Mark W. Zemansky The City College of the City University of New York New York, New York
A Physics Teacher Looks at Chemical Thermodynamics
On October 23, 1973, the author delivered the first Edward A. Guggenheim Memorial Lecture at the University of Reading, where Guggenheim had sewed as Professor of Chemistry and Chairman of the Chemistry Department for many years. The present article is based on the memorial lecture, and contains an analysis of the first law of thermodynamics along with a short discussion of six topics in chemical thermodynamics that deserve a clearer exposition than is found in most chemistry textbooks: (1) Use of the symhol F, (2) Improper use of the word "function," (3) Symbol for the Helmholtz function, (4) Leaving out the word "change," (5) Reference states for tabulated numerical values of thermodynamic functions, and (6) Use of the symbol AG for three different quantities. There have appeared in recent years a hook ( I ) and several articles (2) by physical chemists in which the first law of thermodynamics is presented in an oversimplified manner, stressing only the principle of the conservation of energy, and failing to give an operational definition of internal energy and of heat. The promoters of this point of view pride themselves on the fact that they do not need even to mention the words "work" and "heat" (3). Attention is directed away from the system, and the method is characterized inappropriately by the adjective "global." I should like to exercise the prerogative of one who has taught thermodynamics for many years, and outline what I believe are some of the features of the first law that make it more subtle and more difficult than some people suspect. When thermodynamics was first developed, i t was concerned primarily with the hehavior of a heat engine by whose agency a net amount of work was done by a system (the "working substance") and was therefore chosen to he positive. This sign convention for work dominated the subject of thermodynamics for many years and is still used by engineers. For the purposes of this paper, let us adopt the more physical point of view as follows. If F is the magnitude of the force exerted by an outside agent on the system, and dx is the magnitude of the displacement of the point of application of the force, then, by definition, when F and dx have the same direction the work is equal to dW = Fdx and this work is said to be done on the system. When F and dx are in opposite directions
and this work is said to he done by the system. This is the sign convention universally applied in all textbooks of mechanics because it leads to the result that positiue work produces an increase in the energy of a mechanical system. Let us refer to the schematic diagram shown in Figure 1, in which there is a substance (either a solid, a liquid, or a gas) on which work may he done in a reversihle way by means of the usual frictionless piston-cylinder arrangement, and also may he done irreversibly by rotating a generator and maintaining a current in a resistor emhed572
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Figure 1. Symbolic apparatus for the performance of work on a system.
ded within the material. What has been labeled the system consists of both the substance and the embedded resistor. This svstem is c a ~ a h l eof existine in eauilihrium states describable with the aid of thermohynamic coordinates. of which one is. as alwavs. " . the tem~eratureT. We accept as an experimental fact that adiahatic conditions are achieved iust as well bv maintainine- the surroundings and the system at the same temperature, as by covering the system with a thick layer of asbestos or wool. With the surroundings a t the same temperature as that of the svstem. we mav measure the total adiahatic work Wad in transferring the system from state one to state two along a number of different paths involving the compression or expansion of the system, or the maintenance of a current in the resistor, or both. The fundamental assumption a t the heart of the first law is the statement that the adiahatic work from one to two is independant of the path. I t follows from this that there exists a function U of the thermodynamic coordinates of the system, called the internal energy, whose change U2 - UI is equal to Wad It is to be noted that what is defined is a change in U and that this definition is operational in that the definition itself is a set of rules for setting up a table of values of U, all of which refer to one arbitrarily chosen reference state at which U is set equal to zero. If the temperature of the surroundings is set at a value different from that of the system and the work necessary to transfer the svstem between the same two states one and two is measured, this work W (with no subscript) will he found to be different from the adiahatic work. The difference is attributed to an energy supply or withdrawal due to the difference of tem~eraturebetween the system and its surroundings. This energy transfer is called-heat, in conformity with the calorimetric definition of heat as the energy transferred by virtue of a temperature difference. Thus, by definition Heat = W (adiabatic)- W (non-adiabatic) or
F
.
...
.
increasing Pand x
= VdP Figure 2 . Work in moving a hydrostatic System in a conservative pressure field.
where Q and W are chosen positive when they enter the system and produce an increase in U. Whether an energy transfer is called "heat" or "work" depends on what is taken to be the system ( 4 ) . Thus, if the embedded resistor is included as part of the system, the process of maintaining a current in the resistor is described as the performance of work; whereas, if the resistor is excluded from the system, then the energy transfer from the warm resistor to the cooler material surrounding it constitutes a heat transfer. Thus, in the operational treatment of the first law, the system dominates the scene; the concept of work is all important, and heat is defined as the difference between adiahatic and non-adiabatic work. To apply the first law, one must know what the system is, what its coordinates are, and what mathematical expression for reversible work is appropriate. This last point provides some complication because there are two different mathematical expressions for reversible work dW in connection with at least three systems, namely, a hydrostatic one with coordinates P, V, and T,a paramagnetic rod with coordinates X , M, and T, and a dielectric rod with coordinates E, U , and T.Let us first consider a hydrostatic system, such as a gas. If the gas is contained within the usual cylinder-piston combination, and is changed in volume by an amount dV, everyone agrees that the work is -PdV. (A compression with a negative dvrequires work done on the system, or positive work.) But suppose that the gas is moved from a position in a conservative pressure field where the pressure is P to another position where the pressure is P + dP, as shown in Figure 2. One sees that, in this case, the work is + VdP. Another example of a conservative pressure field is provided by the vertical column of liquid shown in Figure 3. The system is assumed to be a small gas balloon whose density is negligible compared with that of the surrounding liquid, and therefore with a weight that is negligible compared with the buoyant force B. An external force F equal in magnitude to B is needed to keep the system in equilibrium. If the system is lowered an amount dy, the work is easily seen to be VdP. Whether the system is a gas in a statioliary cylinder or whether it is moved in a pressure field, the gas has undergone a change in volume dV and a change in pressure dP. Yet in the first case dW = -PdVand i n t h e second dW =
Figure 3 . Another type of conservative pressure field.
+VdP. The gas does not know which method was used. Which expression for work leads to a correct internal energy function? If dW = -PdV, there exists a function A such that dA = - (PdV),d. Non-adiabatically, we would therefore have
If, on the other hand, d W = +VdP, there exists a function B such that dB = ( V d P h , and non-adiabatically dQ-dB-
VdP
Equating the two expressions, we get
We are therefore confronted with the question as to which is the internal energy, A or B. The answer lies in the physical meaning of the term PV, which is the potential energy shared by the system with the source of the pressure field. It is this potential energy which is liberated when a rod holding a small cork ball below the surface of a liquid is withdrawn and the cork hall rises and gains kinetic energy as a result. Since the term P V arose from the process of moving the system and is not present when the system is stationary, it does not belong in the expression for the internal energy of the system alone. The work term -PdV leads to the correct definition of internal energy, because it is concerned exclusively with an alteration of the internal configuration of the system alone. The same situation exists with regard to a paramagnetic rod. When the rod is stationary in a magnetic field of intensity X and its magnetization M (total magnetic moment) is increased an amount dM, it is a simple matter to show that d W = X d M (5).If, however, the rod is moved a short distance in the c o n s e ~ a t i v emagnetic field of a large permanent magnet, as shown in Figure 4, then it may be seen that the work is -MdX. Again, the same changes of Volume 51, Number 9, September 7974
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573
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over all the energies and particles is obviously the internal energy U and not U + PV. When ZNiei is set equal to U, and the usual calculations of statistical mechanics are carried out, one gets the equation d U = TdS - PdV, implying that dWis -PdVand not +VdP. In the paramagnetic case, however, the energy of a magnetic ion or nucleus consists of two terms (6). thus
Direction of increasing x and X
L
magnet
'L
Magnetic Field
where 6, is the energy when there is no external mametic field and the quantity -gfiXrni is the magnetic energy in the presence of the field. In this case, the interpretation of the sum Figure 4. Work in moving a paramagnetic rod in a magnetic tiaid.
X and M occur in both cases. Which is correct? It has been stated many times that it is a matter of indifference which expression is chosen. Let us see. If dW = x ~ M , there exists a function A such that dA = ( r d M ) , a and non-adiabatically
If, on the other hand d W = -MdX, there exists a function B such that dB = -(Mdx),,, and dQ=dB+MdX Therefore dB+MdX=dA-XdM dB=dA-d(XM) B=A-XM It is well known that a magnetic material of moment M, placed in a field X , has a potential energy -XM. This potential energy is not possessed by the paramagnetic rod alone but is shared with the source of the field. It is hardly a part of the internal energy of the paramagnetic rod alone, and plays a role only when the rod is moved in a mametic field. Therefore, the correct internal enerm is associated with the work term XdM, which is conc&ed exclusively with the internal configuration of the magnetic . particles. The same principles hold for a dielectric which has a total electric polarization 1l when placed in an electric field of intensity E. The three systems which have just been mentioned are summarized in Table 1, in which the kind of work that leads to an internal energy function is designated as "configurational work," whereas the work to move a system in a conservative field is called "field work." Notice that the enthalpy function may be regarded as the sum of the internal energy and the potential enerW. The choice of a suitable internal energy has a profound hearing on the statistical treatment of the systems listed in Table 1. If the particles of a gas, for example, are only weakly interacting, they have kinetic energy only, and if Nt particles have kinetic energy ti, then the sum ZN,ei Table 1. Kinds of Work CoordiSystem
Hydrostatic Paramagnetic Dielectric
574
nates
Configurational work
P , V, T
Field work
-PdV VdP -(PdV)"d = d U XC, M, T XdM -MdX (3CdMhd = d U E, 11, T Ed11 -ndE (Ed"> I = dIJ
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rotential energy
Enthslpy
PV
U
+ PV
-EM
U
- EM
-En
U
- En
cannot be the internal energy alone. Since the second term on the right is - E M , the magnetic potential energy, the internal energy is the first term on the right, so that U = ZN&. The entire sum TNifi is therefore U - XM, which is the magnetic enthalpy. When ZNi6i is set equal to U and the usual calculations of statistical mechanics XdM, are carried out, we get the equation d U = TdS implying that dWis X d M and not - M d x . To summarize: A correct and useful application of the first law is of necessity determined by the peculiarities of the system, involving the proper choice of the expression for work which will lead to an internal energy possessed by the system alone, not shared with an unknown source of a field. Only after the correct internal energy function has been shown to exist can a meaningful thermodynamic definition of heat he arrived at. In 1923, when Lewis and Randall's hook first appeared, a function H - T S was given the name "free energy" and was denoted by the symbol F. At the same time, European physicists and chemists had been calling a different function, namely, U - TS the free energy and denoting it by the symbol F. Using the same name and symbol for two different functions was an intolerable situation that was pointed out to American chemists on many occasions. It took almost 40 years for them to agree to name H - TS after the discoverer, Gihbs, and to denote it by the first letter of Gibbs' name. Of course, all chemists recognized that both the enthalpy H a n d the Gibbs function G are functions of the independent coordinates of the system. The fact that they were functions, however, was obscured or suppressed by using the name "enthalpy" alone, and by using the expression "Gihhs energy," rather than "Gihhs function." Since the quantities H I T and - G / T vary with the temperature in a more convenient manner than H or G alone, i t was thought useful to tabulate values of H I T and -G/T and to give these quantities names. In ahout 1955, it became the habit to call H the enthalpy and H I T the "enthalpy function" (7). Similarly, G was called the Gihhs energy and -G/T the "Gibbs energy function!" I can not understand how this horror managed to escape detection for 15 years and why physical chemists did not immediately recognize that the use of the word "function" was hardly suited to denote division by T. Fortunately, efforts are being made through the activities of a IUPAC Committee to provide more suitable names for H I T and -GIT. 1; the meantime, the Helmholtz function U - TS has had its uos and downs. While American chemists were denoting H--TS by the same symbol F that the Europeans were using for U - TS, it was recommended that F he given up for both functions and that A he the symbol for Helmholtz and G for Gihhs. (The A arose from the fact that the Helmholtz function is an isothermal work function and the German word for work starts with an A , ) Everything was fine, and by the time Guggenheim changed
+
all his Fs to A's, other authors decided to go back to the European custom of using F for Helmholtz function because the G for Gibbs function was taking hold so well. At this point I can imagine that Guggenheim threw up his hands and, muttering "A plague on both your houses," decided to use both symbols a t the same time, concocting the composite symhol AF, which I am sure, stands for "absolutely final." The functions: energy U , entropy S, enthalpy H, Helmholtz function A, and Gibbs function G all refer to single equilibrium states. If a state changes from, say, o n e t o two, then the change in any of these functions is usually written AU, AS, AH, etc. When a system undergoes a pmcess wherein its state changes, there is associated with the process a AU, not a U; a AS, not an S ; etc. Realizing this, how could it have become habit on the part of practically all physical chemists to use such expressions as: "entropy of mixing," "enthalpy of vaporization," "Gibbs energy of reaction,'' etc. Why not "entropy change due to mixing," "enthalpy change due to vaporization," "Gihbs function change due to reaction," etc.? When the entropy change of a system undergoing an isothermal reversible process is measured a t various Ts and denoted by AS?, it is found that, as T approaches zero, AST approaches zero also. In fact this is one of the most useful forms of the third law. The statement, however, that as T approaches zero, S approaches zero, is not a law of thermodynamics, but a convention, as Guggenheim has pointed out in no uncertain terms. Now neither of the functions U nor S is defined in absolute value. One has operational definitions only of AU and A S . If it is desired to tabulate values of U or S, or of the other functions H, A, or G, i t is understood by everyone that the numbers tabulated are all with reference to an arbitrarily chosen reference state a t which the function is arbitrarily set equal to zero. Usually, to find out the reference state a t which the function has been given the value zero one looks for a statement to this effect a t the head or the foot of the table, or in the body of the text, or allows one's eyes to skim over the table itself to see a t what state the function has been set equal to zero. Table 2 shows with the aid of little round superscripts that all values of H, S , and G , are a t states in which the system is a t one atmosphere pressure. A simple remark to this effect would eliminate the necessity for all the superscripts. The subscript T usually means "at constant T," hut here it means "as a function of T." Since it is obvious that H, S , and G , are all functions of T (otherwise, why would there he a column with T as a heading?), the suhscript is quite unnecessary. The quantity H00 is unnecessary if the remark is made that all enthalpies are calculated from a reference state T = 0,P = 1 atm. Notice that the entropy column is not written ST' - So0 although it is also calculated from a reference state T = 0, P = 1 atm. The zero value of So0 is a convention, not a law of nature. I should like to suggest that, with the aid of a few simple remarks or footnotes, Table 3 conveys the same information as Table 2. Perhaps the most interesting part of chemical thermo-
dynamics concerns itself with chemical reactions. Even when the reactants and products are all mixed at a uniform and constant temperature and pressure, the thermodynamic analysis may require a list of symbols with subscripts, superscripts, bars, circumflexes, stars, and primes. The skill and finesse with which physical chemists handle these equations, particularly when applied to real gases and nondilute solutions have always filled me with admiration. There are times, however, that the study of this part of chemical thermodynamics has been rendered obscure and baffling by the use of the same notation to mean three different things. For example, suppose that four gases, MI, Mz, Ma, and Mq are capable of engaging in the reaction where the u's are stoichiometric coefficients. Suppose that these gases constitute a homogeneous mixture a t constant T and P and that, a t any moment, there are ni moles of substance Mi. Then the Gihbs function of the mixture is
where the p's are the chemical potentials. There are three expressions that you will all recognize:
where t is the degree of advancement of the reaction,
where the g's are the molar Gibbs functions of the separate substances, and
In spite of the fact that these three expressions are all different, and that eqns. (1)and (2) are functions of T and P only, whereas eqn. (3) is a function of T, P, and t , they are all designated by the symbol AG. A graph of the G of a mixture of Cs atoms, Cs ions, and electrons versus t is shown in Figure 5, where thetbree expressions may be seen to be quite different. Equation (1) is AG, (2) is &gi, and (3) is ( 8 G l a t ) r . p . I realize that there are reasons why this has come about, but don't you thmk that it is time to reexamine the delta notation with an eye toward following the universal mathematical con-
Table 2. Values of Thermodvnamic Functions
Table 3. Values of Thermodynamic Functions T ('K)
H
S
iJ/molel
iJlmole 'K)
-G 1.Tlmolei
Figure 5. Graph of G versus (for a reacting mixture at constant T and P
Volume 51, Number 9.September 1974 / 575
vention that A G represents the difference between two values of the G of the system that you have specified in the first place. If you make clear in the beginning exactly what system the function G refers to and what cwrdinates G is a function of, then it ought to he relatively simple to restrict the symbol AG to a change in that function G. It has been my experience that, in their use of thermodynamics, physical chemists do not make mistakes. The final answer a t which they arrive is always correct, hecause they really know what they are doing and they do it hundreds of times more often than other people. My only objection is that they feel so little obligation to do things in a manner that is immediately comprehensible to an outsider. When a physical chemist uses thermodynamics, there are many physicists, engineers, and biologists who could profit by understanding, and I should like to emphasize that the physical chemist has a real obligation to
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speak, to write, to compile tables, and to teach in such a manner that he preserves most of the mathematical amenities. This is precisely what Guggenheim did in all his research and textbook writing. By honoring Edward Guggenheim, we affirm the obligation of the physical chemist to emulate his devotion to mathematical soundness, his care in phraseology, his attention to notation and symbolism, and his courage in pursuing his profession. Literature Cited Ill Bent. Henry A,. "The Second Law."Orford Univ. Press, 1966, pp.9-24. (21 Bent, HenryA.J.CHEM.EDUC.,49,4411972l.
C., S y m p o ~ i u mon the Teachinz of Themodynsmics, Chom.Educ..Amer.Chem.Soe.,New York. Sept. 10, 1969.p.4
131 C r a i ~Norman
Divbion of
(41 Zemansky. Mark W., ThePhyairr Teacher. R,295(19701. 15) Zernansky, Mark W.. "Heat and The,modynarnicr." MeGraw-Hill Bmk 1068. P. 62. In this book, the old en*neerinesim convention farwork i r u e d . 161 R d . l 5 l , p.448. (71 Rossini. Frederick 0.. "Thermodynamics and Physics of Matter: Princeton Univ. P m s . 1955. pp. 18 and 62.
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