Energy and Entropy

Abstract

In his Sixth Memoir on The Mechanical Theory of Heat, Clausius explained how the internal energy of a body ($U$), consists of both thermal content ($H$) and ergonal content ($Z$). These represent the energy associated with the motion and the configuration of the body’s particles, respectively. According to the first fundamental theorem of the mechanical theory of heat—now known as the first law of thermodynamics—the internal energy of a body may be changed by either adding heat ($Q$) to the body, or doing external work ($w$) on the body. But the first law of thermodynamics alone is not sufficient to explain the types of processes which tend to occur in nature. Rather, only those processes, or transformations, occur which are characterized by positive (or at best, zero) equivalence-values; any process which has a negative equivalence value must be compensated by another process having an equal or greater positive equivalence value. This is Clausius’ second fundamental theorem of the mechanical theory of heat. Today, it is known as the second law of thermodynamics. In the reading selection below, taken from his Ninth Memoir, Clausius clarifies these ideas by writing them in a succinct mathematical form. Perhaps most notably, he introduces the concept of entropy, denoted by the letter $S$. What is meant by this term? From where was it derived? And what is the connection between entropy and the second law of thermodynamics?

The entropy of the universe tends to a maximum.

—Rudolph Clausius

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9.1 Introduction

In his Sixth Memoir on The Mechanical Theory of Heat, Clausius explained how the internal energy of a body ($U$), consists of both thermal content ($H$) and ergonal content ($Z$). These represent the energy associated with the motion and the configuration of the body’s particles, respectively. According to the first fundamental theorem of the mechanical theory of heat—now known as the first law of thermodynamics—the internal energy of a body may be changed by either adding heat ($Q$) to the body, or doing external work ($w$) on the body. But the first law of thermodynamics alone is not sufficient to explain the types of processes which tend to occur in nature. Rather, only those processes, or transformations, occur which are characterized by positive (or at best, zero) equivalence-values; any process which has a negative equivalence value must be compensated by another process having an equal or greater positive equivalence value. This is Clausius’ second fundamental theorem of the mechanical theory of heat. Today, it is known as the second law of thermodynamics. In the reading selection below, taken from his Ninth Memoir, Clausius clarifies these ideas by writing them in a succinct mathematical form. Perhaps most notably, he introduces the concept of entropy, denoted by the letter $S$. What is meant by this term? From where was it derived? And what is the connection between entropy and the second law of thermodynamics?

9.2 Reading: Clausius, The Mechanical Theory of Heat

Clausius, R., Mechanical Theory of Heat, with its Applications to the Steam-Engine and to the Physical Properties of Bodies, John Van Voorst, London, 1867. Ninth Memoir.

In my former Memoirs on the Mechanical Theory of Heat, my chief object was to secure a firm basis for the theory, and I especially endeavoured to bring the second fundamental theorem, which is much more difficult to understand than the first, to its simplest and at the same time most general form, and to prove the necessary truth thereof. I have pursued special applications so far only as they appeared to me to be either appropriate as examples elucidating the exposition, or to be of some particular interest in practice.

The more the mechanical theory of heat is acknowledged to be correct in its principles, the more frequently endeavours are made in physical and mechanical circles to apply it to different kinds of phenomena, and as the corresponding differential equations must be somewhat differently treated from the ordinarily occurring differential equations of similar forms, difficulties of calculation are frequently encountered which retard progress and occasion errors. Under these circumstances I believe I shall render a service to physicists and mechanicians by bringing the fundamental equations of the mechanical theory of heat from their most general forms to others which, corresponding to special suppositions and being susceptible of direct application to different particular cases, are accordingly more convenient for use.

• (1) The whole mechanical theory of heat rests on two fundamental theorems,—that of the equivalence of heat and work, and that of the equivalence of transformations.

  In order to express the first theorem analytically, let us contemplate any body which changes its condition, and consider the quantity of heat which must be imparted to it during the change. If we denote this quantity of heat by $Q$, a quantity of heat given off by the body being reckoned as a negative quantity of heat absorbed, then the following equation holds for the element $dQ$ of heat absorbed during an infinitesimal change of condition,

  $$ dQ = dU + A\, dW $$

  (9.1)

  Here $U$ denotes the magnitude which I first introduced into the theory of heat in my memoir of 1850, and defined as the sum of the free heat present in the body, and of that consumed by interior work.¹ Since then, however, W. Thomson has proposed the term energy of the body for this magnitude,² which mode of designation I have adopted as one very appropriately chosen; nevertheless, in all cases where the two elements comprised in $U$ require to be separately indicated, we may also retain the phrase thermal and ergonal content, which, as already explained on p. 5, expresses my original definition of $U$ in a rather simpler manner. $w$ denotes the exterior work done during the change of condition of the body, and $A$ the quantity of heat equivalent to the unit of work, or more briefly, the thermal equivalent of work. According to this $AW$ is the exterior work expressed in thermal units, or according to a more convenient terminology recently proposed by me, the exterior ergo. (See Appendix A. to Sixth Memoir.)

  If for the sake of brevity, we denote the exterior ergon by a simple letter,

  $$ w = AW $$

  we can write the foregoing equation as follows,

  $$ dQ = dU + dw. $$

  (9.3)

  In order to express analytically the second fundamental theorem in the simplest manner, let us assume that the changes which the body suffers constitute a cyclical process, whereby the body returns finally to its initial condition. By $dQ$ we will again understand an element of heat absorbed, and $T$ shall denote the temperature, counted from the absolute zero, which the body has at the moment of absorption, or, if different parts of the body have different temperatures, the temperature of the part which absorbs the heat element $dQ$. If we divide the thermal element by the corresponding absolute temperature and integrate the resulting differential expression over the whole cyclical process, then for the integral so formed the relation

  $$\int \frac{dQ}{T} \leq 0$$

  (9.4)

  holds, in which the sign of equality is to be used in cases where all changes of which the cyclical process consists are reversible, whilst the sign $<$ applies to cases where the changes occur in a non-reversible manner.³

• (2) We will first consider more closely the magnitudes occurring in Eq. (9.3) in reference to different kinds of changes of the body.

  The exterior ergon $w$, which is produced whilst the body passes from a given initial condition to another definite one, depends not merely on the initial and final conditions, but also on the nature of the transition.

  In the first place, we have to consider the exterior forces which act on the body, and which are either overcome by, or overcome the forces of the body itself;—the exterior ergon being positive in the former, and negative in the latter case. The question then arises, are these exterior forces, at each moment, the same as, or different from the forces of the body? Now although we may assert that for one force to overcome another, the former must necessarily be the greater; yet since the difference between them may be as small as we please, we may consider the case where absolute equality exists as the limiting case, which, although never reached in reality, must be theoretically considered as possible. When force and counter-force are different, the mode in which the change occurs is not a reversible one.

  In the second place, the change taking place in a reversible manner, the exterior ergon likewise depends upon the intermediate conditions through which the body passes when changing from the initial to the final condition, or, as it may be figuratively expressed, upon the path which the body pursues when passing from its initial to its final condition.

  With the energy $U$ of the body whose element, as well as that of the exterior ergon, enters into the Eq. (9.3), it is quite different. If the initial and final conditions of the body are given, the variation in energy is completely determined, without any knowledge of the way in which the transition from the one condition to the other took place—in fact neither the nature of the passage nor the circumstance of its being made in a reversible or non-reversible manner, has any influence on the contemporaneous change of energy. If, therefore, the initial condition and the corresponding value of the energy be supposed to be given, we may say that the energy is fully defined by the actually existing condition of the body.

  Finally, since the Heat $Q$ which is absorbed by the body during the change of condition is the sum of the change of energy and of the exterior ergon produced, it must like the latter depend upon the way in which the transition of the body from one condition to another takes place.

  Now in order to limit the field of our immediate investigation, we shall always assume, unless the contrary is expressly stated, that we have to do with reversible changes solely.

  The Eq. (9.3) which expresses the first fundamental theorem, holds for reversible as well as for non-reversible changes; hence, in order to apply it specially to reversible changes, we have not to modify it externally in any manner, but merely to understand that by $w$ and $Q$ are meant the exterior ergon and quantity of heat which correspond to reversible changes.

  On applying to reversible changes the relation (9.4) which expresses the second fundamental theorem, we have not only to understand by $Q$ the quantity of heat which relates to reversible changes, but also, instead of the double sign $\leq$, we have simply to employ the sign of equality. We obtain for all reversible cyclical processes, therefore, the equation

  $$ \int \frac{dQ}{T} = 0 $$

  (9.5)

[Articles 3–13 of Clausius’ memoir have here been omitted for the sake of brevity.—K.K.

• (14) All the foregoing considerations had reference to changes which occurred in a reversible manner. We will now also take non-reversible changes into consideration in order briefly to indicate at least the most important features of their treatment.

  In mathematical investigations on non-reversible changes two circumstances, especially, give rise to peculiar determinations of magnitudes. In the first place, the quantities of heat which must be imparted to, or withdrawn from a changeable body are not the same, when these changes occur in a non-reversible manner, as they are when the same changes occur reversibly. In the second place, with each non-reversible change is associated an uncompensated transformation, a knowledge of which is, for certain considerations, of importance.

  In order to be able to exhibit the analytical expressions corresponding to these two circumstances, I must in the first place recall a few magnitudes contained in the equations which I have previously established.

  One of these is connected with the first fundamental theorem, and is the magnitude $U$, contained in Eq. (9.3) and discussed at the beginning of this Memoir; it represents the thermal and ergonal content, or the energy of the body. To determine this magnitude, we must apply the Eq. (9.3), which may be thus written,

  $$ dU = dQ - dw $$

  (9.6)

  or, if we conceive it to be integrated, thus:

  $$ U = U_0 + Q - w $$

  (9.7)

  Herein $U_0$ represents the value of the energy for an arbitrary initial condition of the body, $Q$ denotes the quantity of heat which must be imparted to the body, and $w$ the exterior ergon which is produced whilst the body passes in any manner from its initial to its present condition. As was before stated, the body can be conducted in an infinite number of ways from one condition to another, even when the changes are to be reversible, and of all these ways we may select that one which is most convenient for the calculation.

  The other magnitude to be here noticed is connected with the second fundamental theorem, and is contained in Eq. (9.5). In fact if, as Eq. (9.5) asserts, the integral $\int \frac {dQ}{T}$ vanishes whenever the body, starting from any initial condition, returns thereto after its passage through any other conditions, then the expression $\frac {dQ}{T}$ under the sign of integration must be the complete differential of a magnitude which depends only on the present existing condition of the body, and not upon the way by which it reached the latter. Denoting this magnitude by $S$, we can write

  $$ dS = \frac{dQ}{T}; $$

  (9.8)

  or, if we conceive this equation to be integrated for any reversible process whereby the body can pass from the selected initial condition to its present one, and denote at the same time by $S_0$ the value which the magnitude $S$ has in that initial condition,

  $$ S = S_0 + \int \frac{dQ}{T} $$

  (9.9)

  This equation is to be used in the same way for determining $S$ as Eq. (9.7) was for defining $U$.

  The physical meaning of the magnitude $S$ has been already discussed in the Sixth Memoir. If in the fundamental Eq. (9.4) of the present Memoir, which holds for all changes of condition of the body that occur in a reversible manner, we make a small alteration in the notation, so that the heat taken up by the changing body, instead of the heat given off by it, is reckoned as positive, that equation will assume the form

  $$ \int \frac{dQ}{T} = \int \frac{dH}{T} + \int dZ $$

  (9.10)

  The two integrals on the right are the values for the case under consideration, of two magnitudes first introduced in the Sixth Memoir.

  In the first integral, $H$ denotes the heat actually present in the body, which, as I have proved, depends solely on the temperature of the body and not on the arrangement of its parts. Hence it follows that the expression $\frac {dH}{T}$ is a complete differential, and consequently that if for the passage of the body from its initial condition to its present one we form the integral $\int \frac {dH}{T}$, we shall thereby obtain a magnitude which is perfectly defined by the present condition of the body, without the necessity of knowing in what manner the transition from one condition to the other took place. For reasons which are stated in the Sixth Memoir, I have called this magnitude the transformation-value of the heat present in the body.

  It is natural when integrating, to take, for initial condition, that for which $H \,=\, 0$, in other words, to start from the absolute zero of temperature; for this temperature, however, the integral $\int \frac {dH}{T}$ is infinite, so that to obtain a finite value, we must take an initial condition for which the temperature has a finite value. The integral does not then represent the transformation-value of the entire quantity of heat contained in the body, but only the transformation-value of the excess of heat which the body contains in its present condition over that which it possessed in the initial condition. I have expressed this by calling the integral thus formed the transformation-value of the body’s heat, estimated from a given initial condition. For brevity we will denote this magnitude by $Y$.

  The magnitude $Z$ occurring in the second integral I have called the disgregation of the body. It depends on the arrangement of the particles of the body, and the measure of an increment of disgregation is the equivalence-value of that transformation from ergon to heat which must take place in order to cancel the increment of disgregation, and thus serve as a substitute for that increment. Accordingly we may say that the disgregation is the transformation-value of the existing arrangement of the particles of the body. Since in determining the disgregation we must proceed from some initial condition of the body, we will assume that the initial condition selected for this purpose is the same as that which was selected for the determination of the transformation-value of the heat actually present in the body.

  The sum of the two magnitudes $Y$ and $Z$, just discussed, is the before-mentioned magnitude $S$. To show this, let us return to Eq. (9.10), and assuming, for the sake of generality, that the initial condition, to which the integrals in this equation refer, is not necessarily the same as the initial condition which was selected when determining Y and Z, but that the integrals refer to a change which originated in any manner whatever suited to any special investigation, we may then write the integrals on the right of (9.10) thus:

  $$ \int \frac{dH}{T} = Y - Y_0 \hskip 0.25in \text{ and } \hskip 0.25in \int dZ = Z - Z_0, $$

  wherein $Y_0$ and $Z_0$ are the values of $Y$ and $Z$ which correspond to the initial condition. By these means Eq. (9.10) becomes

  $$ \int \frac{dQ}{T} = Y + Z - (Y_0 + Z_0) $$

  (9.12)

  Putting herein

  $$ Y + Z =S $$

  (9.13)

  and in a corresponding manner

  $$ Y_0 + Z_0 =S_0 $$

  we obtain the equation

  $$ \int \frac{dQ}{T} = S - S_0 $$

  (9.14)

  which is merely a different form of the Eq. (9.9), by which $S$ is determined.

  We might call $S$ the transformational content of the body, just as we termed the magnitude $U$ its thermal and ergonal content. But as I hold it to be better to borrow terms for important magnitudes from the ancient languages, so that they may be adopted unchanged in all modem languages, I propose to call the magnitude $S$ the entropy of the body, from the Greek word $\tau \rho o\pi {\eta }'$, transformation. I have intentionally formed the word entropy so as to be as similar as possible to the word energy; for the two magnitudes to be denoted by these words are so nearly allied in their physical meanings, that a certain similarity in designation appears to be desirable.

  Before proceeding further, let us collect together, for the sake of reference, the magnitudes which have been discussed in the course of this Memoir, and which have either been introduced into science by the mechanical theory of heat, or have obtained thereby a different meaning. They are six in number, and possess in common the property of being defined by the present condition of the body, without the necessity of our knowing the mode in which the body came into this condition: (1) the thermal content, (2) the ergonal content, (3) the sum of the two foregoing, that is to say the thermal and ergonal content, or the energy, (4) the transformation-value of the thermal content, (5) the disgregation, which is to be considered as the transformation-value of the existing arrangement of particles, (6) the sum of the last two, that is to say, the transformational content, or the entropy.

  [Article 15 of Clausius’ memoir has been omitted for the sake of brevity.—K.K.]

• (16) If we now assume that in one of the ways above indicated the magnitudes $U$ and $S$ have been determined for a body in its different conditions, the equations which hold good for non-reversible changes may be at once written down.

  The first fundamental Eq. (9.3), and the Eq. (9.7), resulting from it by integration, which we will arrange thus,

  $$ Q = U - U_0 + w $$

  (9.15)

  hold just as well for non-reversible as for reversible changes; the only difference being, that of the magnitudes standing on the right side, the exterior ergon $w$ has a different value, in the case where a change occurs in a non-reversible manner, from that which it has in the case where the same change occurs in a reversible manner. With respect to the difference $U\, -\, U_0$ this disparity does not exist. It only depends on the initial and final condition, and not on the nature of the transition. Consequently we need only consider the nature of the transition so far as is necessary in order to determine the exterior ergon thereby performed; and on adding this exterior ergon to the difference $U \,-\, U_0$, we obtain the required quantity of heat $Q$ which the body takes up during the transition.

  The uncompensated transformation involved in any non-reversible change may be thus obtained:—

  The expression for the uncompensated transformation which is involved in a cyclical process, is given in Eq. (9.11) of the Fourth Memoir.⁴ If we give to the differential $dQ$ in that equation the opposite sign, a quantity of heat given off by the body to a reservoir of heat being there reckoned positive, whilst here we consider the heat taken up by the body to be positive, it becomes

  $$ N = - \int \frac{dQ}{T} $$

  (9.16)

  If the body has suffered one change or a series of changes, which do not form a cyclical process, but by which it has reached a final condition which is different from the initial condition, we may afterwards supplement this series of changes so as to form a cyclical process, by appending other changes of such a kind as to reconduct the body from its final to its initial condition. We will assume that these newly appended changes, by which the body is brought back to the initial condition, take place in a reversible manner.

  On applying Eq. (9.16) to the cyclical process thus formed, we may divide the integral occurring therein into two parts, of which the first relates to the originally given passage of the body from the initial to the final condition, and the second to the supplemented return from the final to the initial condition. We will write these parts as two separate integrals, and distinguish the second, which relates to the return, by giving to its sign of integration a suffix $r$. Hence Eq. (9.16) becomes

  $$ N = - \int \frac{dQ}{T} - \int_r \frac{dQ}{T}. $$

  Since by hypothesis the return takes place in a reversible manner, we can apply Eq. (9.14) to the second integral, taking care, however, to introduce the difference $S_0 \,-\, S$ instead of $S\,-\,S_0$ (where $S_0$ denotes the entropy in the initial condition, and $S$ the entropy in the final condition), since the integral here in question is to be taken backwards from the final to the initial condition. We have therefore to write

  $$ \int_r \frac{dQ}{T} = S_0 - S. $$

  By this substitution the former equation is transformed into

  $$ N = S - S_0 - \int \frac{dQ}{T}. $$

  (9.17)

  The magnitude $N$ thus determined denotes the uncompensated transformation occurring in the whole cyclical process. But from the theorem, that the sum of the transformations which occur in a reversible change is null, and hence that no uncompensated transformation can arise therein, it follows that the supposed reversible return has contributed nothing to the augmentation of the uncompensated transformation, and the magnitude $N$ represents accordingly the uncompensated transformation which has occurred in the given passage of the body from the initial to the final condition. In the deduced expression, the difference $S\,-\, S_0$ is again perfectly determined when the initial and final conditions are given, and it is only when forming the integral $\int \frac {dQ}{T}$ that the manner in which the passage from one to the other took place must be taken into consideration.

• (17) In conclusion I wish to allude to a subject whose complete treatment could certainly not take place here, the expositions necessary for that purpose being of too wide a range, but relative to which even a brief statement may not be without interest, inasmuch as it will help to show the general importance of the magnitudes which I have introduced when formalizing the second fundamental theorem of the mechanical theory of heat.

  The second fundamental theorem, in the form which I have given to it, asserts that all transformations occurring in nature may take place in a certain direction, which I have assumed as positive, by themselves, that is, without compensation; but that in the opposite, and consequently negative direction, they can only take place in such a manner as to be compensated by simultaneously occurring positive transformations. The application of this theorem to the Universe leads to a conclusion to which W. Thomson first drew attention,⁵ and of which I have spoken in the Eighth Memoir. In fact, if in all the changes of condition occurring in the universe the transformations in one definite direction exceed in magnitude those in the opposite direction, the entire condition of the universe must always continue to change in that first direction, and the universe must consequently approach incessantly a limiting condition.

  The question is, how simply and at the same time definitely to characterize this limiting condition. This can be done by considering, as I have done, transformations as mathematical quantities whose equivalence-values may be calculated, and by algebraical addition united in one sum.

  In my former Memoirs I have performed such calculations relative to the heat present in bodies, and to the arrangement of the particles of the body. For every body two magnitudes have thereby presented themselves—the transformation-value of its thermal content, and its disgregation; the sum of which constitutes its entropy. But with this the matter is not exhausted; radiant heat must also be considered, in other words, the heat distributed in space in the form of advancing oscillations of the æther must be studied, and further, our researches must be extended to motions which cannot be included in the term Heat.

  The treatment of the last might soon be completed, at least so far as relates to the motions of ponderable masses, since allied considerations lead us to the following conclusion. When a mass which is so great that an atom in comparison with it may be considered as infinitely small, moves as a whole, the transformation-value of its motion must also be regarded as infinitesimal when compared with its vis viva; whence it follows that if such a motion by any passive resistance becomes converted into heat, the equivalence-value of the uncompensated transformation thereby occurring will be represented simply by the transformation-value of the heat generated. Radiant heat, on the contrary, cannot be so briefly treated, since it requires certain special considerations in order to be able to state how its transformation-value is to be determined. Although I have already, in the Eighth Memoir above referred to, spoken of radiant heat in connexion with the mechanical theory of heat, I have not alluded to the present question, my sole intention being to prove that no contradiction exists between the laws of radiant heat and an axiom assumed by me in the mechanical theory of heat. I reserve for future consideration the more special application of the mechanical theory of heat, and particularly of the theorem of the equivalence of transformations to radiant heat.

  For the present I will confine myself to the statement of one result. If for the entire universe we conceive the same magnitude to be determined, consistently and with due regard to all circumstances, which for a single body I have called entropy, and if at the same time we introduce the other and simpler conception of energy, we may express in the following manner the fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat.

  • 1 The energy of the universe is constant.

  • 2 The entropy of the universe tends to a maximum.

9.3 Study Questions

Ques. 9.1. What are the fundamental equations of the mechanical theory of heat? How are they expressed mathematically? Which of these equations belong to Clausius? And what condition does Clausius’ fundamental theorem place on the types of processes which can occur in nature?

Ques. 9.2. What is the energy of a body? How are the thermal content and ergonal content of a body related to its energy? Does the energy of a body in a particular state depend on how this state was achieved?

Ques. 9.3. Does the work accomplished by (or on) a body undergoing a transition between an initial and a final state depend upon the nature—the particular “path”—of the transition? What about the heat absorbed by the body? Can work and heat then be expressed mathematically as “complete differentials”? If so, under what specific conditions?

Ques. 9.4. What is the entropy of a body? How are the transformation-value of the thermal content and the disgregation of a body related to its entropy? Does the entropy of a body in a particular state depend on how this state was achieved? In what sense, then, are energy and entropy similar?

Ques. 9.5. Does the heat radiation surrounding a body have entropy? What difficulty does this present? Can the mechanical theory of heat, and Clausius’ theorem in particular, be applied to radiation?

Ques. 9.6. What do the fundamental theorems of the mechanical theory of heat imply about the fate of the universe?

9.4 Exercises

Ex. 9.1 (Entropy of equilibration). In this exercise, we will compute the change in entropy associated with heating up a 1-kg silver block, initially at the temperature of ice-water, using three different methods.

Method 1: :

Suppose we place the silver block in direct thermal contact with a large reservoir of boiling water until its temperature rises to 100 $\relax ^\circ$ C. What is the entropy change of the silver block as a result of this process? Of the reservoir? Of the universe as a whole? Is this method of heating the silver block a reversible process?

Method 2: :

Suppose, instead, we heat up the silver block in two stages. We first let it equilibrate with a reservoir at 50 $\relax ^\circ$ C, then we let it equilibrate with another reservoir at 100 $\relax ^\circ$ C. What is the entropy change of the silver block as a result of this two-step process? Of the 50$\relax ^\circ$ reservoir? Of the 100$\relax ^\circ$ reservoir? Of the universe as a whole? Is this a reversible process?

Question: :

Which of the previous two methods of bringing the silver block to 100$\relax ^\circ$ results in a lower change in entropy of the universe as a whole? Why do you suppose this is? Can you imagine a process whereby you could raise the temperature of the silver block to 100$\relax ^\circ$ reversibly, that is, with no change in the entropy of the universe?

Method 3: :

Finally, suppose we heat up the silver block to 100 $\relax ^\circ$C by allowing heat to flow into it from an enormous boiling water reservoir through a reversible heat engine. In this case, what would be the change in entropy of the silver block? Of the reservoir? Of the universe as a whole? Is this a reversible process?

Ex. 9.2 (Energy, entropy and the carnot cycle). Clausius states that the energy of a substance is comprised of both thermal and ergonal content. These describe, respectively, the kinetic and potential energies of the particles comprising the substance. For an ideal (non-interacting) gas, the ergonal content must be zero. Hence the energy of an ideal gas must be determined strictly by the motion of the gas molecules. Since the temperature of a substance provides a measure of the average kinetic energy of its particles, this implies that the energy of a fixed quantity of ideal gas depends only on its temperature, regardless of its volume or pressure. With this in mind, reconsider the carnot cycle described in Ex. 4.2. Suppose that the working substance inside the cylinder acts as an ideal gas.

• a) How much heat is drawn into the gas from the hot reservoir during the isothermal expansion? How much heat is ejected from the gas into the cold reservoir during the isothermal compression? What about during the other two (adiabatic) processes which make up the carnot cycle?

• b) Recall that the efficiency of any engine cycle is defined by Eq. 5.4. What, then, is the efficiency of this cycle? Is your answer in agreement with Eq. 5.8? Is this what you would expect for a carnot cycle? Why?

• c) What is the change in entropy of (i) the air, (ii) the hot reservoir, and (iii) the cold reservoir, during each of the four processes which make up the carnot cycle? What is the change of entropy of the entire universe during a complete cycle of this engine?

9.5 Vocabulary

• 1. Elucidate

• 2. Exposition

• 3. Figurative

• 4. Contemporaneous

• 5. Impart

• 6. Disparity

• 7. Ponderable

• 8. Infinitesimal

• 9. Allude

• 10. Incessant

• 11. Aether

• 12. Entropy