The two parts of the second law of thermodynamics

Abstract

According to Sommerfeld, the well known Clausius and Kelvin statements of the second law of thermodynamics comprises two parts. The first part includes the Carnot principle that all Carnot engines operating between the same temperatures have the same efficiency. The second part contains the law of increase in entropy. Usually, the two parts are understood as a logical consequence of these statements, including the Carnot principle. Here, we argue that this principle need not be a derived law and may be considered as a fundamental law, without the need of demonstration. To this end we analyze the roots of the second law, which are contained in the memoir of Carnot on the production of work by heat, and its emergence in the papers of Clausius on heat.

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DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2018-0174
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The two parts of the second law of thermodynamics
Mário J. de Oliveira∗1
1Universidade de São Paulo, Instituto de Física, São Paulo, SP, Brasil
Received on June 09, 2017; Revised on July 09, 2018; Accepted on July 18, 2018.
According to Sommerfeld, the well known Clausius and Kelvin statements of the second law of thermodynamics
comprises two parts. The rst part includes the Carnot principle that all Carnot engines operating between the
same temperatures have the same eciency. The second part contains the law of increase in entropy. Usually, the
two parts are understood as a logical consequence of these statements, including the Carnot principle. Here, we
argue that this principle need not be a derived law and may be considered as a fundamental law, without the
need of demonstration. To this end we analyze the roots of the second law, which are contained in the memoir of
Carnot on the production of work by heat, and its emergence in the papers of Clausius on heat.
Keywords: thermodynamics, second law of thermodynamics, Carnot principle, Clausius principle.
1. Introduction
The second law of thermodynamics as conceived by Clau-
sius [1] and Kelvin [2] can be understood as having two
parts, which Sommerfeld [3] called the rst and second
parts of the second law. The rst part includes the Carnot
principle, which states that all engines operating accord-
ing to the Carnot cycle with the same temperatures have
the same eciency, and its main consequence that the
ratio
dQ/T
of the innitesimal heat and the temperature
is an exact dierential, which allows the denition of
entropy. The second part comprises the law of increase
in entropy. In addition to the statements of Clausius
and Kelvin, several other statements of the second law
have appeared. Some are equivalent to the statements
of Clausius and Kelvin, some are equivalent to the rst
part only, and some are equivalent to the second part
only.
The rst aspect that comes to mind regarding the two
parts is that the rst part concerns systems that are in
thermodynamic equilibrium, or more precisely systems
that are undergoing an equilibrium process. The second
part, on the other hand, refers to systems that are not in
equilibrium. If the second law of thermodynamics is un-
derstood as a law about systems undergoing nonequilib-
rium processes, than only the second part should properly
be identied as the second law.
Usually, both the rst and the second part of the second
law are derived from the Clausius or Kelvin statements
of the second law. However, the derivation is not free
from diculties and, in fact, it has been criticized on the
grounds that the reasonings and the hidden assumptions
employed in the derivation are not totally clear [4]. Since
the Carnot principle, of the rst part, is considered to
∗Correspondence email address: oliveira@if.usp.br.
be true for systems in thermodynamic equilibrium, these
diculties can be overcome by assuming that the Carnot
principle is an independent fundamental law, without
the need of demonstration. As to the second part, we
will see later that it can be derived from the fundamental
law that heat spontaneously ow from a hot to a cold
body, which we call Clausius principle.
The independence of the Carnot and Clausius princi-
ples, and thus of the two parts of the second law, has
conceptually certain advantages. Temperature is con-
sidered to be an elementary concept of the theory and
entropy is a derived concept dened by
dS
=
dQ/T
. One
could invert this proposition and assume that entropy is
an elementary concept, and dene temperature from en-
tropy, for systems in equilibrium. In fact, this is the point
of view implicit in the Gibbs approach to equilibrium
thermodynamics [5, 6]. Clausius principle is then used to
show that entropy has the properties of convexity, which
is the hallmark of Gibbs equilibrium thermodynamics.
This is also the point of view adopted by Callen in his
Thermodynamics [7].
Another advantage is found in the development of a
thermodynamics for systems out of equilibrium. Consider-
ing Carnot principle as independent of Clausius principle,
we may dispose of the Carnot principle, which is valid
for systems in equilibrium, without aecting Clausius
principle which continues to be valid. We should also
mention the simplication one gains by teaching the
fundamental laws of thermodynamics according to the
present perspective [8].
From the theoretical point of view, the roots of the sec-
ond law, and of the Carnot and Clausius principles, are
found on the memoirs of Carnot [9] and Clapeyron [10].
The emergence of the second law is found in the rst
paper by Clausius on heat [1]. Although the memoirs
Copyright by Sociedade Brasileira de Física. Printed in Brazil.

e20180174-2 The two parts of the second law of thermodynamics
of Carnot and Clapeyron did not receive a proper at-
tention of the scientic community at the time of their
publications [11], in 1824 and 1834, respectively, they
were recognized later on by Kelvin in 1848 [12] and by
Clausius in 1850 [1]. In the following, we present the
theoretical development of the theory of heat in the pe-
riod around the transition from the caloric theory to
thermodynamics, focusing on the papers of Carnot and
Clausius on heat. At the same time we present our point
of view concerning the two parts of the second law of
thermodynamics.
2. Carnot’s theory
The investigation on the theory of heat before the emer-
gence of thermodynamics comprised the search for laws
related to the thermal behavior of bodies, such as the
adiabatic and thermal expansion of gases, and the deter-
mination of the specic heat of the substances [13, 14].
The production of mechanical work by heat was a sub-
ject that was not addressed from a general theoretical
point of view, despite the fact that it was quite evident
that a gas performs work in an expansion and that the
steam machine, which produces mechanical work from
heat, had already been invented and developed since
the beginning of the eighteenth century [15]. A notable
exception was Carnot who published in 1824 a book on
the subject, entitled Réexions sur la Puissance Motrice
du Feu [9], that is, thoughts on the mechanical work
produced by heat. He was followed on this matter by
Clapeyron, who published in 1834 a memoir where he pre-
sented Carnot’s ideas in analytical form by the use of the
pressure-volume diagram, a convenient expedient since
work is identied as an area in this diagram [10]. The
publications of Carnot and Clapeyron, which revealed
to be relevant for the developement of thermodynamics,
received little attention [11] but were nally recognized
and appreciated by Kelvin in 1848 [12] and by Clausius
in 1850 [1].
The relationship between mechanical work and heat
was approached by Carnot within the prevailing theory
of heat of his time, the caloric theory [13, 14], which
recognizes heat as a material uid called caloric. Accord-
ing to Poisson [16] caloric is an imponderable material
substance contained in every part of a body that passes
from one body to the other without being annihilated or
created. The conservation of caloric in thermal processes
was expressed as a law by Lavoisier and Laplace in their
memoir on heat as follows [17]:
Proposition 1: the amount of free heat always
remains the same in the simple mixture of
bodies.
In addition to the conservation of caloric, which Carnot
took for granted, he grounded his theory on a fundamen-
tal rule concerning the behavior of the caloric. According
to this rule [9],
Proposition 2: caloric ows in such a way
as to restore equilibrium, passing from a hot
body to a cold body.
Equilibrium of the caloric is reached when the temper-
atures of the bodies become equal. The production of
work in a heat machine, explains Carnot, is due to the
transport of caloric from a hot to a cold body, and not to
its consumption. A consequence of this fundamental rule
is that a necessary condition for the production of work
by a heat machine is the existence of bodies at distinct
temperatures inducing a ow of caloric. How the ow
of caloric produces mechanical work was explained by
Carnot by an ingenious manner as we shall see.
A keen observation of Carnot concerning the function-
ing of a heat machine is its cyclic operation. The agent
that produces works, which may be a gas, a vapor or
any other substance susceptible of volume changes, op-
erates in a cycle, absorbing heat from a reservoir at a
higher temperature and delivering heat to a reservoir at
a lower temperature. Carnot makes use of a specic cycle,
consisting of four successive stages. (1) In the rst, the
working agent undergoes an isothermal expansion while
being in contact with a heat reservoir at a temperature
θ1
. (2) In the second, kept in isolation, the working agent
undergoes a volume expansion and a consequent fall of
temperature. (3) In the third, the working agent under-
goes an isothermal compression while being in contact
with a heat reservoir at a temperature
θ2
smaller than
θ1
. (4) In the last stage, the working agent in isolation
undergoes a volume contraction and a consequent rise
of temperature. In stage (1) the working agent receives
a certain quantity of heat
q
from the reservoir, whereas
in stage (3) it delivers the same quantity of heat
q
to
the reservoir, in accordance with the law of conservation
of caloric. In stages (2) and (4) there is no exchange of
caloric between the working agent and the environment.
Carnot asks for the condition for the maximum work
one can extract from a heat machine in regard to a given
quantity of heat. This should occur if any change in
temperature of the working agent is accompanied by the
change of its volume, and a consequent production of
work. If this does not happen, the ow of caloric, due to
the dierence in temperatures, would not produce work
and would be a useless reestablishment of the equilibrium
of the caloric, that is, a direct passage of the caloric from
a hot to a cold body without any other eect. To avoid
the direct passage of caloric, the working agent should
not be in contact with any other body. That is the reason
why in the processes (2) and (4) of the cycle the working
agent is in isolation, meaning that there is no exchange of
heat. Carnot explains that this type of process is the same
as that considered by Laplace to explain the propagation
of sound in air [18], and occurs in a rapid expansion or a
sudden contraction of a gas enclosed on a recipient.
According to our understanding, each stage of the
cycle is carried out in such a way that the working agent
undergoes an equilibrium process. This understanding is
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Oliveira e20180174-3
implicit in the Clapeyron presentation of Carnot cycle
due to his use of the pressure-volume diagram [10]. Any
point of this diagram is understood as an equilibrium
state and a trajectory drawn on it is understood as a
sequence of equilibrium states, that is, an equilibrium
process.
In the following, Carnot argues that the work does
not depend on the substance of the working agent. This
fundamental principle is expressed by Carnot as follows
[9]:
Proposition 3: The motive power of heat is
independent of the agents used to realize it; its
quantity is xed solely by the temperatures of
the bodies between which, nally, the transport
of caloric is eected.
It is implicit in this statement that it refers to the Carnot
cycle described above.
To explain how work is actually generated in a heat
machine operating according to his cycle, Carnot makes
a surprising and striking analogy with a mechanical sys-
tem. When a load descends from a certain height to a
lower height, it performs a quantity of work which is
proportional to the mass of the load and to the dier-
ence in heights. Thus, when a certain quantity of heat
q
”descends” from a high temperature
θ1
to a low tem-
perature
θ2
, the working agent performs a mechanical
work
w
which is proportional to
q
and to the dierence in
temperatures
θ1−θ2
. According to the fundamental prin-
ciple, proposition 3,
w
does not depend on the substance
of the working agent. Carnot does not say explicitly, but
the motivation for his fundamental principle may be the
result of the analogy with the mechanical system for
which the work depends on the mass of the load but not
on the substance of the load.
Carnot argues that for a given dierence in tempera-
ture the work is smaller at high temperatures and greater
at low temperatures so that the dependence on the tem-
peratures is more complex than just being proportional to
the dierence in temperatures. If we denote by
f
(
θ1, θ2
)
the function of the temperatures that describes this de-
pendence, then we may write
w
q=f(θ1, θ2).(1)
This equation, valid for a Carnot cycle, complemented
by the statement that the function
f
is universal, that
is, it does not depend on the substance of the body
undergoing the Carnot cycle, is the expression of the
Carnot fundamental principle. Notice that the ratio on
left-hand side of equation (1) is the eciency of an engine
operating according to a cycle.
3. Fundamental laws
3.1. Law 1
According to Kuhn [19], the law of conservation of energy
was an example of a simultaneous discovered, announced
by several authors between 1842 and 1847, including
Mayer [20, 21] and Joule [22
–
24]. In fact, Mayer and
Joule advanced the idea that the dissipation of a certain
quantity of work
w
always results in the generation of
the same quantity of heat
q
, and conversely, the disap-
pearance of a certain quantity of heat
q
gives rise to the
same quantity of work w, which we write as
w=Jq, (2)
where
J
is a universal constant. This law, which we call
Mayer-Joule principle, was interpreted as the conser-
vation of energy, because heat was identied as being
related to the work of microscopic particles. However,
it should be mentioned that the Mayer-Joule law is a
macroscopic law. The law of conservation of energy as
advanced by Helmholtz [25], on the other hand, was a
microscopic law. It gave support to the Mayer-Joule law
but no one derived the macroscopic law from the mi-
croscopic law
1
From now on we use the notation
W
for
the mechanical work and measure heat
Q
in terms of
mechanical unit, which amounts to say that
Q
=
Jq
, and
the Mayer-Joule law becomes W=Q.
If we consider that a system is undergoing a certain
thermodynamic process, the variation of its energy
U
,
according to the Mayer-Joule law, is written in the form
∆U=Q−W, (3)
where
Q
is the heat introduced into the system and
W
is
the work performed by the system. In dierential form,
dU =dQ −dW. (4)
In the case of a mechanical work
dW
=
pdV
where
p
is the pressure and
V
is the volume of the system. In
accordance with the conservation of energy, ∆
U
does not
depend on the process but just on the nal and initial
states, which means that dU is an exact dierential.
In equation (3), the heat
Q
is interpreted as work
exchanged with the system as much as
W
. The way of
distinguishing these two quantities is to say
W
is the
adiabatic work by the introduction of the concept of
adiabatic wall. The main property of an adiabatic wall
is to forbid the passage of heat. It should be remarked,
that this property is not a denition of adiabatic wall,
because otherwise we would face a circular reasoning.
Adiabatic wall should be understood theoretically as a
primitive concept, without the need of denition [26].
1
In fact, the derivation of macroscopic laws from microscopic laws
was the aim of the kinetic theory advanced by Clausius, Maxwell
and others.
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e20180174-4 The two parts of the second law of thermodynamics
3.2. Law 2.1
The Mayer-Joule law was clearly in conict with Carnot’s
theory because the latter was based on the conservation
of caloric. Joule suggested the abandoning of Carnot’s
theory [24] but Kelvin disapproved this solution [27].
This conict was solved by Clausius [1] by observing that
Carnot principle, expressed by proposition 3, could be
separated into two parts. One of them was the conser-
vation of caloric, implicit in proposition 3, which could
be abandoned. Thus, Clausius retained only the part of
Carnot principle expressed by equation (1), which was a
consequence of proposition 3, with
q
interpreted as the
heat absorbed in a Carnot cycle, which we denote by
Q1
.
Thus, equation (1) becomes
W
Q1
=f(θ1, θ2).(5)
Within Clausius’ theory, equation (5), valid for a Carnot
cycle, complemented by the statement that the function
f
is universal, that is, it does not depend on the substance
of the body undergoing the Carnot cycle, becomes the
expression of the Carnot fundamental principle within
Clausius’ theory.
In accordance with Mayer-Joule law, Clausius writes
Q1−Q2=W, (6)
where
Q2
is the heat delivered by the working agent.
Combining equations (5) and (6), it follows that
Q2/Q1
is a universal function of
θ1
and
θ2
. By a combination of
two Carnot cycles we may conclude that
Q2
Q1
=T(θ2)
T(θ1),(7)
where
T
(
θ
)is a universal function of
θ
, which we may
call absolute temperature. Using the notation
T1
=
T
(
θ1
)
and T2=T(θ2)we may write
Q2
Q1
=T2
T1
.(8)
Employing the convention that heat is positive when
it is absorbed by the working agent and negative when it
is released, then equation (8) may be writen as
Q1/T1
+
Q2/T2
= 0. The generalization of this equation leads to
the result IdQ
T= 0,(9)
valid for any cycle, which means that
dQ/T
is an exact
dierential. Thus there exists a state function
S
, called
entropy by Clausius [28], whose dierential
dS
is given
by
dS =dQ
T.(10)
It should be remarked that all the equations refer to
equilibrium processes, which, in accordance with Clapey-
ron representation [10] are understood as trajectories in
the pressure-volume diagram. Equation (10) with the
understanding that
dS
is an exact dierential and that it
is valid for a system undergoing an equilibrium process,
we call law 2.1. It is the fundamental consequence of the
Carnot principle. If we integrate equation (10) along any
trajectory, we nd
∆S=ZdQ
T.(11)
Again, we remark that equation (11) should be under-
stood as valid for an equilibrium process.
Another way of reaching equation (10) was conceived
by Carathéodory. Instead of using Carnot principle, he
uses a principle stating that [29]: ”In any arbitrary neigh-
borhood of a given initial state, there are states that can
not be arbitrarily reached by an adiabatic process.” In
other words, there are states that can only be reached
with the exchange of heat. This is the case, for instance,
of two states belonging to the same isotherm.
Let us consider an adiabatic work of the general form
dW =X
i
yidXi.(12)
Then, the conservation of energy in dierential form,
given by equation (4), can be writte as
dQ =dU +X
i
yidXi.(13)
Although
dU
and
dXi
are exact dierentials,
dQ
is not
in general. Following Born [30], one has to show that
there exists an integrating factor of
dQ
. If that is the
case, then it suces to postulate that the reciprocal of
this factor is temperature to reach law 2.1 expressed
by equation (10). In the case of more than two term
on the right-hand side of (13), there might not exist an
integrating factor [30]. However, if the condition given by
Carathéodory principle is fullled than the integrating
factor does exist, as demonstrated by Carathéodory [29],
and law 2.1 can also be thought as a consequence of this
principle.
3.3. Law 2.2
In his rst memoir on heat [1], Clausius developed his
theory by assuming two fundamental principles. The rst
was the principle of equivalence of heat and work, ad-
vanced by Mayer and Joule and later identied as the
conservation of energy, called the rst law of thermody-
namics. The second was a new principle developed by
Clausius, inspired on a fundamental rule advanced by
Carnot and related to the production of work in heat
machines, which is proposition 2 stated above. According
to this new principle of Clausius [1],
Proposition 4: [heat] shows everywhere the
tendency to equalize dierences of tempera-
ture, and thus to pass from a warmer body to
a colder one.
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Oliveira e20180174-5
It was rephrased by Kelvin [2], who called the axiom of
Clausius, as:
Proposition 5: It is impossible for a self-acting
machine, unaided by any external agency, to
convey heat from one body to another at a
higher temperature,
which became known as the Clausius statement of the
second law of thermodynamics. Kelvin also introduced
an axiom of his own [2]:
Proposition 6: It is impossible, by means of
inanimate material agency, to derive mechan-
ical eect from any portion of matter by cool-
ing it below the temperature of the coldest of
the surrounding objects,
which became known as Kelvin statement of the second
law. Although the statements have dierent forms, wrote
Kelvin [2], one is a consequence of the other. Later on,
Clausius gave his own statement by writing [31]:
Proposition 7: Heat can never pass from a
colder to a warmer body, if no other related
change occurs at the same time.
It can be rephrased in the following terms. Heat ow
spontaneously from a hotter to a colder body. A brief
statement was also given by Clausius [32]:
Proposition 8: heat cannot by itself pass from
a colder to a hotter body.
To be specic, we call proposition 7, Clausius fundamental
principle.
From the second law, Clausius establishes the inequal-
ity [28, 31]
∆S≥ZdQ
T0,(14)
where ∆
S
is the variation of the entropy of the system
during a process,
dQ
is the innitesimal heat exchanged
with the environment, being positive when it is intro-
duced into the system, and
T0
is the temperature of the
environment, not of the system.
Equation (14), which we call law 2.2, is a direct con-
sequence of the Clausius fundamental principle. To see
this, we proceed as follows. We consider a mechanical
system consisting of two bodies A and B undergoing cer-
tain processes. They are isolated so that the total energy
U
+
U0
=
U0
and the total volume
V
+
V0
=
V0
are
invariant, where unprimed and primed variables refer to
bodies A and B, respectively. To determine the variation
of entropy of each body, we may use equilibrium pro-
cesses because the variation of entropy does not depend
on the process but only on the initial and nal states,
which are assumed to be equilibrium states. Denoting by
γ
and
γ0
the equilibrium processes connecting the initial
and nal states, then the variation of entropy of bodies
A and B are
∆S=Zγ
dQ
T,∆S0=Zγ0
dQ0
T0,(15)
where
dQ
=
dU
+
pdV
is the innitesimal heat received
by body A and
T
is its temperature, whereas
dQ0
=
dU0
+
p0dV 0
is the innitesimal heat received by body B
and T0is its temperature.
The process
γ0
is understood as a trajectory in space
(
U0, V 0
), and the process
γ
is understood as a trajectory
in space (
U, V
). We choose the process
γ0
to be connect to
the process
γ
by means of
U0
=
U0−U
and
V0
=
V0−V
.
The variation of entropy ∆S0is then written as
∆S0=Zγ
dQ0
T0,(16)
where the integral is understood as an integral in space
(
U, V
), obtained by the transformation
U0
=
U0−U
and
V0
=
V0−V
so that
dU0
=
−dU
,
dV 0
=
−dV
, and
dQ0=−dU +p0dV . The total entropy variation is then
∆S+ ∆S0=ZγdQ
T+dQ0
T0.(17)
Up to this point,
γ
can be any process connecting
the initial and nal states. But now, we choose
γ
as
composed by two subprocesses. The rst one, which we
call
α
, is such that
dQ/T
+
dQ0/T 0
= 0 or (1
/−
1
/T 0
)
dU −
(
p/T −p0/T 0
)
dV
= 0, which denes a trajectory in space
(
U, V
). Thus along the subprocess
α
the contribution
to the integral (17) vanishes. The second subprocess,
which we call
β
is such that
dV 0
=
dV
= 0 so that
dQ0
=
−dQ
because in this case
dQ
=
dU
,
dQ0
=
dU0
,
and
dU0
+
dU
= 0. Therefore, we may write the total
variation of the entropy as
∆S+ ∆S0=Zβ
dQ 1
T−1
T0.(18)
Let us suppose that
dQ >
0, that is, the body A re-
ceives a certain quantity of heat from the body B. Taking
into account that integral (18) is performed keeping the
volume of A invariant, implying that the volume of B
is also invariant, then the only possibility from Clausius
fundamental principle is that
T < T 0
, which can be
translated into the inequality
dQ/T −dQ/T 0≥
0. The
same inequality holds if the body A gives heat to the
body B because in this case
dQ <
0, but now
T > T 0
.
We may conclude that the integral in equation (18) is
greater or equal to zero so that
∆S+ ∆S0≥0.(19)
If we use equation (16) to replace ∆
S0
, we reach the
inequality (14).
4. conclusion
We have argued that the Carnot principle can be con-
sidered as a fundamental principle without the need of
demonstration, and thus independent of the Clausius
fundamental principle. The Carnot principle leads to the
DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2018-0174 Revista Brasileira de Ensino de Física, vol. 41, nº 1, e20180174, 2019

e20180174-6 The two parts of the second law of thermodynamics
laws 2.1, represented by equation (10), whereas the Clau-
sius principle leads to law 2.2, represented by the Clausius
inequality (14). These two principles, together with the
conservation of energy are taken as the fundamental laws
of thermodynamics. It is implicit in this approach that
temperature is an elementary concept whereas the en-
tropy is a derived concept, that is, entropy is dened by
the use of law 2.1, which involves the temperature.
Conceptually, the present approach has certain advan-
tages. For example, we may assume that entropy is an
elementary concept holding the property given by equa-
tion (19), which is equivalent to law 2.2. The temperature
is then considered to be a derived concept, dened by
the use of law 2.1. This scheme is in fact used by Callen
in his treatment of equilibrium thermodynamics, where
temperature is dened by
T
=
∂U /∂ S
. It is also implicit
in Gibbs equilibrium thermodynamics in which the con-
vexity properties of entropy is a direct consequence of
equation (19).
The roots of the second law of thermodynamics, from
the theoretical point of view, are found on the memoir
of Carnot on heat. We have made an appreciation of
this memoir pointing out the understanding of Carnot
concerning the production of work by heat, and the ideas
that led him to the main principles of his theory of heat.
Carnot’s theory was based on the conservation of caloric
which was found to be in conict with the Mayer-Joule
law on equivalence of work and heat. We have described
how this conict was solved by Clausius. From Clausius
fundamental principle, we have shown how to derive the
Clausius inequality, the best representation of the second
law of thermodynamics.
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Revista Brasileira de Ensino de Física, vol. 41, nº 1, e20180174, 2019 DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2018-0174