Research: Science and Education
Zeroth Law, Entropy, Equilibrium, and All That Sebastian G. Canagaratna Department of Chemistry and Biochemistry, Ohio Northern University, Ada, OH 45810;
[email protected] Recently Gislason and Craig (1) discussed the question of why two bodies in thermal contact come to a common temperature. They also discussed problems involving systems not in mechanical equilibrium. The discussion and the proofs given by Gislason and Craig are, understandably, colored by their approach to thermodynamics (2–4). Gislason and Craig appear to have eschewed the use of q, the quantity of heat transferred, presumably because their approach does not favor it. They are also critical of the traditional approach to solving such problems and give the impression that these methods are lacking in rigor. An examination of some popular physical chemistry texts as well as some texts on thermodynamics revealed that discussions of problems involving the establishment of pressure or temperature equilibrium are difficult to find. Most students would be more familiar with the traditional approach to thermodynamics than with the approach of Gislason and Craig. It is therefore useful to re-examine the problems discussed by Gislason and Craig using the methods of traditional thermodynamics. In the traditional approaches the use of q is not regarded as creating any problems. Using q, the traditional approach treats the establishment of temperature or pressure equilibrium without any loss of rigor. We also examine how traditional thermodynamics would deal with entropy maximization for some of the problems dealt with by Gislason and Craig. Gislason and Craig’s criticism of the traditional approach to solving problems involving attainment of thermal equilibrium centers both on the concept of equilibrium as well as the formulation and priority of the laws of thermodynamics in the traditional approach. We therefore begin with a very brief review of the salient aspects of equilibrium and the first and second laws, together with a fuller treatment of the zeroth law. The three laws enable us to examine whether the attainment of thermal and mechanical equilibrium can be treated rigorously by the application of the zeroth law, the law of mechanical equilibrium, and the first law. We then discuss the role of the zeroth law in the teaching of thermodynamics. Finally, some examples of entropy maximization treated by Gislason and Craig are treated by the traditional approach. Equilibrium Gislason and Craig state in the note of their article (1): In thermodynamics, whenever the concept of equilibrium is used prior to the development of the second law, this use is provisional. The second law governs equilibrium: ...
They further state (ref 1, p 885) that it is the second law that proves that thermal equilibrium does occur. In the traditional approach, the concept of equilibrium does not depend on the second law; the attainment of thermal equilibrium is dealt with by the zeroth law, not the second law. In fact, temperature is introduced first by the zeroth law, and it
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is then used in the formulation of the second law. It is important to distinguish the concept of equilibrium from the conditions that govern equilibrium. The concept of equilibrium itself only requires that the independent variables of a system be constant in time. For example, we find in Zemansky (5) A state of a system in which Y and X have definite values that remain constant so long as the external conditions are unchanged is called an equilibrium state.
Here, Y and X are independent coordinates that describe the system. To make this definite, suppose we define a sample of a pure substance by its temperature T and pressure p, and these were to remain constant in time. We would say that the system is in equilibrium and we are invoking no particular law in using the word equilibrium. Three Types of Equilibrium There are three aspects to the complete equilibrium of a thermodynamic system (ref 5, p 26, and ref 6). These aspects will be discussed in turn. Mechanics recognizes an equilibrium referred to as mechanical equilibrium. This equilibrium is described in terms of mechanical forces. For continuous systems, mechanical equilibrium is described in terms of pressure. When such a system is in mechanical equilibrium, the pressure of the system is the same throughout. This condition for mechanical equilibrium is referred to as Pascal’s law. If mechanical equilibrium does not exist, a change will take place that will ultimately establish the equality of pressures everywhere. Here we assume the absence of surface, gravitational, electric, and magnetic effects. Thus the law of mechanical equilibrium can predict, without invoking the second law, the direction of change in the attainment of mechanical equilibrium. It is possible for a system to be in mechanical equilibrium and still not be in complete equilibrium. Chemists are aware that slow changes in chemical composition can take place though the pressure and temperature are sensibly the same everywhere. Chemical equilibrium is equilibrium with respect to change of composition. It includes chemical reactions as commonly understood, but in addition it includes phase changes and diffusion due to differences in composition. The condition for chemical equilibrium is governed by the second law of thermodynamics. A third type of equilibrium, thermal equilibrium, has to be introduced because it was recognized that systems in mechanical and chemical equilibrium could still not be in equilibrium when they are put in thermal contact. The importance of the third type of equilibrium was the last to be recognized though the consequences of this type of equilibrium had been intuitively recognized earlier. The zeroth law was formulated after the first and second laws of thermodynamics, but it was considered to be more fundamental to the development of
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Research: Science and Education
traditional thermodynamics than the first and second laws of thermodynamics and was therefore dubbed the “zeroth” law of thermodynamics. Complete (thermodynamic) equilibrium requires all three types of equilibria. Mechanical and thermal equilibria require the equality throughout the system of the pressure and temperature, respectively. The second law deals with the conditions for complete equilibrium. The condition for mechanical equilibrium given by the second law is therefore exactly the same as that given by Pascal’s law. Similarly, the condition for thermal equilibrium given by the second law is exactly the same as that given by the zeroth law. This should not be surprising, since there clearly cannot be a conflict between different laws dealing with the same subject. If we are interested only in thermal equilibrium or in mechanical equilibrium, it is in general not necessary to invoke the second law explicitly. The First and Second Laws The first law is applicable to both equilibrium and nonequilibrium problems, but it does not govern the conditions of equilibrium. In the traditional approach, the concept of work in mechanics and the concept of adiabatic processes coming from the zeroth law are used to introduce a property termed internal energy, U. In some traditional approaches, the first law is used to define a measure of the thermal interaction, called heat, q (5–10). This approach is due to Born (11). Some authors prefer to use a calorimetric definition of heat. The relationship between the two approaches has been discussed recently (12). The concept of work and heat in the Gislason and Craig approach differs somewhat from the usual approach. A discussion of some aspects of this approach has been given recently (13). We will refer to the treatment in refs 5–9 as “traditional approaches”. Though there are slight differences in treatment, they agree as to the values of heat and work to be assigned in any given change. Most of the treatments found in the common physical chemistry texts would be characterized as “traditional”. The second law deals with the potential direction of change and with the conditions for complete (thermodynamic) equilibrium. In the traditional approaches, the concept of temperature (and therefore the zeroth law) is used to introduce an integrating factor for q to define an extensive function termed entropy, S, that (a) remains constant for an isolated system when the changes in the isolated system are reversible and (b) increases when the changes are irreversible. Subject to the conditions imposed on the isolated system, the entropy is a maximum. The second law also helps establish an absolute scale of temperature. In addition to giving the conditions of equilibrium, the second law may be used to make predictions about the stability of the equilibrium state. The second law applied to the thermal stability of systems shows that heat capacities must be positive (ref 7, p 30). The Zeroth Law of Thermodynamics The content of the zeroth law is essentially this: there is a property, which we shall call temperature, that governs a type of equilibrium (to be called thermal equilibrium) that cannot be explained in terms of mechanical forces. Thus the zeroth
law asserts that a complete description of a system must in general include the specification of its temperature (or related non-mechanical variable): we have moved from the domain of mechanics to the domain of thermodynamics. The zeroth law explicitly recognizes the concept of temperature and permits us to establish an empirical temperature scale and thereby measure temperature. Conventionally, the temperature scale is chosen such that physiologically hotter bodies are associated with a higher temperature. Zemansky (ref 5, p 27) describes thermal equilibrium thus: Thermal equilibrium exists when there is no spontaneous change in the coordinates of a system in mechanical and chemical equilibrium when it is separated from its surroundings by a diathermic wall. In thermal equilibrium, all parts of a system are at the same temperature, and this temperature is the same as that of the surroundings. When these conditions are not satisfied, a change of state will take place until thermal equilibrium is reached.
Thermal equilibrium of a body implies a unique temperature for it: every part of it has the same temperature. When we measure the temperature of a body we are using the principle that temperature of the body is equal to that of the thermometer in thermal contact with it. It would be illogical to suppose that two bodies initially at temperatures T1 and T2 and placed in thermal contact could be at two different temperatures when they have reached thermal equilibrium, since the phrase thermal equilibrium implies a unique temperature. What can we infer about thermal interaction? We know from experiment that the internal energy U of any body is an increasing function of T. This implies that heat capacities, C, are positive. Consider the thermal interaction of two samples A and B of the same substance in an adiabatic enclosure but at slightly different temperatures T1 and T2, with T1 > T2 Using qA + qB = 0 and q = CdT it can be shown that the changes in temperature are of opposite sign. Since A and B have a common temperature at equilibrium, it follows that the hotter body gets colder and the colder body gets hotter, and heat flows from the hotter body to the colder body. Since the flow of heat depends only on the temperature difference, we may take this as a general result. Thus we are able to infer the direction in which heat flows without appeal to the second law. Thus Callen (ref 14, p 39) says, in reference to our intuitive notion of temperature: “we should expect that heat should tend to flow from regions of high temperature toward regions of low temperature”. Of course, the second law will confirm this, but we know this result without explicit appeal to it. Is equilibrium necessary for measurement of temperature? In their note Gislason and Craig (1) say The zeroth law enables provisional thermometry and can be stated as follows without using the concept of equilibrium. When two systems (A and C) separately in good thermal contact with a third system (B) cease to show changes in measurable quantities, such as volume or resistance, the two systems (A and C) have the same temperature. As a consequence, the intermediary system (B) can be used as a thermometer. ...
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Research: Science and Education
The words “cease to show changes in measurable quantities, such as volume or resistance” are worth noting: in conjunction with the idea of “good thermal contact” this is completely equivalent to the concept of thermal equilibrium. Even though Gislason and Craig are at pains to avoid mentioning the term thermal equilibrium explicitly, the zeroth law cannot be stated without at least implicitly using the ideas associated with thermal equilibrium. The measurement of temperature by a thermometer means that both the thermometer and the body whose temperature is being measured have the same unique temperature, and this necessarily implies thermal equilibrium. The Role of the Zeroth Law in Teaching Students encounter problems involving two bodies initially at different temperatures being placed in thermal contact early in their physics course in high school. At this stage the emphasis is on working intuitively, rather than being explicitly aware of what laws are being used. There is a need for texts at a higher level to re-work problems of this type with a more detailed explanation. Most of the modern texts (5–10) in physical chemistry mention explicitly the zeroth law. Even if no explicit mention is made of the zeroth law, it is impossible to avoid using the consequences of the zeroth law, since all that the law does is to put the concept of temperature on a firm footing. Indeed, whenever we assume that a body has a unique temperature, we have already used an important result of the zeroth law. In solving problems, it is important for students to know the general laws and principles governing the phenomena being dealt with. The idea that there are three aspects to equilibrium, viz., thermal, mechanical, and chemical, is an important one and should be mentioned at the third-year college level. The result that all bodies in thermal equilibrium have a common temperature and that this is the basis of thermometry should also be discussed explicitly. Knowledge of these principles will allow students to solve problems logically and appreciate the relevance of the laws to problem solving. Gislason and Craig claim (ref 1, p 885) that in the traditional method “the problem is solved by assuming that a single, intermediate temperature is attained”. If the zeroth law has been explicitly introduced, the above statement is not valid. Though the first law does not require a unique final temperature, the zeroth law does. The zeroth law assures us that there will be a common temperature at equilibrium. Furthermore, it is not necessary to assume that the final temperature lies between the two initial temperatures. The final temperature comes out to be the average of the two initial temperatures because we assume that enthalpy, H, increases linearly with T and that both objects have the same mass and specific heat capacities. It is difficult to see why the method should be characterized as “naïve” (Webster: innocently direct but lacking in mental power!). The traditional method seems brief, logically rigorous, direct, and to the point. Entropy Maximization in the Traditional Approach According to the second law, spontaneous processes will lead to an increase in the entropy of the universe. There is some merit in verifying this, as Gislason and Craig have done, for problems involving the establishment of thermal or mechanical
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equilibrium. We have to select, however, only examples where this can be done without the mathematical details obscuring the general principle. The traditional approach uses dSsystem + dSsurroundings > 0 for a spontaneous process. It is also customary for physical chemistry texts to derive the fundamental equations (ref 10, p 105) such as dH ≤ TdS + Vdp. But these equations are used in the texts only for simple deductions such as dH ≤ 0 at constant entropy and pressure. We will use these equations to explore the principle of entropy maximization. It turns out that the same results obtained by Gislason and Craig can be obtained more directly and with much less mathematical manipulation. It is possible to verify that the establishment of thermal equilibrium under very general conditions leads to an increase in the entropy of the universe. According to the zeroth law, two bodies in thermal contact but at different temperatures are not in thermal equilibrium. The second law asserts that the resulting flow of energy from the hotter body to the colder body will always increase the entropy of the universe. We consider two bodies A and B in thermal contact in an adiabatic enclosure. We assume that A and B are in internal mechanical and chemical equilibrium. Let B be at the higher temperature. We will, following Guggenheim (7), use the symbol q to denote heat whether it is finite or infinitesimal. Consider the flow of an infinitesimal quantity of heat q from B to A. Then
dSuniverse
q q TA TB
q
1 1 TA TB
(1) > 0
(2)
Thus, so long as the temperatures are different the entropy of the universe continues to increase; when the temperatures are equal dS = 0 and no further change will take place. This proves that the entropy of the universe is a maximum. Note that apart from the requirement of chemical and mechanical equilibrium, we have not assumed any specific conditions for A and B, that is, we have not required the pressure to be constant or the volume to be constant and so forth. It follows that under all conditions, when thermal equilibrium is attained the entropy of the universe will be a maximum. The above proof concentrates on the essentials. While the examples considered by Gislason and Craig are good exercises in “entropy analysis”, the above proof achieves, without distracting details, the objective of verifying that the final state of thermal equilibrium is a state of maximum entropy of the universe for thermal interactions under all conditions. In the Gislason and Craig approach to thermodynamics, the proof of their eq 1 (as given in their appendix) is far from trivial. In the traditional approach to thermodynamics, if the bodies A and B are in an adiabatic enclosure at constant pressure, then dUA+B = qA+B − pΔVA+B = ‒pΔVA+B so that dHA+B = 0. Their eq 1 follows directly from this. Appropriate fundamental equations may be used to verify that the entropy is maximized in some of the specific cases treated by Gislason and Craig. We first deduce very generally what conditions will lead to equilibrium. We then verify, using the formula for the change in entropy, that if we start
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Research: Science and Education
with the final equilibrium state, any virtual change will result in a decrease in entropy: the change in entropy is zero to the first order in an infinitesimal parameter describing the change (shows it is in equilibrium) and negative to the second order in the same parameter (shows that the entropy is at a maximum). The mathematical complexity is reduced because we are dealing with infinitesimal changes. Our method may be regarded as complementary to that of Gislason and Craig, since we start with the final equilibrium state. Constant Pressure, Equal Heat Capacities Here we treat the case of two identical bodies A and B interacting as constant pressure; this is the “classic case’’ treated by Gislason and Craig. The appropriate equation is the fundamental equation for dH. We assume that B has the higher initial temperature. From dH ≤ TdS + Vdp we get dS ≥ (1∙T)dH − (V∙T)dp. Thus at constant pressure we get TdS ≥ dH. Under the constraint dH = dHA + dHB = 0 the enthalpies of A and B will be such as to maximize the entropy. Since dS = dSA + dSB, we have, at equilibrium
dS tot Cp ln
Tf E T E Cp ln f Tf Tf
= Cp ln 1 +
Cp ln 1
{ C Cp
E Tf
E Tf E Tf
1 −
E Tf
(3)
(4)
Here too we consider A and B in thermal contact at constant pressure in an adiabatic enclosure, the temperature having the unique equilibrium value, but the amounts and heat capacities of A and B being different. This introduces no new problems. When the enthalpy of A is increased by an infinitesimal quantity ε and that of B is decreased by ε, then the magnitude of the change in temperatures of A and B are ε∙CA and ε∙CB, where CA and CB are the two heat capacities. We now have dS tot CA ln Tf F /CA C B ln Tf F /C B Tf Tf CA ln 1
(5)
(6)
This shows that any virtual change at equilibrium under the constraint of constant enthalpy leads to a decrease of entropy: the contemplated change cannot be a spontaneous change. This
F CATf
1 F { CA 2 CATf
2
C B ln 1
CB
1 F 2 C B Tf
F C B Tf
(7)
(8)
2
(9)
This confirms that the final state predicted by the zeroth law is a state of maximum entropy. Note that we did not need to use the value of the final temperature in the proof. Variable-Pressure Problems We now consider the variation of the volumes of the subsystems. Following Callen, (ref 14, p 43) we start with
dS s
p 1 dU dV T T
(10)
which comes from the fundamental equation for U. If our system is an isolated system consisting of two sub-systems 1 and 2 in thermal contact, for all variations of U1 and U2 subject to dU1 + dU2 = 0 and for all variations of V1 and V2 subject to dV1 + dV2 = 0, dS will be greater than 0 for non-equilibrium, and equal to 0 for equilibrium. For any virtual process, since S is already at its maximum, dS will be negative. We now have,
2
2
General Constant-Pressure Problem
dH A dH B 1 1 dS dH B TA TB TB TA
For dS = 0 to be true for any change at equilibrium, the temperatures TA and TB must be equal. As expected, the predictions of the second law and the zeroth law agree. We now verify that the entropy is at its maximum. We will do this by showing that for the equilibrium state, any virtual change that alters the enthalpy of A and B subject to the total change of enthalpy being constant will decrease S. Let us increase HA by an infinitesimal quantity and decrease HB by the same infinitesimal quantity. Since the heat capacities are the same, the temperature of A will increase to Tf + δ and the temperature of B will decrease toTf − δ: this defines the infinitesimal quantity δ. Using eq 2 of Gislason and Craig we see that the total change in entropy dStot from the equilibrium state to the new virtual state is given by
confirms that the initial state corresponding to δ = 0 is a state of maximum entropy.
dS
1 1 dU 1 T1 T2
p1 p 2 dV1 T1 T2
(11)
at equilibrium. Since U and V are independent variables, it follows that the coefficients of the two differentials on the right hand side are separately equal to zero. This gives the result that both systems have the same temperature and pressure at equilibrium. We could of course have obtained the same result from the zeroth law and Pascal’s law. The equilibrium value T of the temperature and the equilibrium value of the volume can be derived using the first law and the gas laws.
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Research: Science and Education
To verify that this is a state of maximum entropy for the case treated in Figure 1 of Gislason and Craig, we note that the equilibrium values of V1 and V2 are both equal to V0∙2, where V0 is the total volume. If we consider increasing the internal energy of 1 and decreasing that of 2 by the same infinitesimal quantity, the temperatures will increase and decrease by the same infinitesimal quantity, say δ. Similarly, consider increasing V1 by an infinitesimal quantity ε from the equilibrium value and decreasing V2 by the same quantity. For this change we have
dS tot n cV ln
T E T E n cV ln T T
V /2 F V /2 F n R ln 0 nR ln 0 V0 / 2 V0 / 2
(12)
where cV is the molar heat capacity at constant volume, and n the amount. Using the same manipulations as for the “classic case”, we can show that all values of δ and ε bring about a decrease in S, confirming that the system is already at its maximum. How Best To Solve Problems One of the stated aims of Gislason and Craig was to help students gain confidence in the use of entropy to solve thermodynamics problems. This, coupled with their poor opinion of the use of the first law for solving problems involving spontaneous change, suggests that they would be in favor of the use of entropy in all problems of the type they have dealt with in their article. They have also been critical of what they refer to as the “local formulation” of the first law for the problems of interest in their article. Are there any advantages in using entropy maximization instead of the zeroth law, first law, and Pascal’s law? Instructors will favor one path over another for the solution of a problem, depending on personal preferences as well as the abilities of their students. However, it is reasonable to require that instructors teach students always to proceed from general principles. The fact that bodies in thermal equilibrium have the same temperature is one such general principle. Often, it is possible to use more than one set of general principles, one set of general principles and laws being a subset of another. It seems prudent in all cases to use the smallest subset that is necessary to solve the problem. All the problems that Gislason and Craig treated are capable of solution with the use of the zeroth law, the first law, and Pascal’s law without loss of rigor. Thus in the variable-pressure problem in their Figure 1, (ref 1, p 887), the most direct approach, using the zeroth law, is that U(1 mol, Ti, 2 atm) + U(1 mol, Ti, 1 atm) = U(1 mol, Tf, p) + U(1 mol, Tf, p). Since the gas is ideal, and U depends on T only, it is clear that Tf = Ti. Calculation of the pressure and volume is now straightforward. The method given by Gislason and Craig obtains the same result, but is considerably more complicated because they use a larger set of general principles to begin with. Even if we were interested in the change in entropy, the above method would be shorter. Favoring the use of the zeroth law, the first law, and Pascal’s law for the sort of problems discussed here does not mean that the principle of entropy maximization is not an important
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one. For general problems involving spontaneous change, we do not have any law other than the second law. The second law not only deals with the direction of spontaneous change, but also predicts that the entropy will be a maximum at equilibrium. We have seen that it is possible to know that heat flows from the hotter body to the cooler body without explicit appeal to the second law. Similarly, the laws of mechanics show that if two gases at different pressures are brought together, gas will flow from the higher pressure to the lower pressure. Here too, the direction of change is obtained without explicit appeal to the second law. The second law will, however, not only predict this direction of change, but will also predict that the entropy will continue to increase during the change. This principle of entropy maximization should certainly be grasped by students and verified in a few simple cases. The attainment of both a unique temperature and a unique pressure gives the condition of thermal and mechanical equilibrium. Problems involving the attainment of only thermal and mechanical equilibrium may be solved using these conditions supplemented by the first law and the necessary equations of state. The second law comes into its own for chemical reactions. For the problems dealt with here, the greater complexity of the second law approach will militate against its adoption as a general method of solution. Literature Cited 1. Gislason, E. A.; Craig , N. C. J. Chem. Educ. 2006, 83, 885–890. 2. Craig, N. C. Entropy Analysis—An Introduction to Chemical Thermodynamics; VCH Publishers: New York, 1992. 3. Gislason, E. A.; Craig , N. C. J. Chem. Educ. 1987, 64, 660–668. 4. Bent, H. A. The Second Law; Oxford University Press: New York, 1965. 5. Zemansky, M. W. Heat and Thermodynamics, 5th ed; Mc-Graw Hill Book Company: New York, 1968; p 5. 6. Levine, Ira N. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 2002; p 4. 7. Guggenheim, E. A. Thermodynamics, 5th revised ed.; NorthHolland Publishing Co.: Amsterdam, 1967. 8. Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry—Matter in Equilibrium, second ed; John Wiley: New York, 2002. 9. Atkins, P.; de Paula, J. Physical Chemistry, Volume 1: Thermodynamics and Kinetics, 8th ed.; W. H. Freeman and Company: New York, 2006. 10. Silbey, R. J.; Alberty, R. A.; Bawendi, M. G. Physical Chemistry, 4th ed.; John Wiley: New York, 2005. 11. Born, M. Phys. Z. 1921, 22, 218. 12. Canagaratna, S. G. Amer. J. Phys. 2005, 73, 299–301. 13. Canagaratna, S. G. Chem. Educator 2004, 10, 5–9. 14. Callen, H. B. Thermodynamics; Wiley: New York, 1960.
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