On the Clausius equality and inequality

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Eur. J. Phys. 32 (2011) 279–286 doi:10.1088/0143-0807/32/2/002

On the Clausius equality and inequality

Joaquim Anacleto

Departamento de Fısica da Escola de Ciencias e Tecnologia da Universidade de Tras-os-Montes eAlto Douro, Apartado 1013, P-5001-801 Vila Real, PortugalandIFIMUP and IN, Institute of Nanoscience and Nanotechnology, Departamento de Fısica daFaculdade de Ciencias da Universidade do Porto, R. Campo Alegre 687, P-4169-007 Porto,Portugal

E-mail: anacleto@utad.pt

Received 23 August 2010, in final form 21 November 2010Published 5 January 2011Online at stacks.iop.org/EJP/32/279

AbstractThis paper deals with subtleties and misunderstandings regarding the Clausiusrelation. We start by demonstrating the relation in a new and simple way,explaining clearly the assumptions made and the extent of its validity. Thenfollows a detailed discussion of some confusions and mistakes often found inthe literature. The addressed points include the issue of temperature in theClausius relation and closely related concepts, such as heat, reversibility andreservoir. The ideas presented in this study are primarily intended for graduatestudents and teachers, and may also be of interest to undergraduate studentswith a solid background in thermodynamics.

1. Introduction

Thermodynamics has several subtleties that make this area of physics fascinating. Themathematics required to deal with its laws and concepts is not too complex, but theaforementioned subtleties are an additional challenge for teaching and have stimulatedinteresting discussions in order to make thermodynamics unambiguous and attractive,reinforcing thus its formative role, in particular, for physics students.

The concept of entropy seems to be the most difficult one to grasp [1–3], possibly becauseits physical meaning is closely related to other difficult concepts, such as that of irreversibility.Entropy is ingrained in the Clausius relation, which is a mathematical formulation of the secondlaw of thermodynamics [4, 5]. Although the aforementioned relation is well established intextbooks and included in higher education curricula for thermodynamics, there remain somedoubts and confusions [6]. Some effort has been made to discuss various concepts, such asheat, work, reservoirs and reversibility, but the discussion presented here is, to the best of ourknowledge, absent from the literature.

0143-0807/11/020279+08$33.00 c© 2011 IOP Publishing Ltd Printed in the UK & the USA 279

280 J Anacleto

Perhaps the most sensitive issue concerning the Clausius inequality is understanding thetemperature that appears in it. In the literature, we find it as the temperature of heat reservoirsthat interact with the system [5], as the temperature of the system itself [6] and also as thetemperature at which the heat is supplied to or removed from the system [7]. This paper aims toaddress this issue by obtaining the Clausius relation in an unusual, clear and simple way. Takinginto account the assumptions made during the demonstration, the usual misunderstandings areexplained and some recommendations are made. Though primarily intended for graduatestudents and teachers, who may find help for their teaching tasks, this study might also beuseful to undergraduate students with a solid background in thermodynamics.

2. The Clausius relation

A thermodynamical process is an interaction between a system and its surroundings, the twobeing separated by a boundary. The energy flow through the boundary is heat Q or work W ,which are positive when they correspond to energy entering into the system. Even thoughthe ideas we aim to address are very general, we assume PVT systems characterized by thepressure P, volume V, temperature T and entropy S.

The system–surroundings interaction is described by [3, 8–10],

T dS − P dV = −Te dSe + Pe dVe, (1)

where the subscript ‘e’ stands for external and will be used henceforward for the surroundings’variables. Equation (1) is more informative than one may realize at first glance, since itencompasses the first and second laws while combining variables of both the system and thesurroundings, allowing the differential of the system entropy dS to be expressed as

dS = 1

T[−Te dSe + (Pe dVe + P dV )] . (2)

The formalism of thermodynamics is greatly simplified if we consider that thesurroundings are made up of reservoirs. Conceptually, a reservoir is a very large systemcapable of exchanging an unlimited amount of energy while suffering no measurable changein its own intensive thermodynamic variables. The processes undergone by the reservoir arealways assumed to be slow, so that dissipative effects do not occur, i.e. the reservoir is a systemalways in complete internal equilibrium, and all processes occurring inside it are reversible[5]. Moreover, depending on whether the reservoir experiences changes in its entropy or not, itis a heat reservoir or a work reservoir, respectively. The main advantage in using reservoirs isthat each term within the square brackets on the right-hand side of (2) gets a precise meaning.The first two terms are the differentials of heat and work [10, 11], i.e.

δQ = −Te dSe, (3)

δW = Pe dVe. (4)

The quantity within the parentheses in (2) is the dissipative work δWD, which is the differencebetween the total work, given by (4), and the configuration work, given by

δWC = −P dV. (5)

Dissipative work is the dissipated mechanical energy and is always greater than or equal tozero [10, 12], i.e.

δWD = Pe dVe + P dV � 0. (6)

On the Clausius equality and inequality 281

Therefore, (2) can be read as

dS = 1

T(δQ + δWD) . (7)

That δWD � 0 does not mean that dissipated energy will end up entirely in the system.In general, it will be divided by the system and the surroundings. Defining α as the fractionof that energy ending up in the system,

0 � α � 1, (8)

the fraction (1 − α) that ends up in the surroundings has to be (negative) heat given by

δQD = − (1 − α) δWD, (9)

which suggests that, like the work, the heat can be written as the sum of two components, onegiven by (9), δQD, and the other is the heat by conduction, δQC, i.e.

δQ = δQC + (α − 1) δWD. (10)

Combining (7) and (10), dS can be rewritten as

dS = δQC

T+

α δWD

T. (11)

Dividing (10) by Te, one obtains

δQ

Te=δQC

Te+

(α − 1) δWD

Te. (12)

By the second law, heat cannot flow spontaneously from lower to higher temperatures,which enables us to write the following implications,

Te > T ⇒ δQC � 0, (13)

Te < T ⇒ δQC � 0, (14)

which together are equivalent to

δQC

T� δQC

Te. (15)

On the other hand, from (8) and since

−1 � (α − 1) � 0, (16)

we haveα δWD

T� (α − 1) δWD

Te. (17)

Using (15) and (17), the left-hand sides of (11) and (12) are related by

dS � δQ

Te, (18)

which applied to a cycle gives the Clausius relation,∮δQ

Te� 0. (19)

A process is reversible if and only if it takes place without increasing the entropy of theuniverse, dS + dSe, which from (3), (10) and (11) is given by

dS + dSe = δQC

(1

T− 1

Te

)+ δWD

(α

T− α − 1

Te

). (20)

282 J Anacleto

Since (α/T − (α − 1)/Te) > 0, the right-hand side of (20) is zero (i.e. the process is reversible)if and only if the proposition

(δQC = 0 or T = Te) and (δWD = 0) (21)

is true, which means that the reversibility is equivalent to the conditions in (21), which in turnare the necessary and sufficient conditions to have the equality in (15) and (17), and hence in(18) and (19). Conversely, the inequality holds for irreversible processes.

Dividing now (10) by T, we get

δQ

T=δQC

T+

(α − 1) δWD

T, (22)

where the second term on the right-hand side, by (8) and (16), satisfies

α δWD

T� (α − 1) δWD

T. (23)

From (11), (22) and (23), it follows that

dS � δQ

T, (24)

which applied to a cycle gives∮δQ

T� 0. (25)

The equality in (24) and (25) is determined only by (23) and holds when δWD = 0, i.e. whendissipative work is zero regardless of whether the process is reversible or not. Conversely, ifδWD > 0, the inequality holds instead.

3. Subtleties and misunderstandings

In the previous section, starting from (1) and using some key ideas, in particular the conceptof reservoir and the positive nature of the dissipative work, the Clausius relation (19) wasestablished in an unusual and simple way. Although the aforementioned relation appearsto be well established in the literature, it is common to find conceptual subtleties andmisunderstandings, which we address below.

3.1. System–surroundings interchangeability

The Clausius relation (19) is a mathematical formulation of the second law, which is closelyrelated to the concept of reversibility. A process is reversible if at the end of the processboth the system and the surroundings can return to their initial states [5]. As (19) refers tocyclical processes, it is not possible to apply it simultaneously to both the system and thesurroundings, except when the equality occurs, i.e. unless the cycle is reversible. In otherwords, while the equality in (19) allows that both the system and the surroundings performcyclic processes, the inequality requires that surroundings undertake an open process, andtherefore the roles of the system and the surroundings become non-interchangeable. Also,interestingly, although relation (19) applies to the system, the temperature therein refers tothat of the surroundings. Typically, manuals mainly describe the system and do not addressthese issues, but the surroundings must be taken into account for a better understanding ofirreversible processes.

On the Clausius equality and inequality 283

3.2. Definition of heat

The quantity δQ in relations (18)–(19) and (24)–(25) is the heat exchanged between the systemand the surroundings. Although the definition of heat has motivated some controversialdiscussions [8, 10, 11, 13–15], the definition of δQ is given by (3), provided that heat isexchanged by means of heat reservoirs.

3.3. The surroundings’ temperature

Perhaps the most common misconceptions are related to the temperature. As shown previously,the temperature in (18) and (19) is the surroundings’ temperature, although this is not alwayssufficiently emphasized [4]. Zemansky [5, p 205] makes this point clear when discussing theClausius inequality: ‘. . . we must point out that T represents the temperature of the reservoiror the auxiliary reversible engine that provides the small quantity of heat δQ. . .’. However,the notation used for both the system and the surroundings’ temperature is the same, whichis didactically inappropriate as it can lead to confusion. Thus the use of different notationsfor the system and surroundings’ temperatures is highly recommended, as is adopted here,denoting them by T and Te, respectively.

3.4. System temperature

In some places, relations (24)–(25) are wrongly stated as the second law, as in [6, equation(11.7), p 84], where we find the following sentence: ‘The formulation (11.7) is limited bythe fact that the temperature T of the system must be defined’ [6, p 85]. Confusion of (18),(19) with (24), (25) is common, but the explanation of this issue is virtually absent from thethermodynamics textbooks. As seen in the previous section, both the aforementioned sets ofrelations are true and are mathematically similar, but differ concerning the physical meaningof the equality, inequality and temperature. In reversible processes, often used to explainconcepts, since Te = T , the two sets of relations (18)–(19) and (24)–(25) are equivalent andthe equality holds in both of them.

3.5. Temperature at which heat is exchanged

Regarding temperature in the Clausius relation, it is common to find sentences like ‘T is thetemperature at which the heat is supplied to the system’ [7, p 74]. This statement needsclarification, since in irreversible heat transfer the temperature at which heat is exchangedmight be undefined. Consider the very common situation of two bodies in thermal contact,one of which (the system) is at temperature T and the other one (surroundings) at temperatureTe, with Te > T , and δQ is the infinitesimal heat exchanged. The boundary is supposedto be structureless with such a low thermal conductivity so that the process is quasi-static.Moreover, as the boundary is also assumed to be very thin, an abrupt change in temperatureacross it occurs so that the temperature at which heat is exchanged becomes undefined. Thesituation is illustrated at the top of figure 1.

The bottom of figure 1 shows an enlargement of the boundary, making it visible nowthat temperature varies continuously from T to Te. In order to relate the irreversibility withthe spatial variation of temperature, the boundary is divided into n layers (with n being aninteger), so that in each one the temperature can be taken as nearly constant, where T1 = T andTn = Tn+1 = Te. The temperature at which heat is exchanged is the temperature of the layerj (1 � j � n) chosen as a new separating element between the system and the surroundings.

284 J Anacleto

Figure 1. Heat exchange between two bodies at different temperatures (top). At bottom is shown aclose-up view of the boundary, which was divided into n layers, so that in each one the temperatureis nearly constant.

So, the system is now composed of all that is left of the layer j , while everything else willbelong to the surroundings.

There are two choices of particular interest: the choice of layer n, for which the temperatureat which heat is exchanged Te and the surroundings behave as a heat reservoirs, since allirreversibilities are within the system. Therefore relations (18)–(19) are valid and if thesystem carries out a cyclic process, one has

∮(δQ/Te) = −�Se < 0. On the other hand,

if layer 1 is chosen, the temperature at which heat is exchanged is T and all irreversibilitiesare relegated to surroundings, which therefore cannot be considered a reservoir. Relations(24)–(25) are now applicable, and for a cyclic process one has

∮(δQ/T ) = 01 − �Se.

In general, for an arbitrary j , the temperature at which heat is exchanged is Tj , and theintegral

∮(δQ/Tj ) will have a value between 0 and −�Se,

−�Se �∮

δQ

Tj

� 0. (26)

However, the change in the surroundings’ entropy for a given cyclic process cannot dependon the choice of the temperature at which heat is exchanged. This suggests that theClausius inequality has to be generalized to take into account the irreversibilities withinthe surroundings. For the present example, note that the following equation holds for any nand j ,

δQ

Tj

+n∑

k=j

(−δQ

Tk

+δQ

Tk+1

)= δQ

Te, (27)

On the Clausius equality and inequality 285

which integrated in a cycle and using (19) gives

∮ ⎡⎣δQ

Tj

+n∑

k=j

(−δQ

Tk

+δQ

Tk+1

)⎤⎦ � 0. (28)

The Clausius relation (19) is only valid when the surroundings are made up of reservoirs,i.e. when Tj = Te; otherwise relation (28) must be used instead. This is in line with thegeneralization of the Clausius inequality made in [16].

The entropy generation δSGEN, which takes place at the boundary, is given by the additiveinverse of the summation å in (27) when j = 1,

δSGEN = δQ

T− δQ

Te=

n∑k=1

(δQ

Tk

− δQ

Tk+1

)� 0. (29)

3.6. Computing change in system entropy

Determining the entropy change presents some conceptual and practical difficulties, especiallyfor irreversible processes. It is stated and emphasized in thermodynamics textbooks that theequation

�S =∫

δQ

T, (30)

where T is the system temperature, can only be applied in reversible processes. This is awidespread but wrong idea.

This ambiguity becomes evident in problems dealing with changes in entropy of asystem whose temperature varies from Ti to Tf , by placing it in contact with a body at adifferent temperature (top of figure 1). Considering the system heat capacity C constant, suchdetermination is usually carried out as [17]

�S =∫

δQ

T=

∫C dT

T= C ln

Tf

Ti

. (31)

The above procedure requires clarification because even though the process is irreversible,integral (30) is used, almost always without any discussion. When some explanation isprovided, typically it is said that the actual irreversible process is replaced by another oneleading the system reversibly to the same final temperature [17, example 20.6, p 693]. Thisis always a possible explanation, but not the only one. What happens, but is virtually neverexplained, is that although the process is irreversible, dissipative work is zero (which isnot a sufficient condition to ensure reversibility), leading to the equality in (24). Thereforeintegral (30) is applicable, which allows the calculation of the variation of entropy usingprocedure (31).

4. Conclusions

Subtle aspects relating to the Clausius equality and inequality were discussed, enabling anew insight into the physics of irreversible processes. An equation describing the system–surroundings interaction, the concept of reservoir and the positive nature of dissipative workhave enabled a theoretical framework which comprises an original and simple demonstrationof the Clausius relation. The aforementioned theoretical framework was then used to analysein detail some common errors and misunderstandings, in particular about the temperatureappearing in the Clausius relation, making it clear that it is the surroundings’ temperature.

286 J Anacleto

Moreover, the entropy generation in a boundary across which heat is flowing was analysed,showing the need for a generalization of the Clausius inequality when the surroundings areno longer made up of heat reservoirs. The clarification of statements (18)–(19) and (24)–(25),followed by a discussion about issues that are sensitive and related to them, is an importantcontribution of this study, which is hoped to be useful to students and teachers because ittouches on topics that, as far as we know, are not satisfactorily explained in textbooks.

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