(eJ ;j pll lecture REFERENCE ST A TES AND RELATIVE VALUES OF INTERNAL ENERGY, ENTHALPY, AND ENTROPY A. G. FREDRICKSON University of Minnesota Minneapolis, MN 55455 MANY PROBLEMS IN chemical engineering ther- modynamics require for their solution the application of the balance equations for moles of various chemical species, (total) energy, and entropy. Fairly general forms of these equations, which are applicable to open, moving and deform­ ing, and unsteady state systems, are (1) A. G. Fredrickson, professor of chemical engineering at the Uni• versity of Minnesota, received his BS and MS at that institution in 1954 and 1956, pnd his PhD from the University of Wisconsin in 1959. He d dt (Ut +Kt+ <I>t) = Q +Wu+ W ex p has been the American editor of Chemical Engineering Science since 1975 and i~ the author of "Principles and Applications of Rheology." (2) + I W q (hq + kq + cpq) His current research area is the dynamics of microbial populations, and q his non-professional interests lie in the fields of natural history, pre­ (3) history, and photography. Here, the energy and entropy transfer rates asso­ action equations must be balanced is expressed by ciated with heat transfer, Q and S, are related to the conqitions the local heat flux vector q and temperature T on the boundary aB of the system B by (4) Q = -f (n·q) and this must be satisfied for all reactions (i) and c)B all elements (k) ; Ej k is the number of atoms of the k t" kind of element in one molecule ( or meric or S = -f i, (n·q) repeating unit) of the j t" chemical species. The oB quantity Wq is the molar flow rate of material at the qt" port of the system; it is positive if material The second law of thermodynamics requires that enters the system at that port but negative if ma­ the entropy generation rate Sa shall be non-nega­ terial leaves the system there. The mole fraction tive, and further, shall be non-negative for every of the j t h chemical species in this stream is de­ subsystem into which B may be divided. The quan­ t h noted by Xq,i• Finally, -Wu and - W ex p are the rates tity a;i is the stoichiometric coefficient of the j at which the system performs useful and expan­ chemical species in the i th reaction, R; is the rate of sion work on the surroundings, and Kt and <I>t de­ the i t" reaction in the system, and the summation note kinetic and potential energy. on i in Eq. (1) is over a set of independent ra­ Eqs. (1-3) are not valid for systems which actions [1]. The fact that all valid chemical re- exchange matter with their surroundings by dif­ fusion across a permeable boundary, but they are © Capyright ChE Division, ASEE, 1983 64 CHEMICAL ENGINEERING EDUCATION easily generalized to include that case also. It can culation of changes of internal energy and entropy. be shown that the results to be deduced here apply The reference quantities do not depend on the tem­ to these more general systems also, but in the perature and pressure of the system, though they interests of simplicity such systems will not be do depend on its composition. Similarly, if we let considered further here. hq rH be the enthalpy of one mole of matter of the The quantities U t and St which appear in Eqs. qt1• process stream when this matter is in the ref­ (2) and (3) are the absolute values of the internal erence state for enthalpy, then we can define a energy and entropy of the system, respectively, relative enthalpy and the quantities hq and Sq which appear in these equations are the absolute values of the enthalpy hq' = hq - hqrH (5c) and entropy of one mole of the q t" process stream, as well as a relative entropy respectively. Thermodynamics provides means for (5d) calculating changes in Ut, St, hq, Sq when the sys­ Sq' = Sq - Sqrs tem and the q n, process stream experience pro­ for the q'·" process stream. It is necessary to place cesses which change their states, but thermody­ the subscript q on the reference state quantities namics does not identify the states in which these hq rH and Sq rs because these quantities depend on quantities are zero. It is sometimes held that the the composition of the q th stream, and this is not in third law of thermodynamics identifies such a general the same as the composition of other pro­ state for the entropy but, within the context of cess streams. However, hq rH and Sqrs do not depend 1 classical thermodynamics at any rate, it is clear on the temperature and pressure of the qt • process that this is not so: the third law, or rather the stream. version of it usually called Nernst's heat theorem, If the relative properties Ut', St', hq', sq' are to asserts only that the entropy changes of certain be useful then they should obey the same balance kinds of systems undergoing certain kinds of equations as do the absolute properties U t, St, hq, isothermal processes approach zero as the tem­ Sq; that is, the relative properties should obey peratures of the processes approach absolute zero [2]. Hence, we do not know how to find the absolute (Ut' +Kt + <I> t) values of U t, St, hq, and Sq, :t However, we need to be able to assign numer­ = Q +Wu + Wexp + I wq (hq' + kq + <pq) (6) ical values to internal energy, entropy, etc. if we q want to tabulate what effects changes of tempera­ dSt' S S "' ' -----at = + G + : Wq Sq (7) ture, pressure, and composition have on them. Hence, reference states for internal energy, where Q, Wu, Wexp, S, and Sa are the same quan­ entropy, and enthalpy are chosen, and values of tities that appear in Eqs. (2) and (3). Clearly, if internal energy, entropy, and enthalpy relative to the relative properties satisfy these balance equa­ the reference states are tabulated. The most fa­ tions they then can be used (correctly) in the cal­ miliar example of this is the steam tables, but the culations of heat and work effects that occur so fre­ utility of enthalpy-composition and related dia­ quently in applied thermodynamics. I have been grams demonstrated in the well-known Hougen, unable to find a recent discussion of the choices of Watson, and Ragatz book [3] make such diagrams reference states which will lead to Eqs. (6) and nearly as familiar as the steam tables. (7), and so provision of such discussion is the ob­ Let Utru be the (absolute) internal energy of jective of this paper. the matter of the system being considered when this matter is in the reference state for internal REFERENCE STATE 1 energy, and let Stt-s be the (absolute) entropy of the matter of the system when this matter is in The Pure Chemical Compounds at Specified Conditions the reference state for entropy. Then the relative of T and p internal energy, Ut', and relative entropy, St', of Although it would be possible to choose refer­ the system are defined by ence states which are solutions we shall not do so Ut' == Ut - U trU (5a) here. Instead, we shall consider only reference states in which only pure substances, unmixed St'= St - StrS (5b) with other substances, are present, and present in and the calculation of Ut' and St' involves only cal- states of aggregation that are at least metastable. SPRING 1983 65 fled by arbitrary choices of the entropies si rs, but If the relative properties ... are if reactions occur, Eq. (10) shows that Eq. (9) to be useful then they should obey the same balance will be satisfied if and only if we chose these en­ equations as do the absolute properties ... tropies such that k ll'. Jj Sj rS = 0 (11) To begin with, we choose the pure substances to be j the pure compounds of which the system and the for each and every independent reaction occurring. process streams are composed. The reference state This shows that if chemical reactions occur, the for internal energy is assumed to involve in the conditions of temperature and pressure for the most general case a different combination of tem­ entropy reference states of the compounds of perature and pressure for each compound, and the which the system is composed cannot be chosen temperature and pressure of the reference state arbitrarily, but must be chosen so that the en­ for internal energy for a given compound are as­ tropies sirs satisfy a number of constraints equal sumed to be different from the temperatur e and to the number of independent reactions which oc­ pressure of the reference state for entropy for that cur. compound. If it is not possible to choose reference Consider now Eq. (8). Let Uiru (hi,n) be the states in this very arbitrary way, we want to find (absolute) molar internal energy (enthalpy) of that out. the pure j th compound when it is in the reference It is easy to show that Eqs. (6) and (7) follow state for internal energy (enthalpy). Then from Eqs. (2) and (3) and the definitions of Eqs. U trU = k NJ UjrU (5) if, and only if, the reference properties satisfy j dUtrU ~ h 0 hqrH = k Xq,j hj rH dt- ~ Wq qrH = (8) j and and so dStrS ~ 0 ~ - ~ Wq Sq rS = (9) dUtrU ~ h ~ ( dNj h ~ ) ---ar- -: Wq qrH = 1 UjrUdt - jrH~ Wq Xq ,J These equations are general; they must be satisfied dN­ for every valid choice of reference states of Ut, H t, = 7(U jrU -hJrn) a{ and St, It might be supposed at first that the time derivatives in these equations are zero, but such is + k R 1 k ll'.