Mnemonic Diagrams for Thermodynamic Systems James M. Phillips Departmem of Physics, University of MissouriKansas City, Kansas City, MO 64110 In 1929, a compact diagram implying considerable information was used by Max Born in his Gottingen lectures on thermodvnamics.' I t first annears in the literature in 1935 in a paper 6y F. 0.Koenig2anti has been utilized in a remarkable text bv H. B. C a l l e ~Since ~ . ~ then. the use of the diaeram in thermodynamics has hecome commonplace for those who find such geometrical representations helpful. The purpose of this note is t o present a new mnemonic diagram (Fig. 1 ) that is a threedimensional extension of the Horn construction. In the energy formulation of the macroscopic thermodynamics of equilibrium (MTE), the internal energy U as a function of the extensive independent variables Xi contains the maximum possible information about a system. The fundamental equation in terms of the generalized extensive parameters is U=U(X,,X ,,..., X,)
(t>rn+2)
(1)
where m is the number of component species of the system. The conjugate intensive parameters are given by
Figure 1. Athraedimensional mnemonic diagram for a simple thermodynamic system. The cube veniws represent the thermodynamic potentials and the adjacent cube faces the corresponding natural variables. Opposing faces are the conjugate pairs of thermodynamic variables. The octahedron edges are guides to the thermodynamic relationships(for details see ref 3.Note the top view is the traditlonel hvoUimensionai Born diagram.
The differential form of the fundamental equation is written
Equation 3 is the sumof the possible interactions a system of interest may have with a reservoir dU=dQdW+dA
7 with the differrntial df, give the thermodynamic derivatives. The partial derivatives of the thermud\.namic potential with respect to the extensive variable Yi are Jfk/JYj= Xi
(4)
where dQ is the heat added to the system, d W is the work done by thesystem, and dA is the diffusive interaction (mass action). Legendre transformations of the fundamental equation result in the other thermodynamic potential of different "natural" variables
(05 i 5 k)
(9)
and correspondingly (h + 1 5 j 5 t ) (10) JfklJXj= Y, The Maxwell relations are determined by changing the order 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 The Jacohian
is the sufficient condition for the existence of a particular tran~formation.'~~ Some of the resulting thermodynamic potentials
are more common than others and have been traditionally given names such as Gibhs, Helmholtz, enthalpy, and Grand potentials. Note, in the case k = t, fb = 0, which gives the GibbsDuhem equation
(JXilaXj) = @Yj/JYi)
i 5 h and j
>k
(12)
and
For systems with more than two conjugate pairs of independent thermodynamic variables, not all of the thermodynamic derivatives and Maxwell relations are represented on the same twodimensional Born diagram. The new threedimensional mnemonic diagram (Fig. 1) extends t h e
' Tisza, L. Generalized Thermodynamics; MIT: Cambridge. MA. 1965~
Each transformed potential is a function of its own "natural" independent variables. Termbyterm comparisons of eq 674
Journal of Chemical Education
Koenig, F. 0.J. Chem. Phys. 1935,3,29. Callen. H. 6. Thermodynamics: An introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics; Wiley: New York, 1960.
written as a function of p and T. The corresponding Maxwell relation reduces to zero equals zero. One of the "natural" variables, in this case pressure, has become dependent. The rules for using the threedimensional diagram are the same as the twodimensional one. These rules are outlined clearly by caller^.^ Notice that the top view of Figure 1is the Born diagram. Figure 2 is the generalized version of the new diagram with XIYl and XzYz as the conjugate pairs of thermodynamic variables. The entropic formulation of thermodynamics can he represented in the fashion of Figure 1 by using the U vertex as the entropy S(U,V,N) and the S face center as U. The conjugate pairs are then U l/T, V PI T, andN plT. The Fvertex becomes the Massieu function Y(lIT,V,N) and then Q vertex the Kramers function q(l1 T,V,pIT). The Planck formulation can also be formed by the G vertex of Figure 1becoming +(lIT,P,N) and the conjugate pairs VIT P , H 1/T, and N p/T. To illustrate the Maxwell relations with Figure 1,consider the simple system

Figure 2. The generalized version of the mnemonic diagram given In Figure 1. representation to three conjugate pairs. This could be all three oossihle tvnes .. of svstemreservoir interactions shown in eq 1. Systrmi u,ith a diffusive inwraction but two conjugate pairsolwork variables uliofit thesymmetry of Figure 1. Figure 1 depicts an octahedron whose vertices are on the face cenrerb of an enclosing rube. The disaonals of the octahedron are arrows connecting conjugate pairs and indicating the signs given in eq 7. The cuhe vertices are the possible thermodynamic potentials whose "natural" variahles are the adjacent cuhe faces. Legendre transformations move the representation from the internal energy corner U(S,V,h9 to the 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 GibbsDuhem zero (eo 8). For Durnoses of iustifvine the related edges ot Figure i, the the;m