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I
Penner 31
Sola Regula Secunda:
A Logical Approach to the
Second Law of Thermodynamics
Margaret K. Penner
Bethel College, Kansas
February 16, 2006
( Margaret K. Penner 2006
Sola Regula Secunda:
A Logical Approach to the
Second Law of Thermodynamics
Margaret K. Penner
Bethel College, Kansas
February 16, 2006
Don S. Lemons, Physics Advisor
Christopher Earles, Math Advisor
Abstract:
We consider the implications of the second law of thermodynamics independently from other physical laws. By applying rigorous logic to our model of the thermodynamic world, we explore the whole domain of possibility to determine how the second law restricts the universe. We carefully consider and compare the precise logical meanings of the various formulations of the second law and finally arrive at a deeper understanding of its content.
Table of Contents
3Table of Contents
4I. Introduction
4A. Philosophical Motivation
6B. Historical Note
7II. Preliminaries
7A. Definitions
12B. Statement of Second Law
13C. “Un-Statement” of the First Law
14III. The Possibility Catalog
18IV. One Law, Two Formulations
18A. The Postulate of Clausius
19B. Equivalence of Formulations
21C. Inequivalence of Common Formulations
24V. Important Consequences
24A. Some Corollaries
27B. The Essence of the Second Law
28VI. Conclusion
29Appendix A: Formulations of the second law
31Appendix B: Possibility Catalog for the Conventional Postulate of Thomson
I. Introduction
A. Philosophical Motivation
The traditional approach to classical equilibrium thermodynamics begins by exploring the consequences of the first law of thermodynamics before stating the second law of thermodynamics and then exploring the results of the two laws combined. In this approach the effects of assuming only the second law – before combining it with the first law – are mentioned briefly if at all. Thus the traditional student of thermodynamics is left with the impression that one cannot draw important conclusions about the physical world from the second law alone. There are legitimate reasons for this: modern thermodynamics believes wholeheartedly in the validity of both the first and second laws, and the bulk of important theoretical results have been based on their combination.
However, the second law remains wholly logically independent, and the logical meaning of any proposition consists of everything that can be deduced from that statement. Thus only by exploring the range of deductions from the second law alone can we deeply understand the logical meaning of the second law of thermodynamics, which is the purpose of this paper.
To this end, I shall focus on the important consequences that one can indeed discover using only the second law of thermodynamics. Some of these consequences have been widely and erroneously thought to logically depend on both the second and first laws. In the process of probing the purely logical depths of the second law, we must allow ourselves to imagine an alternative reality in which the first law does not regulate thermodynamic processes.
Section II will rigorously define the concepts and ideas that we employ to describe and analyze thermodynamic systems. These are then used in section III to describe and explain “The Possibility Catalog,” which should aid the reader in setting aside the “first law intuitions” gained from years of experience with physics in this universe to make the methods of proof employed in this paper more lucid. Sections IV and V contain the proofs of various theorems and corollaries logically dependent on the second law alone.
During the Protestant reformation of the 16th century, the idea of sola scriptura (only scripture) became extremely important, because the people involved desired to construct a faith and a church based only on the biblical word of God in order to more deeply understand the divine inspiration they saw in the Bible. While the Protestants’ call for sola scriptura had a somewhat more serious and consequential object, our quest is logically analogous: constructing a thermodynamic universe based only on the constraints in the second law of thermodynamics so that we can more fully understand the essence of this important physical law. For this reason I have entitled this paper sola regula secunda, Latin for “only second law.” This title shall be our focus and our guide along the way.
B. Historical Note
Historically, the second law of thermodynamics was discovered prior to the first law, the law of conservation of energy. Before heat was accepted as a form of energy, enabling the first law to be articulated, some scientists were working with an alternative view of the nature of heat, supposing it to be a sort of fluid called caloric, which was always conserved. This assumption is entirely consistent with the second law of thermodynamics, as evinced by the memoir of Sadi Carnot (1796-1832), a text influential in articulating the second law while holding fast to the idea of caloric. Thus, this paper also has some historical inspiration: Scientists who were entirely convinced of the second law were able to make important contributions to thermodynamics, while never believing in the first law. This paper seeks to fully explore the range of true results attainable under that set of assumptions without the more complicated, more developed (and in our universe, false) conclusions resulting from an alternative (and in our universe erroneous) theory of heat such as caloric.
II. Preliminaries
A. Definitions
Before exploring the second law and its implications, it is useful to discuss some terms which have specific definitions in this paper and to describe how some important thermodynamic concepts will be translated into schematics, making our abstract discussion and proofs more immediately and visually comprehensible.
A thermodynamic system is that which we choose to observe. The important property of a system is that it is well defined – that we know what matter belongs to the system and what does not. Beyond this definition we shall cease to be concerned with what our system actually contains for two main reasons. First, so much of the stark, logical beauty and enduring validity of classical thermodynamics is a result of its restricted domain. Thus we shall limit the study of our systems to the macroscopic level with no concern for the microscopic mechanisms – the substance and structure of the matter – by which thermodynamic phenomena are carried out. Second, as we free our minds to enter the abstract universe of sola regula secunda it will become increasingly difficult to imagine the details of an “actual” physical system that behaves in ways we shall permit. Therefore we will represent every system schematically in the same way, by a circle (see Fig. 1 for an example of the described schematics). Occasionally we shall label the systems with a number or letter in order to differentiate between systems that behave in meaningfully different ways.
The environment in which a system exists is everything in the universe besides the system. We shall generally simplify the interactions between a system and its environment, or “surroundings,” to include only a very restricted subset of the environment, but these simplifications merely expedite our proofs and in no way diminish the general logical legitimacy of our claims. Mirroring its definition, the environment will be represented in the diagram as everything outside the circle representing the system.
As in other areas of physics, work is defined to be a quantity of force exerted through a distance. We can only talk about work when one body does work on another body. In thermodynamics, work happens as an interaction between the system and its environment. In the interest of verbal conciseness and visual intuition we (and other thermodynamicists) use several different phrases to denote this interaction: “Production of (positive) work” and “appearance of (positive) work (in the environment)” are used to indicate that the system is doing work on the environment, while “consumption of work,” “production of negative work,” and “appearance of negative work (in the environment)” are used to indicate that the environment is doing work on the system. However, it is important for the reader to bear in mind that work is a process that occurs between two bodies and does not exist as a separate entity. In the schematics, work shall be represented by a horizontal arrow labeled with the quantity W going into or out of the system to indicate consumption or production of work respectively.
Heat is the quantity that is transmitted between the system and the environment that is not work, but that changes the state of the system and/or environment in the process. Using the traditional first law the reader may assert that heat and work are merely different forms of energy, that both quantify a transfer of energy from one body to another. However, in the universe of sola regula secunda we can make no such claims; It is just as valid to think of heat as a sort of “igneous fluid,” like the caloricists of the late 18th and early 19th centuries. Thus our definition of heat is left intentionally abstract. Heat shall be represented by a vertical arrow labeled with the quantity Q going into or out of the system to indicate absorption of heat by the system from the environment or rejection of heat by the system into the environment, respectively.
A heat reservoir is a body in the environment that remains at a constant temperature, regardless of how much heat is absorbed from it or rejected into it. At first glance this seems like a brazen simplification, but there are thermodynamic bodies on earth, such as the ocean, that approximate this behavior. Heat reservoirs are a traditional way of analyzing the interactions of classical thermodynamic systems with their environments; when a system’s environment does change temperature, one simply imagines the system interacting with a series of heat reservoirs with infinitesimally small temperature differences. A heat reservoir shall be represented by a box labeled with the empirical temperature of the reservoir (henceforth referred to simply as the “temperature”), T. When two or more reservoirs are present in the same picture, a reservoir with a higher temperature will always be depicted above a reservoir with a lower temperature.
A set of interactions between the system and environment involving work and heat (i.e. a thermodynamic transformation) is said to be a cycle if the state of the system after the transformation is identical to its state before the transformation.
Traditionally a heat engine is thought to be a system that transforms heat into work. The inverse of a heat engine is then called a refrigerator, a system which consumes work, enabling heat to flow from a colder body to a warmer body. In this paper we shall use the term heat engine more generally to denote any system that has up to three interactions with its surroundings: exchange of heat with up to two reservoirs of specified, different temperatures and production of positive or negative work. Under this definition, a “refrigerator” is a heat engine. Classical thermodynamics is unconcerned with the mechanisms whereby these interactions occur, preferring to make statements about the possibilities and restrictions on the amount and direction of heat flow and work production. This shall be our approach as well, occupying the bulk of section V.
While we limit our study to engines operating in cycles between two (or fewer) given heat reservoirs, our claims and proofs could be generalized to more complex engines, because any engine operating between n reservoirs can be analyzed as a combination of n-1 engines, each operating between two reservoirs.
Fig. 1 Schematic of a heat engine operating between two heat reservoirs: a system absorbing QH heat, rejecting QC heat, and producing W work.
A thermodynamic transformation is reversible if, by making an infinitesimally small change in the environment, the transformation can be caused to reverse itself. A reversible process is therefore described as quasi-static, because the system is continually in equilibrium. In this universe, reversibility necessitates using very slow processes to prevent shock waves and avoiding friction and other forms of hysteresis. But we are only concerned with the logical implications of reversibility: that a known reversible engine allows us to simply construct another possible engine, represented by reversing all the arrows in the schematic.
Efficiency is defined as the ratio of work produced W to heat absorbed Q by the system. Efficiency is typically denoted by (, where
.
B. Statement of Second Law
While there are many different ways to verbally express the second law of thermodynamics, and these options shall be discussed in more depth in section IV, here I merely state the version that shall be used throughout this paper as our basic assumption, the fundamental law of the universe governed by sola regula secunda:
The Second Law: It is impossible to make any transformation whose only final result is the exchange of a non-zero amount of heat with fewer than two heat reservoirs and the production of a positive amount of work.
Since the second law is a statement of impotence and all of its consequences statements of impotence, the natural method of proof is contradiction. Rigorous use of logic thus necessitates an auxiliary assumption, which asserts the possibility of a process central to our reasoning and proofs later:
Auxiliary Assumption: For any two heat reservoirs at different temperatures, there can exist an engine operating in a cycle between them, which absorbs heat at the higher temperature, rejects heat at the lower temperature, and produces positive work. There can also exist an engine operating in a cycle between any two heat reservoirs at different temperatures, which absorbs heat at the lower temperature, rejects heat at the higher temperature, and consumes work.
Since these process is widely known to be possible in our universe, which is governed by even more than the second law, it clearly does not violate the second law.
C. “Un-Statement” of the First Law
The first law of thermodynamics has no real bearing on the rest of this paper. However, the reader has some background in classical thermodynamics, not only from physics and chemistry but also from copious experience in this universe where the first law governs our lives and is thus engrained in our worldview. Therefore I think it enlightening at the beginning of our journey into the universe of sola regula secunda to state precisely the assumptions that are usually made that we are not making. This is not to say that the first law is false, even in our constructed universe. To the contrary – it is logically consistent with the second law, but in this paper we shall willfully refuse to draw any conclusions from it or use it in any proof or analysis.
The first law can is given as a combination of three assertions:
· For every thermodynamic system there exists a quantity E called internal energy, which is a function only of the state of the system and not of the processes by which the state was attained.
· The internal energy difference between two states is measured by the work required to transform the system from one state to the other without heat flowing into or out of the system.
· The internal energy change is equal to the amount of heat absorbed by the system (this quantity can be negative) minus the work produced.
And this is concisely notated by
.
Even from a strictly mathematical perspective these equations point easily to the essence of the first law: since quantities of work and heat appear together additively, they must have the same units. They are simply different manifestations of the same physical entity, namely energy, and energy is conserved.
In the universe of sola regula secunda we can make none of the assertions in the above paragraph. We cannot construct a relationship between work and heat, make a statement about how their units relate to each other, or claim that “energy” (which we cannot meaningfully define) is conserved. It truly is a different universe, and we shall spend the next section becoming more intuitively acquainted with it.
III. The Possibility Catalog
The following is an exhaustive catalog of schematics for all processes that a heat engine operating between two or fewer heat reservoirs could conceivably undergo. Recall that according to our definition heat engines are simply systems undergoing up to three interactions with the environment, making an exhaustive list possible. Included are two processes, (a) and (b), that involve only the environment to emphasize that the second law also constrains what happens in the environment. Those processes that can be proven prohibited by the second law alone have been marked with a symbol. Of course engines (k), (l), and (f) directly violate the second law.
The impossibility of those engines so marked either follows directly from the second law or involves a fairly trivial proof. Such proofs are always by contradiction: combining the proposed engine with an engine known to be possible and obtaining a direct violation of the second law means that the proposed engine also violates the second law. The majority of these proofs are left to the reader, and the more complex proofs shall be found elsewhere in this paper (sections IV and V). However, I present the following formal proof to provide a flavor of how they are implemented.
Claim:
Engine (c) violates the second law.
Proof:
Proof is by contradiction.
Assume
is permitted by the second law.
Combine it with engine (o), operating between TC=T and some arbitrary TH>T, which we formally assumed to be possible in section II and adjust QC = Q as follows:
This directly violates the second law. Therefore our assumption must be false, and engine (c) is impossible.
The nature of the second law implies that it can only be used to constrain a universe. That is, heat engines cannot be constructed and proven to be “possible” under only the second law; instead, we can prove certain processes impossible and assert that whatever is not prohibited is allowed. Thus, while I present this list with the confident claim that those processes not marked prohibited are viable heat engines in the universe of sola regula secunda, this assertion is impossible to prove.
We have even more confidence in the possibility of some of the permitted engines: Classical thermodynamics recognizes engines (g), (n), (o), (s), (t) and (v) as entirely valid engines in a universe – such as ours – governed by both the second and first laws (provided, of course, that the appropriate quantities of work and heat are used). Those engines can be built experimentally and thus clearly do not violate the second law.
But the interesting parts of this catalog are those engines that are not permitted by the first law but that are allowed by the second law alone: engines (e), (m), and (u). Those are the processes that differentiate the universe of sola regula secunda from our universe and must be examined further. The second law does not prohibit spontaneous consumption of work, accomplished by engine (e), nor does it prohibit spontaneous consumption of a non-zero amount of work and heat, as in engines (m) and (u). These three engines make it obvious that in this new, less-constrained universe, “energy” is not conserved, which is why we are unable to define the term in a meaningful way.
IV. One Law, Two Formulations
A. The Postulate of Clausius
While there are many different ways to verbally express the second law of thermodynamics, there are two major, conceptually distinct formulations, one attributed to William Thomson, also known as Lord Kelvin (1824-1907), and one to Rudolf Clausius (1822-1888). This section is devoted to the discussion of these wordings and the necessity of rigorous formulation for consistency and logical efficiency in our statements of the second law.
The formulation given in section II is a version of Thomson’s postulate, based on one found in Vanderslice and is printed again below for convenient comparison along with the necessary auxiliary assumption:
Postulate of Thomson: It is impossible to make any transformation whose only final result is the exchange of a non-zero amount of heat with fewer than two heat reservoirs and the production of a positive amount of work.
Auxiliary Assumption: For any two heat reservoirs at different temperatures, there can exist an engine operating in a cycle between them, which absorbs heat at the higher temperature, rejects heat at the lower temperature, and produces positive work. There can also exist an engine operating in a cycle between any two heat reservoirs at different temperatures, which absorbs heat at the lower temperature, rejects heat at the higher temperature, and consumes work.
Or as we can now state more simply, engine (o) is definitely possible between any two different heat reservoirs, as is its reverse, engine (v).
The postulate of Clausius focuses on a different aspect of the processes:
Postulate of Clausius: It is impossible to make any transformation whose only final result is to transfer heat from a body at a given temperature to a body at a higher temperature.
As with the postulate of Thomson, rigor requires the auxiliary assumption to be used with the postulate of Clausius to positively affirm the possibility of two particular heat engines to be used in logical proofs.
B. Equivalence of Formulations
Formal, logical proof of the equivalence of these two postulates is presented below.
Claim 1:
Postulate of Thomson ( Postulate of Clausius.
Proof:
Proof is by contradiction.
Assume the postulate of Clausius does not hold.
Thus
is possible.
Note that this is engine (j) from the possibility catalog, which is known to violate the postulate of Thomson.
Thus our assumption is false, and the postulate of Clausius must be true if the postulate of Thomson is true.
Claim 2:
Postulate of Clausius ( Postulate of Thomson.
Proof:
Proof is by contradiction.
Assume the postulate of Thomson does not hold.
Thus
is possible.
Re-draw the engine so that it absorbs heat from a reservoir drawn below the system, which doesn’t change anything physically or logically. Then combine it with engine (v), operating between TC=T and some arbitrary TH>T, which we formally assumed to be possible, and adjust W= W’ as follows:
This directly violates the postulate of Clausius. Thus our assumption is false, and the postulate of Thomson must be true if the postulate of Clausius is true.
C. Inequivalence of Common Formulations
Looking closely at a variety of college-level physics texts demonstrates the wide variety of formulations of the second law of thermodynamics. A sampling of formal statements of the law found in such texts is compiled in Appendix A. There are three main, conceptually different ways that the second law of thermodynamics is expressed: it can be based on entropy, on the postulate of Clausius, and on the postulate of Thomson.
Because, as previously discussed, in the universe of sola regula secunda we cannot define energy in a meaningful way, it is also impossible to define entropy. The formulations that use entropy to articulate the constraints of the second law are dependent on an assumption of the first law. They are generally concise, elegant, and attractive to the student of thermodynamics as immediately and quantifiably applicable, but as this paper endeavors to discover the essence of the second law alone, we have no use for these formulations.
The formulations based on the postulate of Clausius are straightforward, sometimes depending on a previous definition of engine or refrigerator. However, most interesting to our pursuit of the second law are the formulations based on the postulate of Thomson because with the exception of Vanderslice, all the textbooks I have found state the law in a way equivalent to the following:
Conventional Postulate of Thomson: It is impossible to make any transformation whose only final result is to transform heat into positive work.
Upon close reading and logical reasoning we find that the conventional formulation is not logically equivalent to the version of Thomson given in section II, even though many texts using the conventional formulation state Claim 1 and some present proofs, which make use of the first law. The proof of Claim 1 (Postulate of Thomson ( Postulate of Clausius) fails for this conventional formulation because we cannot prove that engine (j) is impossible. The attempted proof would be similar to the following:
Claim:
Engine (j) violates the conventional postulate of Thomson.
Attempted Proof:
Proof is by contradiction.
Assume is permitted by the conventional postulate of Thomson.
Combine it with engine (o), operating between TC and TH, which we formally assumed to be possible in the auxiliary assumption, and adjust QC’ = QC as follows:
This appears to be a cycle transforming heat into work with no other effects. However, that is only the case if QH’-QH is a positive quantity. If, instead, QH’-QH is a negative quantity, then the cycle spontaneously and simultaneously produces work and heat, as shown below:
This is engine (l), and while it appears ridiculous to our minds, trained in conservation of energy, it does not violate the conventional postulate of Thomson, nor can it be reduced to an engine that does so.
Clearly since the conventional version of Thomson’s postulate is not equivalent to the postulate of Clausius, and the postulate of Clausius is equivalent to our more precise version from section II, it follows that the conventional version of Thomson’s postulate is not equivalent to ours. In fact, this conventional phrasing of Thomson’s postulate on its own is less powerful than our version because it excludes fewer possible processes. The most immediately obvious, differing consequence of the conventional formulation is that engine (l) cannot be shown to be impossible. In addition to those engines permitted by our formulation of the second law, the conventional formulation also allows engines (b), (d), (f), (h), (j), (p), and (r), with some restrictions on the values of QC for engines (j) and (r). See Appendix B for a more comprehensive, visual overview of the possibilities. It is important to note that all engines permitted by the conventional formulation and prohibited by our formulation of the second law alone are also prohibited by the first law.
V. Important Consequences
A. Some Corollaries
In this section I present some important corollaries, which can be proven based entirely upon the second law and the auxiliary assumption. The first five have traditional proofs in the literature based only on the second law, so they are stated here without proof:
Corollary 1:
An engine operating in a cycle between two heat reservoirs will produce positive work only if heat is absorbed at the high temperature and rejected at the lower temperature.
Corollary 2:
No engine operating between two heat reservoirs can have a higher efficiency than a reversible engine operating between the same two reservoirs.
Corollary 3:
All reversible engines operating between the same two reservoirs have the same efficiency.
Corollary 4:
If two reversible engines operate with a common source temperature and different refrigerator temperatures, the engine operating over the larger temperature difference has the higher efficiency.
Corollary 5:
No change in the state of a system connected to a single heat reservoir can produce more work in the surroundings than the same change in state carried out reversibly.
I have found another corollary that depends only the second law, but whose proof has traditionally invoked the first law. I present a new proof, relying only on the second law, making it more logically efficient than the proof given in the literature.
Corollary 6:
The ratio of heat absorbed to heat rejected by a reversible heat engine operating in a cycle between two heat reservoirs is determined entirely by the temperatures of the reservoirs.
Proof:
Proof is by contradiction. Given the reversible engines (1) and (2) as depicted below, assume
.
Adjust the engines so that QC1= QC2. Then, by assumption
.
Since by Corollary 3,
, we have
.
Together, the previous two equations give us
. Choose engine (1) so that W1> W2. Then reversing engine (2) results in the following:
Note that the actual direction of arrows in the combined engine depends on the relative magnitudes of the quantities. We know that W1-W2 >0 by our choice of engine (1), but we don’t know whether QH1-QH2 is positive, negative, or zero. If positive, the result is engine (k), which has been shown to violate the second law. If negative, the result is engine (l), which also violates the second law. If zero, the result is engine (f), again violating the second law. In all three cases our initial assumption is false, and
for two reversible engines, which means that the ratio of heat absorbed to heat rejected by a reversible heat engine operating in a cycle is determined entirely by the temperatures of the two heat reservoirs.
B. The Essence of the Second Law
As the above corollaries and proofs make clear, the second law is concerned with directionality. It does not constrain the quantities of work and heat flowing into and out of the system, but merely the direction in which they flow. The second law, both in “our” version, equivalent to the postulate of Clausius, and in the more limited formulation, serves to exclude certain engines from the realm of possibility based only on whether work is being produced or consumed and heat being absorbed or rejected into warmer or cooler reservoirs. In a sense it is humbling and refreshing to observe such a powerful and consequential law, which lends itself better to analysis and argumentation using diagrams with circles and arrows than to describing relationships using equations and quantities.
VI. Conclusion
I have examined the logical content of the second law alone and found this law, which deals almost exclusively with the direction of processes to be independently more consequential than is often recognized. It has also become apparent that wording has a significant impact on the logical meaning of the law and that different sets of thermodynamic conclusions must be drawn from even slight variations in its formulation.
In most physics texts the second law of thermodynamics takes a secondary role; when the second law is formally stated, the first law has already been introduced, explored, and accepted as assumed. In such a setting it is unnecessary to formulate the law precisely in order to achieve equivalence with the postulate of Clausius. Instead, the first law is invoked extensively in exploring the additional results that become accessible when the second law is added to the thermodynamic paradigm, which has already been fundamentally shaped by the first law. In this setting, exploring the full range of engines excluded from possibility by the second law alone becomes trivial.
So perhaps more than a question of precision in formulation, we are faced with a question of logical efficiency: Is it more important to us as physicists or pedagogues to have laws that are independently in their strongest form or to have laws that have more disjoint domains? In terms of the possibility catalog this question asks if we would rather have each law logically disallow the maximal number of engines on its own or have the minimal number of engines simultaneously disallowed by both laws. The latter possibility points to a valid argument in favor of the conventional postulate of Thomson (not equivalent to Clausius), but in either case, the student of thermodynamics should be aware of the precise role that the second law plays in constraining the universe.
Appendix A: Formulations of the second law
Using Entropy:
“In any spontaneous process there is always an increase in the entropy of the universe.”
“The entropy of an isolated system never decreases. The entropy either increases, until the system reaches equilibrium, or if the system began in equilibrium, stays the same.”
“During any process of nature, the entropy change of the universe – the entropy change of a system plus that of its environment – must be greater than or equal to zero.”
Based on Clausius:
“It is impossible to construct a refrigerator that transfers heat from a cold reservoir to a hot reservoir without aid from an external source.”
“It is impossible for a refrigerator working in a cycle to produce no other effect than the transfer of heat from a colder body to a hotter body.”
“An engine operating in a cycle cannot transfer heat from a cold reservoir to a hot reservoir without some other effect on its environment.”
“It is not possible for any cyclical machine to convey heat continuously from one body to another at a higher temperature without, at the same time, producing some other (compensating) effect.”
Based on Thomson:
“A transformation whose only final result is to transform into work heat extracted from a source that is at the same temperature throughout is impossible.”
“It is impossible to construct a heat engine that, when operating in a cycle, completely converts heat into work.”
“It is impossible for an engine working in a cycle to produce no other effect than that of extracting heat from a reservoir and performing an equivalent amount of work.”
“An engine operating in a cycle cannot transform heat into work without some other effect on its environment.”
“It is impossible to make any transformation whose only final result is the exchange of a non-zero amount of heat with less than two heat reservoirs and the appearance of a positive amount of work in the surroundings.”
Appendix B: Possibility Catalog for the
Conventional Postulate of Thomson
QC
T
T
Q
Q
T
Q
Q
T
W
W
QH
W
QH
TH
TH
TC
QH
QC
QC
QH
TC
TH
QC
QH
TC
TH
QC
QH
TC
TH
T
Q
W
TH
Q
T
W
Q
T
W
Q
T
TC
QC
QH
TC
TH
QC
QH
TC
TH
QC
QH
TC
TH
QC
QH
TC
TH
QC
QH
TC
TH
QC
QH
TC
TH
QC
QH
TC
TH
(a)
W
W
W
W
W
W
W
W
(b)
(f)
(e)
(d)
(c)
(k)
(l)
(m)
(n)
(o)
(p)
(j)
(i)
(h)
(g)
(t)
(u)
(s)
(r)
(q)
(v)
QH
W
TH
W
Q
T
TC
T
W
QC
QC
(o)
Q
TC
Q
W
Q
QH
TH
1
2
TH
TC
1&2
QH
TC
W
QH
TH
Q
T
TC
1
W’
QC
QH
TH
Q
W
2
1&2
TH
TC
Q+QC
QH
TH
TH
TC
QC
W
1&2
QH’
QH’-QH
W
2
1
W
QC’
TC
TC
QH
TH
1&2
QC
QH
QH-QH’
TC
TH
2
1
TC
QC2
W1
QC1
QH1
TH
W2
QH2
TH
TC
QH1-QH2
W1-W2
TC
TH
1&2
2
1
W1
QH1
W2
QC2
QH2
TC
TH
(s)
W
QC
QC1
TC
W
QC
QH
TH
QH
QC
(g)
TC
T
Q
T
Q
T
Q
T
Q
W
(a)
(b)
(e)
(d)
(c)
W
(f)
(k)
T
Q
W
(l)
T
Q
W
(m)
TH
QH
QC
W
(p)
TC
TH
QH
QC
W
(q)
TC
TH
QH
QC
W
(r)
TC
TH
QH
QC
W
(t)
TC
TH
QH
QC
W
(u)
TC
TH
QH
QC
W
(v)
TC
TH
QH
QC
(h)
TC
TH
QH
QC
(i)
TC
TH
QH
QC
(j)
TC
T
Q
(n)
W
� J.T. Vanderslice, H. W. Schamp, Jr., and E.A. Mason, Thermodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1966), pp. 31-36 is the only text we encountered that devoted significant effort to exploring the consequences of the second law alone. That section was the inspiration for this paper, which is a more rigorous treatment of the topic with additional results.
� S. Carnot, Reflections on the Motive Power of Fire (Chez Bachelier, Paris, 1824).
� Based on J.T. Vanderslice, H. W. Schamp, Jr., and E.A. Mason, Thermodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1966), p. 29.
� Based on J.T. Vanderslice, H. W. Schamp, Jr., and E.A. Mason, Thermodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1966), p. 22.
� For n engines, n-1 such adjustments (of heat or work) can always be made, as shown in E. Fermi, Thermodynamics (Dover, New York, 1956), p. 37.
� Based on J.T. Vanderslice, H. W. Schamp, Jr., and E.A. Mason, Thermodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1966), p. 29.
� Based on Based E. Fermi, Thermodynamics (Dover, New York, 1936), p.30.
� To be entirely rigorous we continue to make use of the auxillary assumption, as we did with the earlier formulation.
� Based on Vanderslice, 32-36.
� S. S. Zumdahl and S. A. Zumdahl, Chemistry (Houghton Mifflin, Boston, 2000), p. 798.
� R. D. Knight, Physics for scientists and Engineers (Pearson Education, San Fransisco, 2004), p. 566.
� A. Van Heuvelen, Physics: A General Introduction (Little, Brown and Company, Boston, 1982), 213.
� J. Sanny and W. Moebs, University Physics (Wm. C. Brown, Dubuque, IA, 1996), p. 381.
� P. A. Tipler, College Physics (Worth, New York, 1987), p. 329.
� H. C. Ohanian, Physics (W. W. Norton & Company, New York, 1989), p. 560.
� D. Halliday and R. Resnick, Fundamentals of Physics (John Wiley & Sons, New York, 1986), p. 439.
� D. Halliday and R. Resnick, Fundamentals of Physics (John Wiley & Sons, New York, 1986), p. 439.
� J. Sanny and W. Moebs, University Physics (Wm. C. Brown, Dubuque, IA, 1996), p. 380.
� P. A. Tipler, College Physics (Worth, New York, 1987), p. 328.
� H. C. Ohanian, Physics (W. W. Norton & Company, New York, 1989), p. 559.
� J.T. Vanderslice, H. W. Schamp, Jr., and E.A. Mason, Thermodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1966), p. 29.