7/29/2019 article_phillips_1987.pdf

1/2

Mnemonic Diagrams for Thermodynamic SystemsJamesM. PhillipsDepartmem of Physics, University of Missouri-Kansas City, Kansas City,MO 64110

In 1929, a compact diagram implying considerable infor-mation was used by Max Born in his Gottingen lectures onthermodvnamics.' I t first annears in the literature in 1935 ina paper 6y F. 0. oenig2 anti has been utilized in a remark-able text bv H. B. C a l l e ~ ~ . ~ince then. th e use of the diaeramin thermodynamics has hecome commonplace for those whofind such geometrical representations helpful. The purposeof this note is to present a new mnemonic diagram (Fig. 1 )that is a three-dimensional extension of the Horn construc-tion.In th e energy formulation of the macroscopic thermody-namics of equilibrium (MTE), the internal energy U as afunction of the extensive independent variables Xi containsthe maximum possible information about a system. Thefundamental equation in terms of the generalized extensiveparameters is

U=U(X,,X ,,..., X,) ( t > r n + 2 ) (1)where m is th e number of component species of the system.Th e conjugate intensive parameters a re given by

The differential form of the fundamental equation is written

Equation 3 is the sumof the possible interactions a system ofinterest may have with a reservoirdU=dQ-dW+dA (4 )

where dQ is the heat added to the system, dW is the workdone by thesys tem, and dA is the diffusive interaction (massaction).Legendre transformations of the fundamental equationresult in the other thermodynamic potential of different"natural" variables

The Jacohian

is the sufficient condition for the existence of a particulartran~formation. '~~Some of the resulting thermodynamic potentials

are more common than others and have been traditionallygiven names such as Gibhs, Helmholtz, enthalpy, and Grandpotentials. Note, in the case k = t, fb = 0, which gives theGibbs-Duhem equation

Each transformed potential is a function of its own "natur-al" independent variables. Term-by-term comparisons of eq

Figure 1.Athrae-dimensional mnemonic diagramfor a simple thermodynamicsystem. The cube veniws represent the thermodynamic potentials and theadjacent cube faces the corresponding natural variables. Opposing faces arethe conjugate pairs of thermodynamic variables. The octahedron edges areguides to the thermodynamic relationships for details see ref3. ote the topview is the traditlonel hvoUimensionaiBorndiagram.

7 with the differrntial df, give the thermodynamic deriva-tives. The partial derivatives of the thermud\.namic poten-tial with respect to the extensive variable Yi are

Jfk/JYj= -Xi (05 5 ) (9 )and correspondingly

JfklJXj= Y, ( h+ 1 5 5 ) (10)The Maxwell relations are determined by changing theorder of differentiation with respect to a pair of "natural"variables while holding all others constant. For example,

a2fk/ax,ay = a2fk/a~,axj (11)Substitution of eqs 9 and 10 into eq 11gives

-(JXilaXj) = @Yj/JYi) i 5 h and > k (12)

and

For systems with more than two conjugate pairs of inde-pendent thermodynamic variables, not all of the thermody-namic derivatives and Maxwell relations are represented onthe same two-dimensional Born diagram. The new three-dimensional mnemonic diagram (Fig. 1) extends the

'Tisza, L. Generalized Thermodynamics; MIT: Cambridge. MA.1965~Koenig, F. 0. . Chem. Phys. 1935,3 ,29 .Callen. H. 6. Thermodynamics: An introduction to the PhysicalTheories of Equilibrium Thermostatics and Irreversible Thermody-namics; Wiley: New York, 1960.674 Journal of Chemical Education

7/29/2019 article_phillips_1987.pdf

2/2

Figure 2. Thegeneralized versionof the mnem onic diagram given In Figure 1.

representation to three conjugate pairs. This could be allthree oossihle tvnes of svstem-reservoir interactions shown.in eq 1.Systrmi u,ith a diffusive inwraction but two conju-gate pairsolwork variables uliofi t thesymmetry of Figure 1.Figure 1 depicts an octahedron whose vertices are on theface cenrerb of an enclosing rube. The disaonals of the octa-hedron are arrows connecting conjugate pairs and indicatingthe signs given in eq 7. The cuhe vertices are the possiblethermodynamic potentials whose "natura l" variahles are theadjacent cuhe faces. Legendre transformations move therepresentation from the internal energy corner U(S,V,h9 tothe other named potentials [(F(T,V,N), Helmholtz;G(T,P,N), Gihhs; H(S,P,N) ,enthalpy; and Q(T,V,p), grand.The G' vertex is, by Legendre transformation, the Gibbs-Duhem zero (eo 8). For Durnoses of iustifvine the relatededges o t Figure i, he the;m