Thermodynamic Partial Derivatives and Experimentally Measurable Quantities Gentil A. Estevez Department of Physics, University of Central Florida, Orlando, FL 32816 Kai Yang Physics Department, Rensselaer Polytechnic Institute, Troy, NY 12181 Basab B. Dasgupta RCA-Consumer Electronics Division, Indianapolis, IN 46201 As is well known, equations of state today seldom are determined from pressure, volume, and temperature measurements, but are rather derived from measurements of equations of state derivatives such as the bulk modulus (via sound velocities), heat capacities, thermal expansivities, and other quantities. Assuming, however, that a theoretical model for an equation of state can be found, a practical situation that is often considered is (.I .) : (1) . . Giuen the eauation of state that describes a thermodynamicsystem,andthe s~ecificheat, determine several oartial derivatives associated with the system. A related but somewhat more general problem is: (2) Given a macroscopic system, express thermodynamic partial derivatives that are inaccessible to direct measurement (e.g., the Joule-Thomson coefficient), in terms of macroscopic quantities that are conveniently measurable in a laboratory. Although inherently interesting from a mathematical viewpoint, problem (1) above is of a quite limited practical use. Indeed, the several expressions sought depend on partial derivatives that are not only unknown, but are also not directly observable experimentally. Aware of the practical difficulties that generations of thermodynamics students have had with the computation of partial derivatives of arbitrary thermodynamic variables, in terms of experimentally determined quantities, we have attempted to reduce this problem to a simple formalism. The main goal of this article is to propose a flow-chart-type figure and related formulas to obtain exnressions for the derivatives of the old thermodynamic function in terms of the new variables. the new function and its derivatives and derivatives of the old function, involving only measurable quantities. The figure would be of interest to students of physical chemistry in helping to organize the students' thinking of difficult derivations. Review of Fundamental Equations For pedagogical reasons we begin by briefly reviewing several standard mathematical identities. Assume a set of five variables (x, y, z, u, w ) is given such that f(x, y, z) = 0, with u and w functions of any two variables among the remaining three parameters of the set, i.e., only two variables are independent. Under these conditions relations 14 in the Appendix can be readily estahlished (2). Among the wellknown experimentally defined quantities are the pressure p, temperature T, and volume V. Other quantities are the thermodynamic response functions, namely, the heat capacities, either Cvand Cp, the isothermal and adiabatic compressibilities, KT and Ks, the constant-volume heat of pressure variation, r v , and the thermal expansion coefficient, NT. I t is useful for the treatment that follows to introduce three types of symbols: (1)a and P to denote the experimentally measurable variables p , T, or V, (2) "a" and "b" to
890
Journal of Chemical Education
denote either the internal energy E or the entropy S, and (3) B, C, and D which will be reserved for the enthalpy H, or either of the thermodynamic state functions, A (Helmholtz free energy) or G (Gibbs potential). We consider first what we regard as an important type of partial derivative. Indeed, several other derivatives can be expressed in terms of this type. 1. a. as la^)^ (i) We take @ = T. From the Maxwell relations (equality of mixed second partial derivatives) (2) we can immediately write eq 5 and 6 in the Appendix. (ii) Next, we take 0 = V. From the defining equation for the specific heat, Cv, we obtain eq 7 in the Appendix. In determining an expression for (aSlap)v, eq 2 will be employed with V = T. Thus eq 8 in the Appendix is obtained. (iii) a. When B = p the partial derivative to he transformed is (aS/aT),. The equation of state can be employed to relate one of the variables of the argument of S t o the other two. Assuming that the volume can be expressed as a function of p and T, then
where we have made use of eqs 4, 6, and 7. Furthermore, since Cp = T(aS/aT),, the last expression above develops into a well-known relationship between Cp and Cv, which is eq 9 in the Appendix. Notice that, because of eq 9, any thermodynamic derivative that can be expressed in terms of Cv, can be also written in terms of Cp. In the present article we will only employ Cv. b. (aslaw, Employing the relation (see eq 2)
together with eq 9 yields eq 10 in the Appendix. In general, for isothermal (constant T)processes, the partial derivatives of one arhitrary dependent thermodynamic variable does not depend on Cv. The deduction of eqs 8-10 turned out to be slightly involved. However, the procedure to obtain them is fairly simple (namely, transforming them into the forms (aS/dN)T and (aS/aT)v). Having obtained all partial derivatives of the type (as/ we now turn to the computation of another important type of derivatives, namely ( a E l a ~ ) ~ . 2. ( ~ E I ~ N ) ~ Employing the mathematical identity, eq 4, the following result is readily obtained
I
I
LT.
I I I I
II I
The meaningofthe several symbolsappearing in this figure are asfoiiows:n and Bdenotellwexperimemallymeasurabievariabiesp, T, and V. lhe quantities denoted by "a" and " brepresent,each, either the internal energy Eor theentropy S. Finally. thecapitalletters B. 6,and Dare employed todenotelbenthaipy H, aeither tree eneravl. of the memwdvnamic sale functions GlGibbs wtenlkl). w A IWmholtz . -.. The row numbers in the table indicate the total number of times mat B.. c.. D.. "a", and " b appear as variables in a panialderivative.Column numbers indicate the total number of times b t 8, C, and Dappear in aglvenderivative.LT denotes a Legendre transformation. me numben alongside llw m w s represent the equation horn the text. employed in going from one step to the next.
With the definitions of temperature and pressure it is possible to recast this expression as eq 11 of the Appendix. From eq 11 it follows that problems of the type (aElacr)8can be transformed into problems of the type (aS/aa)8,which have already been discussed in section 1above. 3. A thermodynamic derivative contains three variables of the set Lp, V ,T ,E, S , A, G, H).Since A, G, and Hmay be involved, we define next the Legendre transformation (LT) expressions from which these potential functions are obtained. These are eqs 12-14 of the Appendix. In several cases, the entropy S may appear explicitly in the final result for a partial derivative of interest. It is thus convenient to give here an expression for S. Again, for pedagogicalreasons, we begin by writing the entropy in the differential form:
dS = (JS/JT),dT+ (dS/JV),dV Noting that ( a S / a n v = CVIT and ( d S l a V )= ~ (aplanvand integrating the resulting expression yields eq 15 of the Appendix. Use of Dlagram
In this short section we outline the use of the flow chart to compute thermodynamic partial derivatives. The procedure for finding thermodynamic derivatives is given by the flowchart presented in the figure. Although from a practical
viewpoint several of the derivatives shown in that diagram are not useful, they have been included for the sake of completeness. Employing the flow chart, all derivatives involving A, G, and H are obtained in terns of quantities that can be measured in the laboratory, such asp, V , T , C, r v , (BVIaT),, and
(JVI~P)T. The row numbers in the figure represent the total number of times that the quantities B, C, D, "a", and "b" appear in a particular partial derivative. Likewise, the column numbers represent the total number of the quantities B, C, and D present in a given derivative. Notice that all steps in the figure are marked by arrows. In each step, the relevant equation from the text is referenced by its equation number. To facilitate the use of the figure we have constructed an Appendix where we have presented all auxiliary equations that are needed. Example: The Joule-Thomson Coefllclenl
As an illustration of the use and relative utility of the flow chart, the deduction of the Joule-Thomson coefficient (3,4) PJT = ( a T / a p ) ~will , be considered. This thermodynamic quantity is a special case of derivatives of the type (aB/an)B. Since the partial derivative in question contains only B, it can thus be fonnd in the figure under row number 1 and column number 1. Volume 66
Number 11 November 1989
89 1
To compute ( a T l a p ) ~we follow the prescription in the figure that eq 3 be employed to modify the original partial derivative into a product. (JTIJp), = -(JTIJH),(JHIJP)~
(16)
The next step according to the figure consists of using eq 1to transform the result iust obtained into a ratio. This is easilv done with the result that (JTlap)~ = -[(JHIJP)~(JHIJT)~I
(17)
Notice the presence of Cp in the denominator of eq 17. Following the algorithm given in the figure a Legendre transformation is now used for the enthalpy H, i.e., H = E pV. This leads immediately to
+
The figure tells us to employ next eq 4 (or equivalently eq 11) to change the last result above. We thus get prn = (JTIJp), = [T(JSlJp),
- Vl/Cp
(19)
Notice that there is no need to employ eq 5 t o convert eq 19 to a form where only measurable quantities appear, the reason being that it has been already expressed in terms of experimentally determined quantities. This was done at an earlier step when we recognized that by definition C, = (aHl 8'0,. The relationship between Cp and Cv given by eq 9 can, of course, be utilized to express the Joule-Thomson coefficient in terms of Cv. The desired final result then reads PJT = l[T(aVIJnp
- VI/[Cv + T ( J ~ l J T ) v ( a V l J n , l l
(20)
It is worth pointing out that in the traditional way the derivation of the Joule-Thomson coefficient would require finding (aVlaS)~,which is a somewhat difficult a task to accomplish. In fact, this derivative expressed in terms of measurable quantities cannot be deduced employing Maxwell relations. From row 1and column 0 of the figure, and . utilizing eqs 1 and 6, it followsthat (avlas), = K T I ~ TGoing back to the Joule-Thomson coefficient given by eq 20, we observe that it can be readily expressed in terms of the Vand T partial derivatives of the equation of state explicit in p. This is achieved by direct application of eq 3 to change (aV1 XI'),. The resulting equation has been recently employed together with the van der Waals, Redlich-Kwong (9,and Beattie-Bridgeman (6) equations of state to measure the values of PJT for several gases (7). The algorithm employed in this paper to obtain a particular thermodynamic partial derivative may not he the shortest one found in the-literature (8-10);hoGever, this method is useful because in all cases that we have considered we have been able to express arbitrary partial derivatives in terms of quantities involving only measurable quanrities. Finally, we mention that the extension of the current procedure 6 systems with more than one component is straightforward but tedious. Acknowledgment Dedicated to Kerson Huang and John David Jackson with admiration and affection. We want to express our sincere thanks to Wilfrido Solan~Torres,Alfonso DiazJimBnez, Victor Santiago, and Peilian Lee for their kind encouragement in so many ways. Financial support for this work was provided in part by a grant to GAE from Universidad Francisco Jose de Caldas, Bogot6, Colombia. Literature Cited 1. For s student's pcmeptionof what therrnodynamirr is. a*.:
Andrew, F. C. Thermodynomiea:mneipk8 and Applicolio~:Wiley-1ntusci.n~ New Ymk. 1 9 7 %3: ~ see
also Barrow. G. M.Physical Chemistry, 5th ed.; MeGraw-Hill: New Yoxk, 1988.
2. Husng.K.SfoliaficalM~rhonLs.2ndBd.;Wiley:NewYo1k,19S7.Arirnpl~methodfor deriving Maxwii relations has been given by Ritchie. D. J. Am. J Phys. 1968.36, 760-770.
892
Journal of Chemical Education
3. Rybolt, T. R. J. Ckm. E d w . 1981,SB.621. 4. Naggle, J. T Phyairol Chomisfry; Little, Bra-: Boeton, 1985: pp 105-110. 5. Redlieh. 0.: Kwong, J. N. S. Chem Rsu. 1949. d4.233. 6. Besttie, J. A,, Bridpeman, 0. C. J.Am. Chern. Soc. 1928,50,31333140. 7. Halpern. A. M.; Gorashti, S.J. Chom. Edw. 1986.63.1Wl. 8. See, for example, Gilmore, R.J. J Chem. Phys. 1981, 75, 5964-5970. Seedso Mohan, G. Am. J Phys. 1968,37,912918. 9. Gilrnore, R.J. Cotostrophe Theory for Scienlisla and Engl'nglm; Wiley New York,
1981:... 00 235-240. 10. Andresen, B.: Berm R. S.;Gilmore, R.; &g,
E.; Salamon. P. Phyr. Reu. A. 1988.37,
845-846.
Appendix: Auxlllary Equations Needed for the Proper Use of the Figure Note that the first 15 equations are referenced by the equation numbers that they would have in the text. The last sin equations are simply the definitions of the thermodynamic "response functions" for a single-component substance.
