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THE]THERMODORM, a Mnemonic Octahedron of Thermodynamics [jn a classroom THE]THERMODORM, A Mnemonic Octahedron of Thermodynamics k (8) d(- PV) =-SdT - PdV - l NidJJi = dU[T,µi] A. F. GANGI, N. E. LAMPING, AND P. T. i=l EUBANK and Texas A&M University k (9) College Station, Texas 77843 O =- SdT + VdP - l NidJJi • dU[T,P,JJi ] , i=l The development, use, and construction of the the last relation is also called the Gibbs-Duhem THERMODORM*1, a mnemonic octahedron in equation. U, H, A, G, TS-PV, TS, and -PV will be termed the energy functions with S V P T terms of the energy functions of thermodynamics ' ' ' ' and their variables is described. This easily con­ N1 and µ.1 as their independent variables. The structed, pocket-sized device provides at a glance natural variables of U are S, V, and N1 while the complete set of Maxwell relations and energy those of G are T, P, and Ni, etc.6 function derivatives for k component systems. A total of 7 (k + 2) Maxwell equations result from cross differentiation of Equations (2-8) of THEORETICAL BACKGROUND the type a2u _ a2u For a homogeneous phase of k components in asav = avas ( 10) an open system a fundamental equation of ther­ to yield, for example, modynamics2,3 is given by the expression for the internal energy of the system : H: )v N = - ff ls N ( lll ' i ' i U • U(S,V,Ni); i = 1,2, . .. , k (1) gross differentiation of Equation (9) yields re­ The total differential of the internal energy is: lations of the type k dU • TdS - PdV + l JJidN. , (2) i =l 1. [ ~ )T JJ = - f* lPJJ = ~ (l2) where the intensive variables (the temperature ' i ' i since the phrase rule allows but k 1 independent 'I', the pressure, P, and the chemical potential, + intensive variables from the potentials P, T, and /J.i) are defined by the partial derivatives of the µ.1, Thus, internal energy.4 By means of Legendre transformations, the [* l p JJ = [ *l T = [ :~~ ) = O' (13) internal energy U (S,V,Nd can be converted into ' i , µi P • T , µj the other thermodynamic energy functions :5 the since [dTJp = 0, etc. enthalpy H(S,P,Ni), the Helmholtz free energy ,/J.i A(T,V,Ni), the Gibbs free energy G(T,P,N;) , etc. A total of 7 (k + 2) derivatives may be formed Their differentials are: by differentiation of the seven nonzero energy k functions with respect to their natural variables. (3) dH = TdS + VdP + l JJ idNi = dU[P]' i::l Of the derivatives for TS - PV, TS, -PV, two are k (4) [ •~~S)J : Tand _[ •(a~V ) J : -P, (14) dA - - SdT - PdV + I µ dNi = dU[T]' P,µi T,µi i=l i k as a consequence of the phase rule. Other deriva­ (5) dG =- SdT + VdP + l µ, dNi = dU[T,P] i=l 1 tives are consistent with the Maxwell relations. k For example, (6) d(TS-PV) Tds - PdV - N ,dJJi dU[JJi l = I 1 = i=l V • [· ~!S) l = s [ *) (15) k S,JJi S,JJi (7) d(TS) - TdS + VdP - l Nidµi = dU[P,JJi J i =l which agrees with the Maxwell relation, '''"Thermodynamic Octahedral Display Of the Rela­ ar } [ av ) v (16) [ µ= asPJJ=s w 5 tions of Maxwell" ' i ' i 30 CHEMICAL ENGINEERING EDUCATION A. F. Gangi received his BS, MS and PhD ('60) in N. E. Lamping received his BS in Geology from the Physics from the University of California at Los Angeles. University of Notre Dame, an MS in Geodetic Sciences While at UCLA he was associated with the Institute of from Ohio State University and a PhD ('70) in Geophysics Geophysics. His doctoral thesis subject was the diffraction from Texas A&M University. His PhD thesis subject was of -elastic waves by wedges. He was manager of the An­ "The Mohorovicic Discontinuity as a Phase Transition." tenna Department at Space-General Corp. in El Monte, Major Lamping has been awarded the Air Force Com­ California before becoming an Associate Professor of mendation Medal twice; once in 1962 for meritorious ser­ Geophysics at MIT. Presently he is a Professor in the vice and in 1968 for outstanding achievement. He presently Geophysics Department at Texas A&M University. His is Chief of the Cratering Geophysics Section at the Air research interests are theoretical seismology, properties Force Weapons Laboratory, Kirtland AFB, New Mexico. of the earth's interior and theoretical geophysics. P. T. Eubank was a graduate from Rose Polytechnic Institute and received his PhD degree from Northwestern Of the derivatives for H, A, and G, we find University under Professor J. M. Smith in 1961. He has taught thermodynamics and transport phenomena for c1 = [ :~1 ) = µ1 c1n the past ten years at Texas A&M University and is on P,T,Nj the research staff of the Thermodynamics. Research Center where Gi is the partial molar Gibbs energy. there. He is presently pursuing his principal research interest, the physical and thermodynamic properties of DEVELOPMENT AND USE polar vapors, under the direction of Professor J . S. Row­ linson at Imperial College, London, England. (right) The Maxwell relations and energy function derivatives can be remembered by means of the 6 7 thermodynamic squares • with arrows pointing along the diagonals as shown in Figure 1 for the energy functions U, A, G, and H flanked by their natural variables (excluding N1). The four com­ mon Maxwell relations are found using this square. For example, the Maxwell relation [¥s t N • [ *ls N (lB) p , i ' i s H is obtained from this square using the following Figure I .-The Common Figure 2.-The Constant procedure. We take the derivative of the variable Thermodynamic Square. Temperature Square. in the top left corner with respect to the variable where now the sign is negative because one ar­ in the bottom left corner holding constant the row points upward and the other downward. We variable in the bottom right corner and on the have placed N1 at the center of this square since face. We equate this to the derivative of the these four Maxwell relations are for the constant variable on the top right corner with respect to Ni. the variable on the bottom right corner holding The square also provides the derivatives of constant the variables in the bottom left corner the energy functions such as and on the face. The sign is determined by the arrows; the sign is positive if both arrows point [¾¥L,N i = T, [¥vtNi = -P, [¾¥-] p ,Ni = -S either upward or downward. Using this proce­ The sign is positive if the variable is at the head dure, rotation of the square 90 ° clockwise yields of the arrow and otherwise negative. There are six such squares, one for a fixed [* )T N = - [ *t N (19) ' 1 ' i value of each of the independent variables. Fig- WINTER 1972 31 ure 2, the constant temperature square, yields Maxwell relations such as (20) aµi ] = [ ~v ] = v i T [ ap Ni ,T Ni P,T,Nj (the partial molar volume) and energy function TS- PV 0 derivatives such as We now place the eight energy functions (in­ cluding the zero Gibbs-Duhem function) on the -PV T triangular sides of a regular octahedron with Figure 3.-The Constant Figure 4.-The Energy the six independent variables located at the Chemical Potential Square. THERMODORM. points. The energy functions are placed on the fabricated. The points of the octahedron have top pyramid of the octahedron so as to reproduce been planned normal to the major axes to pro­ the square of Figure 1 when sighting directly vide small squares for lettering the independent into N1 point. The bottom pyramid of the octahe­ variables. A common soldering gun with a fine dron reproduces the square of Figure 3 when point is adequate for lettering. The arrows, we look directly into the µ. 1 point. On the plane S ~ T, Ni ~ µ.1 and P ~ V, can be set bisecting the top and bottom pyramids of the directly in the mold if the device is cast. Other­ octahedron the U surface intersects the (TS-PV) wise, a negative sign should be placed before S, surface since they share V and S as natural vari­ Ni, and P. The minus sign is then used when ables - likewise for G and zero which share P, forming Maxwell relations and energy deriva­ T, and etc. All six mnemonic squares may be tives but is ignored where it appears in the de­ seen on the THERMODORM. nominator of a partial derivative. A collapsable paper THERMODORM can be CONSTRUCTION easily constructed from the pattern provided by Figure 4 is a pictorial representation of a Figure 5. THERMODORMS can be also made for clear plastic model which can be quickly and easily S(U,V,N1), V(U,S,N1), and N1(U,S,V) and the functions obtained by Legendre tranformations ® @ Figure 5.-Paper Mnemonic THERMODORM. \ Assembly Instructions \ 1. Cut along solid lines. \ 2. Score and fold down .along dashed lines. 3. Connect sides with same letters. ' © A.F. Gangi, N.E. Lamping, 1971 34 CHEMICAL ENGINEERING EDUCATION analogous to that for the energy functions. Trans­ forms of the entropy are termed Massieu func­ [eJ na book reviews tions which yield relations upon cross differentia­ tion analogous to those of Maxwell. Processes and Systems in Industrial Chemistry, SUMMARY H. P. Meissner, Prentice-Hall, Englewood Cliffs, N.J. (1971), 386pp., $14.95. The theoretical basis, use, and construction of a mnemonic octahedron for the representation of the energy functions and their derivatives to­ This book is primarily an introductory dis­ gether with the Maxwell equations, has been pre­ cussion of reaction kinetics and associated ther­ sented.
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