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Chapter 9
Mnemonic Devices for Thermodynamic Relationships Ray C. Dudek* Department of Chemistry, Wittenberg University, Ward Street at North Wittenberg Avenue, Springfield, Ohio 45387, United States *E-mail:
[email protected] Students are presented with over a dozen equations in learning thermodynamics which can be overwhelming when seen the first time. Mnemonic devices are employed so that remembering these relationships is much easier, and this also makes the equations more accessible. The use of mnemonic devices can be particularly valuable on standardized exams where time is limited and equations are not given. This chapter will give an overview of 3 mnemonic devices. The first is an Energy Pie developed by Fieberg and Girard (Fieberg, J, E.; Girard, C. A. Mnemonic Device for Relating the Eight Thermodynamic State Variables: The Energy Pie. J. Chem. Educ. 2011, 88, 1544-1546). The second is an energy square created by Rodriguez and Brainard (Rodriquez, J.; Brainard, A. J. An Improved Mnemonic Diagram for Thermodynamic Relationships. J. Chem. Educ. 1989, 66, 495-496). The last one, on which more detail will be presented, is called VAT-VUS and is the creation of the author.
Introduction A standard course in thermodynamics presents a high number of equations which are new to the typical undergraduate student. These thermodynamic equations involve different forms of energy and the independent variables upon which they depend. More specifically, the equations first define the Internal Energy (U), Enthalpy (H), Helmholtz Energy (A), and Gibbs Free Energy (G), as shown in Figure 1a, and then relate them to changes in pressure (p), temperature © 2018 American Chemical Society Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
(T), volume (V) or entropy (S) as seen in Figure 1b. Next, Figure 1c, the partial derivatives of the equations in 1b result in the new relations presented in Figure 1c. Finally, Figure 1d, the Maxwell Relations arise from relating two partial derivatives taken from the original energy equations. Derivations of these relationships are shown in nearly every physical chemistry textbook, so they are not presented in this text.
Figure 1. a-d. The thermodynamic relationships that will be solved for using mnemonic devices.
While instructors in physical chemistry would recommend against trying to memorize all 20 equations, and instead advise to deriving them as needed, most students do not find these derivations readily accessible. As a result, students routinely look them up each time they are needed, or if not possible, often incorrectly remember the needed relationship. This becomes particularly problematic on standardized exams (for example ACS, GRE, or MCAT) where the equations are not commonly given and are not available to the student. To assist with recalling these 20 thermodynamic relationships mnemonic devices have been employed. Another feature of these mnemonic devices is that they help visually demonstrate the interconnectedness of the variables they employ. The use of mnemonic devices in assisting in memory enhancement has been documented (1). Mnemonic devices relating these variables were first published by Koeing (2) over eighty years ago, and have appeared in numerous textbooks (3–5) and journal articles (6–11) since then. These mnemonic devices have gotten increasingly sophisticated and feature more of the equations shown in Figures 1 than the original one published by Koeing. All mnemonic devices use pattern 132 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
recognition, where often one equation is memorized as an example of how the device works, and then the pattern is repeated employing the other variables to obtain the remaining equations. Each mnemonic device employs some feature to indicate the negative signs in the resulting equations. In this chapter, three of the more contemporary mnemonic devices will be discussed: two that have been recently published (6, 7), and one that is the creation of this chapter’s author. How to obtain the thermodynamic relationships from each mnemonic device will be presented in the remainder of this chapter, though they are usually easier to understand in the figures and following the numbers than from the text alone. Students are shown how to use the third mnemonic device presented (VAT-VUS) only after they have learned the thermodynamic relationships seen the derivations of the Maxwell equations. When first exposed to the mnemonic device, students are excited to see that there is one method to visualize and remember all of the equations. The students are encouraged to memorize one sample expression and use it to assist in recalling how the mnemonic device works.
Mnemonic Devices Energy Pie Developed by Fieberg and Girard (6), the Energy Pie was published in 2011. Rather than using a square arrangement of the variables, a circular format with three distinct slices is used, as shown in Figure 2. To remember the arrangement of the variables, the authors recommend using a second mnemonic of “An UnderHanded Guide To Study Physical Variables”. The lines separating the circle into three sections designate the negative signs in the equations.
Figure 2. Using the Energy Pie to obtain a fundamental equation. (Adapted with permission from reference (6). Copyright 2011 Ameircan Chemical Society.) 133 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
The first set of equations (Figure 1a) is obtained for the energy relations, by seeing which variable are in the adjacent regions. For example, U = A + TS as the two terms on the right side of the equation are in the regions next to the term on the left. Hence, it is easy to see that G = A + PV, and H = A + TS + PV. The next set of equations (Figure 1b) can be found by drawing lines from the energy term to the other four variables. As shown in Figure 2, the line drawn is used to determine the equation: dU = TdS – PdV. Where the drawn line crosses a section, a negative sign is appears in the final equation. All of the differential expressions are found by using connecting lines, though the user has to remember to go from the center to the outside for dA, and in reverse (outside to the center) for dH. The Maxwell relations can be realized by putting the sequence of variables from the previous relation into the following form: . The ± sign depends on the number of times a line section is crossed: 0 or 2 times remains positive, and 1 or 3 times becomes negative.
HPG-HSU The next mnemonic device, published in 1989 by Rodriguez and Brainard (7), puts the variables in the more traditional square arrangement, seen in Figure 3. Each of the four thermodynamic energy quantities are in the corners and the independent variables are located at the center of each side. Two arrows are drawn to help determine the positive or negative sign used in each equation. No name was attached to this mnemonic device, so for the purposes of this publication it will be referred to as “HPG-HSU”, which is the order of the variables across the bottom of the square, and then up the left side. This naming convention is also used for the last mnemonic device “VAT-VUS”.
Figure 3. a. Using the HPG-HSU to obtain the energy definitions (3a), the fundamental equations (3b), and the Maxwell equations (3c). (Adapted with permission from reference (7). Copyright 1989 Ameircan Chemical Society.) 134 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
The energy definitions (Figure 1a) are obtained by starting in any corner, as the variable on the left side of the equation, and then going clockwise to the next corner for the first variable on the right side. The last term comes from going to the variable adjacent in the counter-clockwise direction, and multiplying it with the one on the opposide side of the square. If one goes against the direction of the arrow that connects the two terms, and negative sign is added to the last term on the right. For example, as shown in Figure 3a, start at the first term on the left of the equation, U, and the proceed closewise to the next corner, A, for the first term of the right side of the eqution. Then go counter-clockwise to the next variable, S, and then go across the square for the last variable, T. By going with the arrow between S and T, no negative sign was applied. This pattern could be repeated by starting at G, going clockwise to H, and then counter-clockwise to T, and finally to S. In this second example, a negative sign is applied (going against the arrow between T and S), to give the final equation G = H – ST. The differential form for each energy term (Figure 1b) is obtained by taking the derivative of the adjacent variables and multiplying it by the variable on the opposite side of the square. As before, going against the arrow when connecting the derivative and the variable, it is multiplied by a negative sign. An example of this, shown in Figure 3b, is to start at one corner with U for the left side of the equation (dU). Then go to an adjacent variable, V, which will be the derivative, and that is multiplied by the variable on the opposide side of the square, P. Going against the arrow results in negative sign for the PdV term. Repeat the process with the other adjacent variable, S (again the derivative), and the opposite side of the square for the multiplier, T. The final equation is dU = TdS – PdV. A second example is to start with G for the left side of the equation. The right side will be dT multiplied by S, and dP multiplied by V. The SdT term will be negative. The final equation for the second example is dG = -SdT + VdP. The authors chose not to present how to obtain the derivative expressions in Figure 1c, and instead suggest that the students derive them from the differential form of the energy equations. Finally, the Maxwell Relations are obtained by starting at any of the center variables and going clockwise or counterclockwise for the other two variables on one side of the equation. The other side of the equation begins by going to the next center variable, and reversing direction for the remaining two variables. Unfortunately, no mention is made as to how to determine the negative signs in the Maxwell Relations. An example of using HPG-HSU to obtain a Maxwell Relation, shown in Figure 3c, is to start at V and then go clockwise to the next center term, T, for the partial derivative, ∂V/∂T. The next center term in the clockwise direction, is the constant of the expression, P. Continuing in the clockwise direction for the first term of the second partial derivative, S. This time go counter-clockwise for the rest of the expression. So the partial derivative of S is with respect to P, with T being the constant. The resulting expression is:
135 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
A second example would be to start with P and the derivative would be with respect to S with V being the constant. The second term would start with T as the derivative with respect to V and this time S would be constant. One of the partial derivatives will be negative, though it is not clearn from this mnemonic device how that is obtained. The overall final expression is:
VAT-VUS This final mnemonic device rotates the variables used in the previous one, such that the energy values are in the middle of each side, and the independent variables are located in each corner. This is similar to the orientation used by Callen (3), but all 20 equations presented in Figure 1 will be obtained. The name of this mnemonic device indicates how the letters should be arranged, with V-A-T across the bottom, and V-U-S along the left side. Then “Hills over Ground” is used to place the two remaining energy terms, and finally a somewhat childish, though memorable “Pee in the corner” for the final independent variable. As opposed to the previous mnemonic device of HPG-HSU, the layout of the variables is relatively easy for the students to remember. A potential drawback is that VAT-VUS uses two sets of arrows to designate the signs in the equations, one for the equations in Figure 1a & 1b, and the other for the equations in Figure 1c & 1d. Though, similar to all mnemonic devices, if the students remember one sample equation on how to use the device, the location of the arrows become apparent. For the energy equations and the differential expressions of the energy equations (Figure 1a & b), two diagonal arrows are draw inside the square, both pointing to the upper corners. To obtain the first set of equations (Figure 1a), start with any energy term located in the center of a side, and go counterclockwise to the next center for the first term on the right side. Next, go to the adjacent counterclockwise corner, and then diagonally across the square for the final two variables that make up the last in the expression. As with the previous mnemonic device, if going against the diagonal arrow a negative sign is applied to the final term. An example of this, shown in Figure 4a, is to start with U for the left side of the equation, and proceed counter-clockwise to A for the first term of the right side of the equation. Next, continue counter-clockwise to the adjacent corner, T, which is multiplied by the term diagonally opposite, S. Since one went with the arrow between T and S, no negative term is applied. The overall equation is U = A + TS. A second example is to begin A for the left side of the equation. This will be equal to next clockwise center, G, and the next clockwise corner, P, which is multipled by the diagonal term V. The second term will be negative, thus yielding A = G – PV. By applying this process a total of four energy expressions are found.
136 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Figure 4. Using VAT-VUS to obtain the energy definitions (4a), the fundamental equations (4b), the partial derivative relations (4c), and the Maxwell equations (4d).
For the differential expressions of the energy (Figure 1b), start at any center energy term, and go to the opposite corners for the two differentials. Finally, travel from the differential terms across the square to the independent variables that are adjacent to the original energy term. If going against an arrow, apply a negative sign. An example of obtaining differential expressions with VAT-VUS, shown in Figure 4b, is to start with the center term U, as the left side of the equation. This will be equal to two terms. The first is the one opposite corner, T, multiplied by the derivative of the diagonally opposide variable, S. The second is the other opposite corner, P, multiplied by the derivative of the variable opposite it, V. The second term will have a negative since one travels agains the arrow connecting P and V. The overall expression is dU = TdS – PdV. A second example is to find the differential expression for H. Here one opposite corner is T, which multiplies the derivative of S. The other opposite corner is V, which multplies the deriviatve of P. Travelling with the arrows means both of these terms are positive. The resulting differential expression for H is dH = TdS + VdP. By repeating this process one can obtain a total of four energy expessions.
137 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
The last two sets of equations (Figure 1c & d), two arrows are drawn outside of the square, along the left and right sides. Each of the arrows points downwards. The partial differential relations can be found by starting at any of the energy terms in the center of the sides, and then going to either adjacent, independent variable for the partial derivative. Next, go to the other adjacent, independent variable, to indicate what is constant in the partial derivative expression. Finally, continue in the same direction travelled as going from the first to the second independent variable (clockwise or counterclockwise), and go to the next corner; the independent variable in the corner will be what the partial derivative expression is equal to. An example of the partial derivative relation, shown in Figure 4c, is starting with U and going down to V for the partial derivative expression ∂U/∂V. Then go to the other variable adjacent to the first one for the constant of the expression, S. This partial derivative is equal to the variable found by continuing in the same direction to the next corner, P. Since the transition from V to S was against the . arrow, a negative sign is applied. The overall expression is: For a second example, one can start at U again, but this time head to S first for the partial derivative ∂U/∂S. The constant for the expression will be V. By continuing in the same direction to the next corner, this partial derivative is equal to T. Here the transition from S to V was with the arrow, so there is no negative . Using this process, a total of sign. This time the final derivative is: eight partial derivative expressions can be found. The Maxwell Relations (Figure 1d) use only the corners of the VAT-VUS diagram, and the same arrow arrangement as for the partial derivative relations. Each Maxwell Relation comprises of two partial derivatives, and their constant, set equal to each other. As such, there are six variables employed in each of the four Maxwell Relations. When using VAT-VUS to find a Maxwell Relation, start in any corner, and travel either clockwise or counter clockwise. The original corner , and the next two corners encountered make up the first partial derivative: where the numbers 1,2, and 3 refer to the order encountered when travelling across each corner. For the second partial derivative, continue in the same direction to the last, unused corner, and then travel in the opposite direction for the remaining three independent variables (4, 5, and 6). These independent variables are placed in the partial derivative expression as: . Negative signs arise if one travels from 1-2-3, or 4-5-6 against an arrow. If both partial derivatives are negative, the negative signs will obviously cancel. An example of a Maxwell relation from the mnemonic device, shown in Figure 4d, is to start at V and going counter-clockwise head to the next variable T for the partial derivative expression, dV/dT. The constant for this partial derivative is found by going in the same direction to the next corner to obtain P. To start the next partial derivative, continue counter-clockwise to the next corner, 138 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
S. Now reverse directions for the partial derivative term, P. So this second partial derivate is ∂S/∂P. Continue counter-clockwise for the constant of T. In going from T to P, one goes against the arrow, yielding a negative sign. The overall expression is . A second example is to start again with V but go the clockwise to S to obtain ∂V/∂S. Continuing clockwise to get the constant P. Continue to the next corner for T, and reverse direction to obtain P. Finally, continue counter-clockwise for the constant of S. Since one went against the arrows in going from V to S and from T to P, the two negative signs will cancel. . A total of eight Maxwell Relations can This final expression is be obtained from repeating this procedure, but four are reciprocals, so there are only four unique Maxwell Relations.
Conclusion Mnemonic devices present an easily accessible way to remember the physical relationships used in thermodynamics. Three mnemonic devices were presented, each having positive and negative attributes. Instructors and students should practice using the mnemonic devices to see which they find most suitable. All of them make learning and remembering physical chemistry relationships easier.
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