8. A. Schaefer
Thermodynamic Potentials
University of Melbourne Point Cook, Victoria, Australia
T h e historical development of the concept of force has relevance for thermodynamics. From being a convenient way of connecting changes of velocity with changes of distance for Kepler, force took on the status of an independent fundamental property of nature after Newton. As expressed by Maxwell, "the existence of a force reveals itself in motion . . . force is that which alters the motion of a mass" (1). Developing almost simultaneously, a third stage culminated with Mach. The existence of force as a causal agent was rejected; force and mass were reduced to purely mathematical expressions relating certain measurements in space and time (8). Similar attitudes may he taken to the concept of energy, especially potential energy, and it may be ultimately an arbitrary decision to accept energy as more fundamental than force; modern relativity seems to support this latter view. I n contemporary mechanics force plays the role of an intermediary connecting mass and acceleration, an invention to satisfv our desire for exvlanation. I t is a proportionality factor with the statement "force equals mass times acceleration" being a nominal definition for functional purposes, and in which it is assumed that mass and acceleration can be given independent meanings ( 2 ) . Force is thus a dependent function of momentum and time. Some modern textbooks of mechanics admit the doubtful character of force as a causal operator, but regard the concept as extremely useful in analysing a great variety of mechanical situations (3). I n thermodynamics on the other hand, i t is by no means unusual to regard pressure a3 applied to a system to cause a change of volume, temperature gradients as c a u s k g heat to flow, and the chemical potential as tending to cause a change in chemical comvosition (A. the ,., 5 ,) . As a conseouence.,~ enerar change v ~ e a r sto be the effect of the causal - a .. oper&on of a thermodynamic potential. I n mechanics, force is seldom measured directly nor does it often appear as the answer to a problem, since the main'purpose of the calculation is to discover a mass or velocity, in which process force acts as am eliminated intermediary ( 2 ) . In thermodynamics, however, pressure and temperature are regarded as important parameters of the system, and are often sought as the answers to calculations. The shift of emnhasis in both mechanics and thermodynamics from force to energy, the latter still somewhat abstract, can be justified if the energy of a system is a more useful or more fundamental concept, or is more easily measured. The concept of force or potential, whether real or imaginary, has been and continues to be useful, but an examination of its meaning and logical position in thermodynamics ~
~
is indicated. It should be noted that this use of the Ford potential does not refer to free energy; there is in this case an unfortunately confused usage. A course of lectures on thermodynamics might well begin with a discussion on the nature of forcc, arriving with reservations (2) at the operational definition f = ma = mdu/dt
(1)
If the force operates over a. distance dv, we have jdr
=
m(dr/dt)dv
=
mudw
(2)
so that integrating JfciT
= m Judo
=
l/~mu2= Ek
(3)
where En is the kinetic energy. For an infinitesimal increase in energy of a body of constant mass fdr = dE
Here as an operational definition, we have force in terms of energy and another conjugate parameter. If the energy concerned is linear, that is translational mechanical energy, and is considered in relation to a linear quantity (distance), then the force is linear and mechanical. The definition becomes: linear mechanical force is the distance rate of change of linear mechanical energy. The Thermodynamic Potentials
An extension of the discussion would lead easily to the statement: expansive mechanical force is equal to the rate of change of mechanical expansion energy with volume u = -dE/dV (6) where the sign is directional, indicating a decrease of volume for an increase of energy, or that p is a compressive force opposite in direction to expansion. A generalization might be immediately attempted, that a potential of any kind is operationally defined as the rate of change of energy of a similar kind with an independent conjugate parameter, all other related variables being held constant.
~
+Z
=
(7)
@ES/bz),,,
The potential +, in terms of an associated energy E,, is equal to the rate of change of this zth kind of energy, with the change in the independent conjugate variable x, all other variables which could effect an energy change being held constant. This makes the 'potential a functionof the energy. Volume 47, Number 1 I , November 1970
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I t is clear that the quantity E, could also be called work. I n the equation
the force f is a covariant vector satisfyingtheequation, and is conveniently defined as the space rate of change of energy (Z). There are not, however, different kinds of energy, but rather different manifestations of a generalized energy observed as work forms, which depend on the associated matter and space coordinate. I t follows that the kind of potential in equations such as eqn. (7) will depend on these same coordinates. This arrangement enables the term energy to be reserved for the total internal energy, U. I t s increment is then measurable in terms of the several forms of work that may be done on the system (6), thus
For a specified uni-variant system, eqn. (7) can then be written and this can be applied to particular cases. For example, an electrical potential, 8, can be defined G = dU/dQ
(11)
When defining the potential as a function of the energy of the system, d U is written rather than dW(E) or electrical work, the energy of the system is generalized and is not spoken of as "work content." When calculating the electrical work applied to the system to increase the energy we can write d W ( E ) = EdQ
=
dU
(12)
which reads, the electrical work applied to the system and disappearing a t the boundary, is equal to the multiple of the electrical potential and the charge transferred, and this is equal to the increase in energy of the system. The operating variable dQ is the electric charge transferred to a system that can accommodate electrical charges only, that is where V, n, and S are constant, the system being non-mechanical, non-chemical and nou-thermal. During the process of change in the energy of the system an appropriate charge transfeks, and work forms are observed a t the boundary during the transfer, but once the charge has entered the system, no matter how the work was measured, only energy is said to exist or increase in the system. Energy and the Thermodynamic Pofenfials
Taking the attitude from mechanics that energy is more fundamental than force and applying it through analogy to thermodynamics, the thermodynamic potentials can be written as functions of the total internal energy. Thus
These definitions are the starting points for a number of approaches to the subject (7-10). Temperature, 746
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Journal o f Chemicol Education
chemical potential, etc., have no independent existence as properties of matter, but are the terms used to describe the energy-entropy or energy-particle number relationships. Their use in this form illustrates what is meant by an intensive property, since each describes the "concentration" of energy relative to s fundamental property of matter, uiz., entropy or particle number or electric charge, etc. The energy change associated with an entropy change can certainly be said to exist, but not as an independent entity. The description of phenomena by the use of any particular word such as temperature or force is a matter of convenience. I n each of the above equations there is implied a perception of energy in a particular way! with a particular set of measuring instruments. The energy perceived via a thermometer is energy conveyed by a thermal transfer mode, i.e., random particle motion, and observed by doing some kind of work. This position is not entirely free of difficulties. The thermodynamic potential has been defined as a function of energy and a thermodynamic charge. Energy and charge are here considered to be fundamental properties of the physical universe. However, there is no direct way of measuring energy. Indeed only changes can be measured in the energy of a system, and these changes in terms of work observed a t a boundary. Electrical work can be measured directly with a Joule meter. I n the case of chemical work the reaction must be arranged to produce electrical or thermal work. The latter may be measured in an ice calorimeter, the melting ice is an intermediary, calibration is achieved electrically, and the chemically produced heat determined by direct comparison. With similar arrangements the several work forms can in principle be measured without requiring the independent determination of the intensive and extensive variables. I n the case of chemical work, the indirect methods are often easier. Equation (9) can he expanded to read
so that the increase in energy of the system can be measured in terms of the several work forms applied to the system and disappearing a t its boundaries. These are the mechanical work of expansion dW(M), electrical work dW(E), the work of a chemical reaction dW(C) and by analogy, thermal work dW(T). Heat, however, is not the same as mechanical work, but has the same mathematical form and energetic significance, and is observed in terms of mechanical changes in matter. Thermal work used in this way is open to some criticism (If), b u t is considered to be equivalent to the classical heat energy, or energy transferred in the heat mode, or energy change associated with thermal changes. By analogy with electrical charge, one can regard the change in energy, measured by the amount of work appearing or disappearing a t the system boundary, as due to the increment of a physical and measurable quantity such as length, volume, electric charge, number of a chemical species or entropy, each of which is added as a charge to the system. The entropy increment d S is thus a thermal charge and the volume increment -dV is the mechanical charge. Equation (17) can now be written
where the potentials are defined by eqns. (13)-(16) and each work form is the product of the appropriate potential and charge. The potentials are proportionality factors and not causal agents, as can be demonstrated by writing eqn. (18) in the form
Equation (19) is a statement of the first law for a thermomecbanical-electro-chemical system ( T , M, E, C ) , with appropriate boundaries in contact with the surroundings.
Deflnition of Transferable Charge
If these potentials are to be defined in terms of the internal energy and the transferable charge, it is clear that the latter should be defined independently. This is not difficult in the case of electrical, chemical, or mechanical charge but in the case of thermal charge, the entropy increment has been defined classically as a function of heat and temperature, the latter regarded as an independent property of matter. Thus dS = q/T = dU/T (Const. V) (20) To define entropy independently, recourse must be had to statistical mechanics, whence a statistical thermal potential can be defined and then related to thermodynamic, kinetic, and empirical concepts of temperature arriving a t eqn. (13) without using analogy. For a thermal system we can then write T = dU/dS (21) and on rearrangement TdS = dU = dW(T) = q (22) which can be used as an operational definition of heat based on independent definitions of energy and entropy, thus q = (bu/as)~,".~dS (23) Heat "flow" can now he explained as the transfer of entropy from system I (or surroundings) to system 11, the entropy in system I being associated with more energy than that of system 11. Using the concept of a n energy "concentration," a decrease of energy in system I and an increase in system I1 occurs if [Vll > [Uln
(24)
[UI = (au/as) = d u / d s
(25)
For a thermal system and
- [Uln = (dU/dS)r
(dU/dS)n
(26)
which, multiplied by d S becomes G[U]dS = (dU/dS)rdS - (dU/dS)ndS
(27)
[Ulr
d S = qI
-
- qII
and dS/dt = ( q ~- q~r)/dt that is, the classical heat transferred is proportional to
the entropy transferred, and heat flux is proportional to entropy flux. Energy loss and gain by thermal systems is usually .explained as due to an energy flow called heat, whereas here it is described as a transfer of a thermal charge from one system to another in the direction I to 11, where the energy concentrations are related as in eqn. (24). Such a process would depend on the nature of the systems involved, and with which of the transferable charges the energy was associated. It is clearly evident that a 12-V battery could not increase the energy of a liter of air by transferring electrical charge to it, nor could i t charge up a flat 6-V battery by transferring entropy to it. If the energy concentration is relative to the entropy as in eqn. (25), describing a thermal system, then energy loss and gain can only occur by the transfer of entropy. If, on the other hand, the energy concentration is relative to electrical charge, then only by a transfer of electrical charge, called a "flow of electricity," can energy changes occur. If system I were thermal and system I1 were electrical, no transfer processes of either kind could occur and consequently no changes in energy. If hoth systems were thermo-electric, energy gain or loss can take place involving the exchange of work forms, called classically "interconversion of energy." I n such cases the transfer concerns two kinds of thermodynamic charge and in order to maintain the principle of microscopic reversibility the charges must be linked or coupled. The Onsager reciprocity relationship can he expressed qualitatively as the transfer of thermodynamically coupled charges. Heat flow equations have the same general mathematical form as the equations of fluid or electrical flow, hut this does not imply that heat is the same as electricity or that they are hoth fluids. The flow concept of continuous exchange as a function of time invites us to think of the process as involving a fluid. Certainly a quantity of some kind transfers from one system to another and material mechanisms are undoubtedly operating. It is, therefore, not unsatisfactory to refer to a thermal, electrical, or even a mechanical charge transfer by the flow terminology. This is not to be considered as a reversion to "caloric theory" in the discredited sense, but a phenomenological statement based on the physiological perception of a macroscopic system. Vector Properties
If these thermodynamic potentials are analogous to mechanical force, they might be expected to exhibit vectorial properties. This is well enough understood for electricity but some discussion is required in the case of temperature and chemical potential. Force as a vector can be resolved into effects in different directions. This does not mean that force has some kind of split directionality, like rays of different strength emanating from a source. It means that if energy is rationalized for a path x in a medium where other paths x' and x" exist, then it is possible to find the energy rationalized along these paths also. The properties of geometric space permit the calculation of the motions of a particle along several other paths which are related to each other by geometric spatial coefficients (cosines). I n the case of a perfect isotropic crystal, the transport Volume 47, Number 1 I, November 1970
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747
of a thermodynamic charge is equally possible in any direction, hut in the case of an anisotropic crystal the transport of the charge along the minor and major axes will occur through paths with different physical properties, and hence a t rates dependent on those properties. The resolved parts of mechanical force vectors are related by spatial coefficients, and by analogy the resolved parts of thermal or electrical potentials are related by material medium coefficients. These are well-known as thermal or electrical conductances which are different for the different paths. The vector concept resides properly in a consideration of the path rather than the potential. I n the case of thermal conductance a quantity called thermal diffusivity is often used, and then has values depending on the available paths (12). The vectorial nature of chemical potential may not, a t first sight he evident, but the diierent growth and solution rates associated with the several axes of a crystal provides a basis for the idea. At equilibrium the chemical potentials in these regions will evidently be the same, but during the reaction (chemical charge flow) there will be directional factors operating. Gibbs discussed the equilibrium between the solid and its solution in terms of the surface tension and the chemical potential (13). Surface tension, density, conductance, crystal arrangement, and activity coefficients are examples of the operating physical medium coefficients which provide the measure of the path differentiation enabling the calculation of the charge transfer along the path. The nucleation of liquids from the vapor has been studied in terms of components of p in the translational, rotational, and internal degrees of freedom. The embryo for the process is considered as a supermolecule with the chemical potential so distributed (14). The thermodynamic potentials exhibit a vector property during a transfer process, but once equilibrium is attained, this effect ceases. Whether or not forces and charges exist, or whether energy is real or more fundamental than force, the
748
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Journal of Chemical Education
above outline of thermodynamics is consistent with mechanics and relativity, and approaches all forms of work from the same viewpoint. To obtain the maximum benefit from a course of thermodynamics, the prior knowledge and understanding of a student must he used and confusion avoided. Hence it is undesirable to use thermodynamic potentials as causative agents. Rather they should be regarded as parameters indicating that the transfer of some conjugate charge can occur. Potentials should be defined and treated as proportionality factors relating the energy change and the charge transferred. The classical use of the term temperature as a property of matter obscures its relation with energy (9). The full development of this approach requires an independent definition of entropy and the incorporation of statistical ideas and constitutes a departure from the classical attitude. Thermodynamics can in this way be shown to be closely related to all branches of mechanics instead of being a separate discipline. Literature Cited
M*xwem, .IC., . "Matter and Motion." Dover Publiaations, New Yark. 1965. pp. 27. 79. JAMMER. M., "Conoepts of Force." Harvard University Press. Cambridge Mass., $67, pp. 205-9. 226. 244-8. 255. Y o m a , H. D., Fundamentals of Mechanics and Heat." M c G r s r Hill Book Co.,New York, 1964, p. 124. ZEMANBKY, M. W., "Heat and Thermodynamiea," McGraw-Hill Book Co., New York. 1957,PV. 44,53,80. VANWYLEN.G. J., AND SONNTAO. R. E., "Fundamentala of Clasaioal Thermodynamies;' John Wilcy & sons, Ino. New York. 19.55, pp. 73, 482. (6) VEINIY, A. I., "Thermodynamics." Published for the U. S. National Aeronautios and Space Administration and t h e Nationai Science Foundation. Washington D. C., by Israel Program for Scientific Tr&nslations,Jerusalem. 1964, pp. 6, 7, 17. (7) M*cn*e. D.. J. CHEM.EDDO.,23,366 (1946). and 32, 172 (1955). 0. K.,"St&tisticd M e ~ h a n i c ~Thermodynamics , and Kinetics," (8) RICE, W. H. Freeman and Co., Shn Francisco, 1967, p. 89. (9) L*ND*u. L. D.. A N D LIFSHITZ,E. M.. " S t & t i ~ t i dPhysim, Volume 5 of B Course of Theoretical Physics." Pergamon Press, London. 1959, p. 4 M . 3&4. (10) C n m e ~ .H. B., "Thermodynarnias," John Wiley & Sona, Inc., New York, 1963, p. 31. (11) M*cDoacnm. F. H.. J. Phus. Chem..44,713 (1944). M., "Heat Transfer," John Wiiev & Sons. Inc.. New York, (12) JACOB, 1958, Vol. 1. DP. 11, 96. (13) Gmss, J . W.,"Seientifio Papers. Volume 1,Thermodynamics," Dover Publications, 1961, pp. 314. 365. W. E.."Chemistry of theSolid State," Butterwarths, London, (14) G*RNER, 1955, p. 160.
