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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
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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.
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Table of Contents
Table of Contents .............................................................................................................. 3
I. Introduction ................................................................................................................... 4
A. Philosophical Motivation ........................................................................................... 4
B. Historical Note ........................................................................................................... 6
II. Preliminaries ................................................................................................................ 7
A. Definitions .................................................................................................................. 7
B. Statement of Second Law ......................................................................................... 12
C. “Un-Statement” of the First Law ............................................................................. 13
III. The Possibility Catalog ............................................................................................ 14
IV. One Law, Two Formulations .................................................................................. 18
A. The Postulate of Clausius ......................................................................................... 18
B. Equivalence of Formulations ................................................................................... 19
C. Inequivalence of Common Formulations ................................................................ 21
V. Important Consequences .......................................................................................... 24
A. Some Corollaries ...................................................................................................... 24
B. The Essence of the Second Law ............................................................................... 27
VI. Conclusion ................................................................................................................. 28
Appendix A: Formulations of the second law .............................................................. 29
Appendix B: Possibility Catalog for the Conventional Postulate of Thomson ......... 31
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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.1 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
1
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.
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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.
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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.2 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.
2
S. Carnot, Reflections on the Motive Power of Fire (Chez Bachelier, Paris, 1824).
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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
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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
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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
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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.
TH
QH
W
QC
TC
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
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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


W
.
QH
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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.3
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.
3
Based on J.T. Vanderslice, H. W. Schamp, Jr., and E.A. Mason, Thermodynamics
(Prentice-Hall, Englewood Cliffs, NJ, 1966), p. 29.
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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. 4
And this is concisely notated by
E  Q W .

Based on J.T. Vanderslice, H. W. Schamp, Jr., and E.A. Mason, Thermodynamics
(Prentice-Hall, Englewood Cliffs, NJ, 1966), p. 22.
4
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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
course engines (k), (l), and (f) directly violate the second law.
symbol. Of
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T
T
T
Q
Q
Q
(a)
(b)
(d)
W
(e)
(f)
TH
TH
TH
QH
QH
QH
QH
QC
QC
QC
QC
TC
TC
(g)
TC
(h)
T
(j)
T
Q
W
(k)
W
(l)
TH
(m)
QC
TC
TC
(r)
TH
TH
QH
W
QC
QH
W
QC
TC
(t)
TH
QH
W
QC
TC
QC
TC
(u)
W
QC
(q)
TH
QH
W
QC
(p)
QH
TH
QH
QC
(o)
W
(n)
W
TC
Q
W
TH
QH
W
TC
T
Q
W
TH
QH
TC
(i)
T
Q
(s)
Q
(c)
W
TH
T
TC
(v)
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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.
T
Q
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:5
TH
QH
QH
W
1
W
2
Q
1&2
Q
TC
5
TH
TC
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.
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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.
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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.6
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).
6
Based on J.T. Vanderslice, H. W. Schamp, Jr., and E.A. Mason, Thermodynamics
(Prentice-Hall, Englewood Cliffs, NJ, 1966), p. 29.
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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.7
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.
TH
QH
QC
TC
Note that this is engine (j) from the possibility catalog, which is known to
violate the postulate of Thomson.
7
Based on Based E. Fermi, Thermodynamics (Dover, New York, 1936), p.30.
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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.
T
Q
W
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:
TH
TH
W’
W
1
2
Q
QH
QH
1&2
QC
TC
Q+QC
TC
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.
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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. 8
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
8
To be entirely rigorous we continue to make use of the auxillary assumption, as we did
with the earlier formulation.
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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
TH
of Thomson.
QH
QC
TC
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:
TH
TH
QH’
W
2
1
QC’
TC
QH’-QH
QH
W
1&2
QC
TC
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,
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instead, QH’-QH is a negative quantity, then the cycle spontaneously and
simultaneously produces work and heat, as shown below:
TH
QH-QH’
W
1&2
TC
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.
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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:9
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.
9
Based on Vanderslice, 32-36.
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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
QH1 QH 2
.

QC1 QC 2
TH

QH2
W2
QH1
W1
1
2
QC2
QC1
TC
Adjust the engines so that QC1= QC2. Then, by assumption
QH1  QH 2 .
Since by Corollary 3, 1  2, we have


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W1
W
 2 .
QH1 QH 2
Together, the previous two equations give us W1  W2 . Choose engine (1)
so that W1> W2. Thenreversing engine (2) results in the following:

TH
TH
QH1
QH2
W2
W1
1
2
QC1
QH1-QH2
W1-W2
1&2
QC2
TC
TC
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
QH1 QH 2
for two reversible engines, which means that the ratio of

QC1 QC 2
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.
Penner 27
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.
Penner 28
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.
Penner 29
Appendix A: Formulations of the second law
Using Entropy:
“In any spontaneous process there is always an increase in the entropy of the universe.”10
“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.”11
“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.”12
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.”13
“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.”14
“An engine operating in a cycle cannot transfer heat from a cold reservoir to a hot
reservoir without some other effect on its environment.”15
“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.”16
10
S. S. Zumdahl and S. A. Zumdahl, Chemistry (Houghton Mifflin, Boston, 2000), p.
798.
11
R. D. Knight, Physics for scientists and Engineers (Pearson Education, San Fransisco,
2004), p. 566.
12
A. Van Heuvelen, Physics: A General Introduction (Little, Brown and Company,
Boston, 1982), 213.
13
J. Sanny and W. Moebs, University Physics (Wm. C. Brown, Dubuque, IA, 1996), p.
381.
14
P. A. Tipler, College Physics (Worth, New York, 1987), p. 329.
15
H. C. Ohanian, Physics (W. W. Norton & Company, New York, 1989), p. 560.
16
D. Halliday and R. Resnick, Fundamentals of Physics (John Wiley & Sons, New York,
1986), p. 439.
Penner 30
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.”17
“It is impossible to construct a heat engine that, when operating in a cycle, completely
converts heat into work.”18
“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.”19
“An engine operating in a cycle cannot transform heat into work without some other
effect on its environment.”20
“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.”21
17
D. Halliday and R. Resnick, Fundamentals of Physics (John Wiley & Sons, New York,
1986), p. 439.
18
J. Sanny and W. Moebs, University Physics (Wm. C. Brown, Dubuque, IA, 1996), p.
380.
19
P. A. Tipler, College Physics (Worth, New York, 1987), p. 328.
20
H. C. Ohanian, Physics (W. W. Norton & Company, New York, 1989), p. 559.
21
J.T. Vanderslice, H. W. Schamp, Jr., and E.A. Mason, Thermodynamics (Prentice-Hall,
Englewood Cliffs, NJ, 1966), p. 29.
Penner 31
Appendix B: Possibility Catalog for the
Conventional Postulate of Thomson
T
T
T
Q
Q
Q
(a)
(b)
(d)
W
(e)
(f)
TH
TH
TH
QH
QH
QH
QH
QC
QC
QC
QC
TC
TC
(g)
TC
(h)
T
(j)
T
Q
W
(k)
W
(l)
TH
(m)
QC
TC
TC
(r)
TH
TH
QH
W
QC
QH
W
QC
TC
(t)
TH
QH
W
QC
TC
QC
TC
(u)
W
QC
(q)
TH
QH
W
QC
TC
QH
TH
QH
W
(p)
W
(n)
QC
TC
Q
W
TH
QH
W
(o)
T
Q
W
TH
QH
TC
(i)
T
Q
(s)
Q
(c)
W
TH
T
TC
(v)