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Mnemonic Device for Relating the Eight Thermodynamic State Variables: The Energy Pie Jeffrey E. Fieberg* and Charles A. Girard Chemistry Program, Centre College, Danville, Kentucky 40422, United States ABSTRACT: A mnemonic device, the energy pie, is presented that provides relationships between thermodynamic potentials (U, H, G, and A) and other sets of variables that carry energy units, TS and PV. Methods are also presented in which the differential expressions for the potentials and the corresponding Maxwell relations follow from the energy pie. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Mnemonics/Rote Learning, Thermodynamics
O
ne of the difficulties that students encounter while studying thermodynamics is remembering the relationships between thermodynamic variables including the state variables pressure, P; volume, V; temperature, T; internal energy, U (or E); enthalpy, H; entropy, S; Gibbs energy, G; and Helmholtz energy, A. Difficulties arise from the sheer quantity of variables and the multitude of relationships between these variables, including differential expressions and Maxwell relations (for a tabular summary of thermodynamic formulas, see Pitzer’s Thermodynamics1). A mnemonic diagram that relates the variables was first introduced by Born and published by Koenig2 and appears in two textbooks.3,4 Additional mnemonics, including two that use triangles, were presented at the Symposium on the Teaching of Thermodynamics at the 1962 national meeting of the American Chemical Society and published in this Journal.5 Other diagrams have more recently appeared.610 This article describes a mnemonic device, the energy pie (Figure 1), that provides students with a tool that relates all of the aforementioned variables and allows determination of the differential expressions of the potentials as well as their corresponding Maxwell relations. The energy pie is applicable for one-component systems in which all external fields can be ignored. The benefits of the energy pie lie in its simplicity and its ease of use to quickly arrive at a multitude of equations.
thermodynamic variables in the energy pie, one can remember that the variables on the lines spell UHG, and the pieces themselves build the phrase Tis a peeve. Alternatively, the placement of the variables can be recalled from various phrases invented by the students, such as “An Umbrella Has Got To Shelter People Valiantly” or “An Under-Handed Guide To Study Physical Variables”—that is, a mnemonic for a mnemonic! This mnemonic device is called the energy pie because all of the displayed parts have units of energy, either inherently (U, A, G, H) or because two variables are coupled (TS and PV) and their units, when multiplied, have units of energy. A side benefit of introducing the energy pie is that it reinforces that the products TS and PV carry energy units. Equations that relate the four thermodynamic potentials are determined in the following way: any potential that straddles two pieces of the pie is equal to the sum of those two pie pieces. For example, U straddles the line between the pie pieces containing A and TS. Therefore, U ¼ A þ TS The definition of the Helmholtz energy follows A ¼ U TS
ð2Þ
Likewise, G straddles the line between the pie pieces containing A and PV. Therefore,
’ USING THE ENERGY PIE The energy pie is divided into three pieces (see Figure 1). One piece contains a label for the Helmholtz energy, A, and the other two pieces include the labels TS and PV. Two variables, U and G, each straddle the line between two pieces of pie. The enthalpy variable, H, is in the middle of the pie and straddles all three pie pieces. The separation of enthalpy from all other energies is one of the primary advantages of the energy pie in comparison to other published mnemonics. To recall placement of the Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.
ð1Þ
G ¼ A þ PV
ð3Þ
Similarly, because H straddles all three pieces of the pie, it is equal to the sum of all three pieces: H ¼ A þ TS þ PV
ð4Þ
Published: August 01, 2011 1544
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Figure 2. Derivation of the differential form of U and the relevant Maxwell relationship from the energy pie. Figure 1. The energy pie diagram.
Table 1. The Four Principal Thermodynamic Potentials, Their Differential Expressions, and Their Corresponding Maxwell Relations Thermodynamic
Differential
Potential
Expression
U
dU = T dS P dV
H
dH = T dS + V dP
A
dA = S dT P dV
G
dG = V dP S dT
Corresponding Maxwell Relation ∂T ∂P ¼ ∂V S ∂S V ∂T ∂V ¼ ∂P S ∂S P ∂S ∂P ¼ ∂V T ∂T V ∂V ∂S ¼ ∂T P ∂P T
Figure 3. Derivation of the differential form of G and the relevant Maxwell relationship from the energy pie.
Alternatively, because H represents all three pieces of the pie, it can also equal the sum of U (two pieces of the pie) and PV (the third piece), H ¼ U þ PV
ð5Þ
which is the definition of enthalpy. Other important equations, such as G ¼ H TS
ð6Þ
can easily be determined: G (the right two pieces) is equal to H (all 3 pieces) minus TS (the third piece). By utilizing the “fundamental equation”11 dU ¼ T dS P dV
ð7Þ
the differential expressions for the other three potentials (Table 1) may be solved in terms of each potential’s “natural variables”.12 In addition, the four Maxwell relations (Table 1) follow by carrying out the exactness test (cross partial derivatives) on each differential expression. It cannot be overemphasized that students in physical chemistry should be able to derive the differential expressions and the Maxwell relations through an understanding of total derivatives, partial derivatives, and the exactness test. The energy pie, however, may also be used to determine the differential forms of the potentials in the following manner. The differential expressions of the potentials, dU, dH, dA, and dG, each contain the variables T, S, V, and P (Table 1). The order of
Figure 4. Derivation of the differential form of A and the relevant Maxwell relationship from the energy pie.
the variables in a differential expression is determined by drawing “energy paths” that emanate from each potential (see Figures 25). The numbers, “1,” “2”, “3,” and “4” label each variable in the order that they are traversed. The assigned numbers are also used to determine the Maxwell relations (vide infra). The differential expression for each potential (generalized as dX) is found by implementing the following equation for a given energy path emanating from a chosen potential, X: dX ¼ ( ð1Þðd2Þ ( ð3Þðd4Þ
ð8Þ
A minus sign is used when crossing the line between two pie pieces. Therefore, the numbers and signs are determined for each potential in the following manner. For the two potentials that straddle two pieces of the pie (U and G), the energy path is drawn straight through all four variables beginning with the pair closest to the potential—left to right for U, or right to left for G (see Figures 2 and 3). For those variables adjacent to the 1545
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Table 1, it must be remembered that the energy paths go toward H (see Figure 5). Note that eq 13 works because it is essentially carrying out the exactness test for a given differential expression of a potential.
Figure 5. Derivation of the differential form of H and the relevant Maxwell relationship from the energy pie.
potential, the sign of the term will be positive (no lines have been crossed); a minus sign before the last term (that includes variables 3 and 4) is used because a line is crossed to reach them. Using the energy path in Figure 2 and eq 8, the differential expression for dU is dU ¼ T dS P dV
ð9Þ
the “fundamental equation”. The energy path to obtain the differential expression for dG is determined in the same manner (using Figure 3 and eq 8): 11
dG ¼ V dP S dT
ð10Þ
For the differential expression for the Helmholtz energy, A, two energy paths are drawn toward the outside of the pie, beginning with the variables closest to A (Figure 4). Because lines must be crossed for each energy path, both differential terms have negative signs. Therefore, dA ¼ S dT P dV
ð11Þ
Because H straddles all three pieces of the pie, the two differential terms will have positive signs (no lines need to be crossed). A modification is necessary, however, to determine the correct differential expression for dH—the paths finish at H instead of begin with H. This “exception” is easily remembered, however, because it occurs only for the unique variable in the center of the pie. Therefore (as seen in Figure 5), dH ¼ T dS þ V dP
ð12Þ
The four Maxwell relations can be found by applying the exactness test to the differential expression for each potential. Alternatively, each Maxwell relation may be determined by applying the following formula to the energy paths ∂1 ∂3 ¼ ( ð13Þ ∂4 2 ∂2 4 where the numbers are those assigned in the energy paths (in the order that the variables are traversed). The positive sign is used if an even number of total lines is crossed (0 or 2) and the negative sign is used if an odd number of total lines is crossed (1). For example, the Maxwell relation that corresponds to the differential expression for U, where 1 = T, 2 = S, 3 = P, and 4 = V (recall Figure 2) is ∂T ∂P ¼ ð14Þ ∂V S ∂S V The negative sign is used because one line was crossed during the energy path. When determining the second Maxwell relation in
’ DISCUSSION Mnemonic devices are often helpful to students when learning new material, such as “OIL RIG”—oxidation is loss (of electrons) and reduction is gain (of electrons)—for the definitions of oxidation and reduction. The energy pie has aided students in learning thermodynamics at this college since 1968. Although positive credit cannot be given to one individual for “discovering” the arrangement of variables in the energy pie, Professor Lyle R. Dawson at the University of Kentucky used the pie in 1957. Recently, the power of the energy pie to determine the differential expressions and Maxwell relations via the aforementioned “energy paths” has been extended. Former students have used the pie in graduate and medical schools. As a fun bonus on final exams, the following has been asked, “If you were to choose one part of the energy pie for consumption, which part would it be and why?” Entertaining answers have included, “H because then I get to eat the entire pie!” and “A because I need the maximum work to get through finals week!” Recall that the Helmholtz function, as a differential, dA, represents the maximum amount of work that may be done by a system.12 ’ SUMMARY The energy pie is a mnemonic device that has made it easier for students to recall the relationships between the eight thermodynamic state variables for a simple, one-component system. The energy pie may also be used to write the differential expressions and Maxwell relations derived from each thermodynamic potential. Although it is important for students to understand the mathematics used to derive the differential expressions and corresponding Maxwell relations, the energy pie has benefitted our students in two primary ways: (i) access to a multitude of equations and relations from one simple diagram and (ii) the speed at which these relations may be arrived (which is helpful during timed exams) because of the simplicity of the diagram and energy paths.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: jeff.fi
[email protected].
’ REFERENCES (1) Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill: New York, 1995; pp 586593. (2) Koenig, F. O. J. Chem. Phys. 1935, 3, 29–35. (3) Callen, H. B. Thermodynamics, 2nd ed.; Wiley: New York, 1985; pp 183186. (4) Laidler, K. J.; Meiser, J. H.; Sanctuary, B. C. Physical Chemistry, 4th ed.; Houghton Mifflin Company: Boston, MA, 2003; p 129. (5) Radley, E. T. J. Chem. Educ. 1963, 40, 261. (6) Phillips, J. M. J. Chem. Educ. 1987, 64, 674–675. (7) Rodriguez, J.; Brainard, A. J. J. Chem. Educ. 1989, 66, 495–496. (8) Pogliani, L; La Mesa, C. J. Chem. Educ. 1992, 69, 808–809. (9) Chaston, S. J. Chem. Educ. 1999, 76, 216–220. (10) Pogliani, L. J. Chem. Educ. 2006, 83, 155–158. (11) Atkins, P. W.; de Paula, J. Physical Chemistry, 8th ed.; W. H. Freeman and Company: New York, 2006; p 103. (12) McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, CA, 1997; pp 896899. 1546
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