated with the conversion of work into heat by the rubbing of a coin on a board, and so on.

Reversible Processes

A substance in thermal equilibrium is generally under the influence of external agencies. Thus surrounding substances confine a given substance to a certain region of space, and they exert upon the given substance a definite constant pressure; surrounding substances are at the same temperature as the given substance and according to the atomic theory the molecules of the given substance rebound from the surrounding substance with their motion on the average unchanged; surrounding substances may exert constant magnetic or electric influences on the given substance; and so on. If the external influences which act upon a fluid in thermal equilibrium are made to change very slowly, causing the pressure, volume and temperature of the fluid to pass very slowly through a continuous series of values and in general involving the doing of work upon or by the fluid and the giving of heat to or taking of heat from the fluid, then the fluid will pass slowly through a process consisting of a continuous series of states of thermal equilibrium. Such a process is called a reversible process, for the reason that the fluid will pass through the same series of states in reverse order if the external influences are changed slowly in reverse sense.

Irreversible Processes

When a substance is settling or tending to settle to thermal equilibrium it may be said to undergo a process. Such a process can not be arrested or held at any stage short of complete thermal equilibrium, but it always and inevitably proceeds towards that state. Such a process may, therefore, be called a sweeping process or simply a sweep. The settling of a closed system to thermal equilibrium may be called a simple sweep. For example, the equilibrium of a mixture of oxygen and hydrogen in a closed vessel may be disturbed by a minute spark, and the explosion of the gases together with the subsequent settling of the water vapor to a quiescent state constitutes a simple sweep. The equilibrium of a gas confined under high pressure in one half of a two-chambered vessel may be disturbed by opening a cock which connects the two chambers, and the rush of gas into the empty chamber constitutes a simple sweep. When external influences change continuously a substance in its tendency to settle to thermal equilibrium never catches up with the changing conditions, but trails along behind them, and we have what may be called a trailing sweep. Thus the rapid heating or cooling of a gas in a vessel is a trailing sweep. So long as heat is given to or taken from the gas at a perceptible rate there will be perceptible differences of temperature in different parts of the gas; and the gas in its tendency to settle to thermal equilibrium never catches up with the increasing or decreasing temperature of the walls of the containing vessel.

A substance may be subjected to external action which, although permanent or unvarying, is incompatible with thermal equilibrium. When such is the case the substance settles to a permanent or unvarying state which is not a state of thermal equilibrium. Such a state of a substance may be called a steady sweep. For example, the two faces of a slab or the two ends of a wire may be kept permanently at different temperatures, and when this is done the slab or wire settles to an unvarying state which is by no means a state of thermal equilibrium. Heat flows through the slab or along the wire from the region of high temperature to the region of low temperature. This flow of heat through the slab or along the wire is an irreversible process and it constitutes a steady sweep. The ends of a wire may be connected to a battery or dynamo so that a constant electric current flows through the wire and the heat which is generated in the wire by the current may be steadily carried away by a stream of water or air. Under these conditions the wire settles to an unvarying state which is by no means a state of thermal equilibrium, the battery or dynamo does work on the wire and this work reappears steadily as heat in the wire.

Thermodynamic Degeneration

Every one must admit that the impetuous character of a sweeping process suggests a certain havoc, a certain degeneration in the substance or system in which the sweep takes place. Consider, for example, a charge of gun-powder which has been exploded; if it is exploded in a large empty vessel, everything is there after the explosion, all of the energy is there and all of the material substance is there, but it can not be exploded a second time! The man on the street has heard much during recent years of the conservation of energy and of the conservation of matter, and the old proverb that "you can't eat your cake and have it" presents to his mind a question which in its less familiar forms, as relating to engines for example, he tries in vain to rationalize in terms of these principles of conservation. Nearly all of the intuitive sense of the man on the street concerning these matters (and he has a great deal) is involved in the second law of thermodynamics, which is not a law of conservation at all. It is a law of waste.

At this point of our discussion it is necessary to use the word degeneration so as to express more or less tentatively the idea that every sweeping process brings about a definite amount of degeneration, an amount that can be expressed numerically just as one speaks of so many pounds of sugar or so many yards of cloth. Thus a certain amount of degeneration is brought about when a compressed gas escapes through an orifice, a certain amount of degeneration is brought about when heat flows from a region of high temperature to a region of low temperature, a certain amount of degeneration is brought about when work is converted into heat by friction or by the flow of an electric current through a wire, and so on.

In a simple sweep the degeneration lies wholly in the relation between the initial and final states of the substance. This is necessarily the case because no outside substance is affected in any way by the sweep, no work is done on or by the substance which undergoes the sweep and no heat is given to or taken from it. In a trailing sweep the degeneration may lie partly in the relation between the initial and final states of the substance which undergoes the sweep, partly in the conversion of work into heat, and partly in the flow of heat from a high temperature region to a low temperature region. In a steady sweep, however, the substance which undergoes the sweep remains entirely unchanged as the sweep progresses, and the degeneration lies wholly in the conversion of work into heat, in the transfer of heat from a region of high temperature to a region of low temperature, or in both. Therefore the idea of thermodynamic degeneration as a measurable quantity can be reached in the simplest possible manner by a careful scrutiny of a steady sweep.

Proposition (a).—The thermodynamic degeneration which is represented by the direct conversion of work into heat at a given temperature is proportional to the quantity of work so converted. Consider, for example, a steady flow of electric current through a wire from which the heat is abstracted so that the temperature remains constant. This process is steady, that is to say, it remains unchanged during successive intervals of time, and therefore any result of the process must be proportional to the time which elapses, that is to say, the amount of degeneration occurring in a given interval of time is proportional to the time, but the amount of work which is degenerated into heat is also proportional to the time. Therefore the amount of degeneration is proportional to the amount of work converted into heat at the given temperature.

Proposition (b).—The thermodynamic degeneration which is represented by the transfer of heat from a given high temperature T1 to a given low temperature T2 is proportional to the quantity of heat transferred. Consider a steady flow of heat from temperature T1 to temperature T2 constituting a steady sweep, a sweep which remains entirely unchanged in character in successive intervals of time. Any result of this sweep must be proportional to the lapse of time, and therefore the degeneration which takes place in a given interval of time is proportional to the time; but the quantity of heat transferred is also proportional to the time, therefore, the amount of degeneration is proportional to the quantity of heat transferred from temperature T1 to temperature T2.

Kelvin's Definition of Temperature Ratio

The definition of the ratio of two temperatures previously given was understood to be a provisional definition. We are now in a position to propose a definition of the ratio of two temperatures which is independent of the physical properties of any particular substance. This definition will remain somewhat vague, however, until the action of the steam engine is discussed in the later sections of this article. According to proposition (a) above, the thermodynamic degeneration which is involved in the conversion of work into heat at a given temperature is proportional to the amount of work so converted and the proportionality factor depends upon the temperature only. Therefore, we may write

 ${\displaystyle \phi ''=m_{1}W}$ (2) and ${\displaystyle \phi ''=m_{2}W}$ (3)

where ${\displaystyle \phi }$ is the degeneration involved in the conversion of an amount of work W into heat at temperature T1 and ${\displaystyle \phi ''}$ is the degeneration involved in the conversion of an amount of work W into heat at temperature T2, and m1 and m2 are factors which depend only upon T1 and T2, respectively. The amount of work W having been converted into heat at temperature T1; imagine the heat to flow to a lower temperature T2, thus involving an additional amount of degeneration according to proposition (b) above. The conversion of work W into heat at temperature T2 and the subsequent flow of this heat to a lower temperature T2 gives the same result as would be produced by the conversion of the work into heat at the lower temperature directly. Therefore the lower the temperature at which work is converted into heat the greater the amount of degeneration involved. That is to say, the factor m2 in equation (3) is larger in value than the factor m1 in equation (2), temperature ${\displaystyle T_{1}}$ being higher[6] than temperature ${\displaystyle T_{2}}$. Therefore, since ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ depend only upon ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$, respectively, it is permissible to adopt the equation

 ${\displaystyle T_{1}/T_{2}=m_{2}/m_{1}T}$ (4)

as the definition of the ratio ${\displaystyle T_{1}/T_{2}}$. This definition of temperature ratios is originally due to Lord Kelvin.

Another way to express the definition which is involved in equation (4) is to consider that the factor ${\displaystyle m_{1}}$ is the smaller the higher the temperature ${\displaystyle T_{1}}$ so that we may adopt ${\displaystyle k/m_{1}}$ as the measure of the temperature ${\displaystyle T_{1}}$ and ${\displaystyle k/m_{2}}$ as the measure of the temperature ${\displaystyle T_{2}}$, giving

 ${\displaystyle m_{1}=k/T_{1}}$ (5) and ${\displaystyle m_{2}=k/T_{2}}$ (6)

where ${\displaystyle \phi }$ is an indeterminate constant. Therefore equations (2) and (3) may be written in the general form

 ${\displaystyle \phi =kW/T}$ (7)

where ${\displaystyle \phi }$ is the thermodynamic degeneration involved in the conversion of an amount of work W into heat at temperature T, and k is an indeterminate constant.

The ratio of two temperatures as defined by equation (4) is very nearly the same as the ratio of two temperatures as measured by the gas thermometer, and therefore gas thermometer temperatures may be used throughout this discussion without appreciable error.[7]

Since the factor k in equation (7) is indeterminate, we may use as our unit of thermodynamic degeneration the amount which is involved in the conversion of one unit of work into heat at a temperature of one degree on the "absolute" scale; then the value of k is unity and equation (7) becomes

 ${\displaystyle \phi =W/T}$ (8)

in which ${\displaystyle W}$ is expressed in joules and ${\displaystyle T}$ in degrees centigrade; and ${\displaystyle \phi }$ is expressed in terms of joules per degree. Thus one joule per degree is the degeneration involved in the conversion of one joule of work into heat at 1° C. on the absolute scale, or the amount involved in the conversion of 1,000 joules into heat at 1,000° C. on the absolute scale.

To convert an amount of work ${\displaystyle W}$ into heat at temperature ${\displaystyle T_{1}}$ involves ${\displaystyle W/T_{1}}$ units of degeneration, to convert the same amount of work into heat at temperature ${\displaystyle T_{2}}$ involves ${\displaystyle W/T_{2}}$ units of degeneration, and therefore to transfer an amount of heat equal to ${\displaystyle W}$ from temperature ${\displaystyle T_{1}}$ to temperature ${\displaystyle T_{2}}$ must involve an amount of degeneration equal to the excess of ${\displaystyle W/T_{2}}$ over ${\displaystyle W/T_{1}}$ or an amount equal to ${\displaystyle W(1/T_{2}-1/T_{1})}$, or ${\displaystyle H(1/T_{2}-1/T_{1})}$, where ${\displaystyle H}$ is the amount of heat transferred.

The Second Law of Thermodynamics

(a) The thermodynamic degeneration which accompanies a sweeping or irreversible process can not be directly repaired, nor can it be repaired by any means without compensation.

This is an entirely general statement of the second law of thermodynamics. The direct repair of the degeneration due to the sweeping process means the undoing of the havoc wrought by the process by allowing the sweep to perform itself backwards, an idea which is exactly as absurd as the idea of allowing a burned house to unburn itself. Following are several specialized statements of the second law of thermodynamics.

(b) Heat can not pass directly from a cold body to a hot body, nor can heat be transferred from a cold body to a hot body by any means without compensation.

(c) Heat can not be converted directly into work, nor can heat be converted into work by any means without compensation.

The direct conversion of heat into work would be the simple reverse of any of the ordinary sweeping processes which involve the degeneration of work into heat, that is, the direct conversion of work into heat would be to allow the sweeping process to perform itself backwards. For example, work is degenerated into heat in the bearing of a rotating shaft, and we all know that to reverse the motion of the shaft does not cause the bearing to grow cold and the heat so lost to appear as work helping to drive the shaft. That would be a rotary engine indeed! There is an important general theorem in thermodynamics to the effect that if two sweeping processes ${\displaystyle A}$ and ${\displaystyle B}$ involve the same amount of degeneration, and if either of the processes, say ${\displaystyle A}$, has been allowed to perform its sweep, then by a lever arrangement, as it were, the process ${\displaystyle B}$ can be carefully let down, and the havoc wrought by the sweep of ${\displaystyle A}$ can be undone. The result of this operation, however, would be to leave the system ${\displaystyle B}$ in the condition in which it would be degenerated if the process ${\displaystyle B}$ had been allowed to sweep instead of being let down. This is very much as if, having two similar houses ${\displaystyle A}$ and ${\displaystyle B}$, one of which ${\displaystyle A}$ has been burned, we could rig up a mechanism which would let down ${\displaystyle B}$ to ashes and cause ${\displaystyle A}$ to be restored in the original actual materials of which it was first constructed. This is of course impossible in the case of the two houses, but it is possible in every known case of thermodynamic degeneration. This general theorem is as thoroughly established as any generalization in physics, and if it is true and if we ever find a way to convert heat into work unconditionally and without cost or compensation, then it will be proved indirectly that a shaft can be driven by heating one of the bearings in which it rotates, for direct conversion of work into heat by one process must be according to this general theorem equivalent to and replaceable by the reverse of any ordinary sweeping process which converts work into heat.

(d) A gas cannot pass directly from a region of low pressure into a region of high pressure, nor can a gas be transferred from a region of low pressure to a region of high pressure by any means without compensation.

Imagine a gas squirting itself backwards through a nozzle into a high-pressure reservoir! The repeated statement of self-evident facts concerning direct repair in these statements of the second law of thermodynamics may seem ridiculous to the intelligent reader, but the second law of thermodynamics is a statement of a fact which every one knows coupled with a generalizing clause which, once thoroughly understood, is almost if not quite self-evident.

Here is one more statement of the second law of thermodynamics, the oldest English version of it:

Humpty Dumpty sat on a wall,
Humpty Dumpty had a great fall,
All the king's horses and all the king's men
Can not put Humpty Dumpty together again.

This is perhaps the most sensible of all the statements of the second law, for which we will allow it to pass for the moment, inasmuch as it ignores direct repair and refers at once to the most powerful of external means. It is important, however, to remember that in Humpty Dumpty's case we are concerned with structural degeneration, not with the much simpler kind of degeneration in a structureless fluid due to turbulence.

Of all the generalizations of physics, the second law of thermodynamics is certainly the most deeply seated in the common sense of men, and one of the most humorous of children's verses refers to the man whose wondrous wisdom enabled him to circumvent it by direct repair:

There was a man in our town,
And he was wondrous wise;
He jumped into a bramble bush
And scratched out both his eyes.
And when he found his eyes were out,
With all his might and main,
He jumped into another bush
And scratched them in again.

Let us return to the fourth statement (d) and consider with the help of an example what is meant by compensation in its thermodynamic sense. A gas can be transferred from a region of low pressure to a region of high pressure by means of a pump, and the work that is done in driving the pump, even supposing the pump to be frictionless, is all converted into heat. This conversion of work into heat is the necessary cost or compensation for the transfer of the gas from the low pressure region to a high pressure region.

Consider the second statement (b). In an artificial ice factory heat is continually abstracted from the freezing-room and transferred to the warm outside air; but to accomplish this result, even by an ideally perfect frictionless mechanism, a certain amount of work is required to drive the ammonia pump and this work is converted into heat. This conversion of work into heat compensates for the transfer of heat from the freezing-room to the warm region outside.

Consider the third statement (c). In ordinary steam engines, heat is converted into work, but to accomplish this transformation a large quantity of heat must be supplied to the engine at high temperature, and some of this heat (about nine tenths of it in the very best of steam engines) must be let down, as it were, to the low temperature of the exhaust to compensate for the conversion of the remainder into work.

Heat Engines

An engine, or to be more specific, a heat engine is a machine for converting heat into mechanical work. The engine is supplied with heat at a high temperature, it transforms a portion of this heat into work, and it delivers the remainder of the heat to a low temperature region. Figure 1 is a diagram for fixing in the reader's mind the various temperatures and quantities of heat and work which are involved in the operation of an engine. The boiler is at high temperature ${\displaystyle T_{1}}$ and the condenser is at low temperature ${\displaystyle T_{2}}$. When the engine is operated for a time, a quantity of heat H r is taken from the boiler, an amount of heat ${\displaystyle W}$ is developed by the engine (in excess of the work required to drive the feed water pump), and the quantity of heat ${\displaystyle H_{2}}$ is delivered to the condenser.

According to the first law of thermodynamics, the work ${\displaystyle W}$ must be equal to ${\displaystyle H_{1}-H_{2}}$, both quantities of heat being expressed in energy units. Therefore

 ${\displaystyle W=H_{1}-H_{2}}$ (9)

As far as the net result is concerned the operation of the steam engine may be thought of as (a) the conversion into work of the whole of the heat H1 from temperature T1, and (b) the reconversion of a

Fig. 1.

portion ${\displaystyle H_{2}}$ of this work into heat at temperature ${\displaystyle T_{2}}$. The regeneration[8] associated with process (a) is equal to ${\displaystyle H_{1}/T_{1}}$ according to equation (8), and the degeneration associated with process (b) is equal to ${\displaystyle H_{2}/T_{2}}$ according to equation (8). If the operation of the engine involves sweeping processes, then the degeneration ${\displaystyle H_{2}/T_{2}}$ must exceed the regeneration ${\displaystyle H_{1}/T_{1}}$, that is, we must have

 style="text-align:center;"|${\displaystyle H_{2}/T_{2}>H_{1}/T_{1}}$ (10)

or, substituting the value of ${\displaystyle H_{2}}$ from equation (9) and solving for ${\displaystyle W}$, we have

 ${\displaystyle W<{\frac {T_{1}-T_{2}}{T_{1}}}H_{1}}$ (11)

The fractional part ${\displaystyle [(T_{1}-T_{2})/T_{1}]}$ of the heat ${\displaystyle H_{1}}$ which is converted into work by the engine is called the efficiency of the engine, and the inequality (11) shows that the efficiency of any engine working between temperatures ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ must be less than ${\displaystyle [(T_{1}-T_{2})/T_{1}]}$ whatever the nature of the working fluid and whatever the design of the engine.

The Perfect Engine.—An engine involving no irreversible or sweeping processes in its operation would be called a perfect engine. In such an engine the degeneration ${\displaystyle H_{2}/T_{2}}$ above mentioned would be equal to the regeneration ${\displaystyle H_{1}/T_{1}}$ so that for a perfect engine we should have

 ${\displaystyle H_{2}/T_{2}=H_{1}/T_{1}}$ (12) or ${\displaystyle T_{1}/T_{2}=H_{1}/H_{2}}$ (13)

Substituting the value of ${\displaystyle H_{2}}$ from equation (9) in equation (12) or (13), and solving for ${\displaystyle W}$, we have

 ${\displaystyle W={\frac {T_{1}-T_{2}}{T}}_{1}H_{1}}$}} (14)

This equation shows that the efficiency of any perfect engine working between the temperatures ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ would be equal to ${\displaystyle [(T_{1}-T_{2})/T_{1}]}$.

Lord Kelvin's definition of the ratio of two temperatures may be understood with the help of equation (13) in which ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ are the amounts of heat taken in and given out by a perfect engine during a given time, and ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are the temperatures between which the engine is working.

Efficiencies of Engines in Practise.—A fraction of the heat which is delivered to an engine with the steam which drives the engine is converted into work. In order that this fraction may be large, the ratio ${\displaystyle T_{1}/T_{2}}$ must be as large as possible, and sweeping processes must be obviated as much as possible in the operation of the engine; ${\displaystyle T_{1}}$ being the temperature of the steam supplied to the engine, and ${\displaystyle T_{2}}$ being the temperature of the exhaust. The ratio of the initial temperature to the final temperature of the expansion steam or gas in an engine depends upon the ratio of the initial volume to the final volume of the steam or gas.

In order that moderately small cylinders[9] may be used for the development of a given amount of power, the initial pressure of the steam or gas must be high; and in order that the final temperature may not be lower than atmospheric or available condenser water temperatures, the initial temperature must be high. The first point concerning high initial pressures is exemplified in the operation of the ordinary gas or gasoline engine in which the mixed charge of gas and air is highly compressed before it is exploded.

In the gas engine the initial temperature is the temperature of the mixture of air and gas immediately after the explosion and it may be from 1,200° to 1,700° C. on the absolute scale; and the temperature is reduced by expansion to perhaps half this value. The temperature and pressure of the steam which is supplied to a steam engine is seldom higher than 190° C. (463° on the absolute scale) and about 175 pounds per square inch, respectively. Any higher temperature and pressure involves a great deal of danger in the boiler. The lowest condenser temperature in steam engine practise is about 50° C. (323° on the absolute scale).

The sweeping processes which take place in a steam engine are as follows:

(a) Friction between the moving parts of the engine. This friction results in the immediate reconversion into heat of a portion of the mechanical energy developed by the engine.

(b) Wire drawing. If the pipes and passages traversed by the steam through the boiler to the engine are small, the pressure in the cylinder with open ports will be lower than boiler pressure, so that the entering steam passes from a region of high pressure into a region of low pressure. Also as the cut-off valve closes, steam will rush into the cylinder through a narrowing aperture. This effect is called wire drawing, and to provide against loss of efficiency from this cause the pipes must be of ample size and the cut-off valve must operate very quickly.

(c) Radiation. The cooling of pipes and cylinders by the giving of heat to surrounding cooler bodies is a sweeping process, and is to be obviated as much as possible by covering the pipes and the cylinder with a thick coating of porous insulating material.

(d) Cylinder condensation. As a charge of steam in the cylinder expands it cools and cools the cylinder and piston, so that when steam is next admitted it heats cylinder and piston up again and is itself cooled. This effect can not be eliminated, but it can be largely reduced by providing separate passages for the ingress and egress of steam; and by using a series of cylinders of increasing size of which the smallest cylinder takes steam directly from the boiler and exhausts into the next large cylinder, which in turn exhausts into a still larger cylinder, and so on. In this way the range of temperature in each cylinder is small and the effects of cylinder condensation are greatly reduced. A steam engine in which expansion of the steam takes place in two stages (in two cylinders) is called a compound engine. An engine in which the expansion of the steam takes place in three stages (in three cylinders) is called a triple expansion engine.

The loss of efficiency due to cylinder condensation is greatly reduced by the use of superheated steam because the exchange of heat between the steam and the cylinder walls is very greatly reduced when the steam does not condense. Thus, S. LeRoy Brown has found that heat is imparted to a metal surface about twenty-four times as fast by condensing steam than by a gas at the same temperature.

(e) Effect of high piston velocity. If the piston speed is too great, the pressure of the expanding steam becomes ineffective because the portions of the steam near the moving piston are expanded and cooled before the remote parts of the steam are affected. This effect is negligible at the highest piston velocities which are mechanically feasible.

(f) Puffing. When the steam at the end of a stroke is still at a pressure which exceeds the pressure in the condenser (or which exceeds the pressure of the outside air when no condenser is used), it rushes through the exhaust port as a sharp puff. Puffing is to be avoided by sufficiently reducing the steam pressure by expansion in the cylinder.

The greatest items of waste in the ordinary sense of actual loss of heat are (a) the incomplete combustion of the fuel and (b) the carrying away of great quantities of heat in the flue gases. The economic use of fuel for the production of mechanical power requires, therefore, a properly designed furnace and intelligent and careful stoking to insure complete combustion, and it requires a sufficient exposure of boiler surface and frequent cleaning of the same to facilitate the flow of heat from the hot gases into the boiler.

The most pronounced sweeping process which intervenes between the completed combustion and the final exhaust of the steam is the flow of heat from the very high temperature of the fire in the furnace to' the moderately low temperature of the water in the boiler, and the greatest waste in the operation of the steam engine in the sense of loss of availability of heat for conversion into work is involved in this sweeping process, and it can hardly be avoided in the steam engine because of the danger involved in the generation of steam at very high pressure in a large boiler.

The best gas engines convert about 30 per cent, of the heat of the fuel into mechanical work. The best steam engines convert about 10 or 12 per cent, of the heat of the fuel into mechanical work. The ordinary run of steam engines convert only 4 or 5 per cent, of the heat of the fuel into mechanical work.

8. To convert an amount of work ${\displaystyle W}$ into heat at a given temperature involves an amount of degeneration, and to convert the heat into work involves the same amount of what may be called thermodynamic regeneration.