A Method for Relating
Reino W. Hakala'
Howard
University Washington, D. C.
Thermodynamic First Derivatives
For homogeneous systems of constant composition in which the only variable potential factors of energy are the temperature and the pressure, the most convenient way to relate thermodynamic first order partial derivatives to C, (bV/bT),, (bV/bP),, P. V, T, and S is by the use of Bridgman's table.? When it is desired to relate a given derivative to derivatives other than Cp, (bV/bT)'p, and (bV/bP),, Shaw's tablea provides the best available approach. It should be noted, however, that Shaw's table is not actually necessary when Bridgman's table is available, for, as was pointed out by Bridgman, the relationship among any four derivatives, in general, each of them expressed in terms of C,, (bV/bT)p, and (bV/bP)T, can be obtained by solving these expressions simultaneously so as to eliminate Cp, (bV/bT)p, and (bV/bP)T. Presented before the Division of Chemical Education at the 144th National ACS Meeting, April, 1963 by Dr. 0.T. Benfey on behalf of the author, who is very grateful to Dr. Benfey far this courtesy. ' Present address: Syracuse University, Syracuse, N. Y. BRIDGMAN, P. W.. Phys. Rev., 3,273(1914). a SHAW, A. N., Phil. Trans. Roy. Sac. London, A234,299 (1935). ' BENT,H.A., J . C h m . Phys., 21,1408(1953). 8 P ~R. C.,~J . Phys. ~ C h~m . , 56, ~ 799 (1952). ~ ~ ~ ,
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B~idgman'stable is thus equivalent to Shaw's. When neither of these tables is a t hand, two other completely general procedures are available, in addition to the use of first order partial derivative transformation formulas. One is a systematic algebraic method4which can be used to generate the same results as are directly given by Bridgman's table. The o t l ~ e r , ~ like Shaw's method, uses Jacohians and can be ueed to generate Shaw's table. The present author has, however, found another general method to be more convenient. This will now be presented. (Other methods which have been published are either of mnemonic value only or are of limited application and so shall not be considered further.) The method which is to be presented uses theorems on Jacohians, so we shall begin with a brief review of the definit,ion and relevant properties of Jacobians and their relationship to the fundamentals of thermodyn a m i c ~ . ~These properties and relationships are sufficiently simple in form to be readily memorized. A Jacobian of two dependent (x, y) and two independent (u, v ) variables, usually denoted by either J ( z , y) or b(x, y)/b(u, v ) , is defined as the determinant
The symmetry between the shorthand notation and the form of the determinant should be carefully noted, for this symmetry aids in memorizing the definition. It should also be noted that the shorthand symbol J(x,, y) does not explictly indicate the independent variables. Among the properties of a Jacobian of two independent variables which can be deduced from its definition are:
therefore contains eight members-the state variables P, V , T, S, and the energy variables E, H, A, and F, there are a total of
different Jacobians, or 28 diierent pairs of Jacobians, to consider, not taking into account the independent variables of the Jacobians. To be able to handle so many Jacobians with facility, we shall classify these Jacobians into the following four sets:
Tertiary
I n the above relationships it is understood that the order in which the independent variables occur is the same for all of the J's. Employing all but the last of the above properties, the fundamental equations of thermodynamics dE dH dA dF
= = = =
TdS - PdV TdS VdP -SdT - PdV -SdT VdP
+ +
can be rewritten in the Jacobian notation,
where y represents any thermodynamic variables. These equations are readily generated at will from the easily remembered mnemonic pattern
I n Jacobian notation, Maxwell's four relations all take the single, simple form J(P, V ) = J(T, S )
which is readily memorized as it involves natural pairs of intensive and extensive thermodynamic variables which occur in the same order (intensive, extensive) in both terms. Bridgman and Shaw included heat and reversible expansion work in the set of fundamental thermodynamic variables, but this is unnecessary since
aQ,,
=
.,
T ~ SawpV,,, , = P ~ V
As our set of fundamental thermodynamic variables 6See also the following textbooks: MARGENAU, H., AND MURPHY, G. M., "The Mathematics of Physics and Chemistry,"
...
D. Van Nostrand Co.., Ino.., New York. 1943. DD. 17-24: REID. C. E., "Principles of Chemical ~heirnodynamics," ~einhold 0. A., Publishing C o p , New York, 1960, pp. 249-259; HOUGEN, and WATSON, K. M., "Chemical Process Principles," Part 11, John Wiley and Sons, Inc., New York, 1947, pp. 465-71; SHERT. K., AND REED, C. E., "Applied Mathematics in ChemiWOOD, cal Engineering," McGraw-Hill Book Co., Inc., New York, 1939, S., "Introduction to Theoretical Physical pp. 17Q-82; GOLDEN, Chemistry," Addison-Wesley Publishing Co., Inc., Reading, Mass., 1961, pp. 31-3.
100 / Journol of Chemicol Educofion
"
x, y ='E,'H,'A;
This classification is simple to remember because it is systematic: The primary (lo) set contains no energy variables; the secondary (2') set contains both state variables and energy variables, the energy variables being listed last; the inverted secondary (2-a) set inverts the order of the two k i d s of variables; and the tertiary (3') set contains only energy variables. Thus the importance of the energy variables increases in the order l o< 2" < 2-0 < 3'. We are now ready to apply these easily memorized definitions and properties to the derivation of equivalent expressions for thermodynamic first order partial derivatives. Any given thermodynamic first order derivative can be found in terms of primary .Jacobians by means of the following four steps: 1. Express the given partial derivative as a ratio of two Jacobians. This is done by utilizing the property
2. Find any 3" Jacobians in terms of 2O Jacobians. This is accomplished by a direct application of the appropriate fundamental equations of thermodynamics in Jacobian notation. 3. Invert any 2" Jacobians. Use is made in this step of the property
4. Find any 2-a Jacobians in terms of l oJacobians. This is achieved by means of the fundamental equations of thermodynamics in Jacobian form. If algebraic simplification is possible subsequent to any of the above steps, it should be carried out before proceeding to the next step. Whenever applicable, the null theorem J(z, z )
=
0
is to be used. The result of the above transformation process is a general solution which holds for any pair of independent variables. The next step is to assume a given pair of independent variables, and then to find all of the primary Jacobians in terms of first order partial derivatives having these independent variables. These relationships are found by use of the property
which is a combination of the properties
The result is a particular solution for the chosen pair of independent variables. By way of illustration, we shall find ( b E / d V ) , in terms of the independent variables T and P, and also T and V . First, we shall find the general solution, following the four steps given above. The numbers a t the right below refer to these four steps.
considering the complexity of the result. It is remarkable that there is very little to commit to memory in this method. The above procedure is of course not limited to T and P a s the independent variables. Any of the other 27 possible pairs could have been taken instead. By way of example, we shall next find the particular solution for ( b E / b V ) , in terms of the independent variables T and V. The derivation follows: ~ ( pS ,)
b?&!(?T) , V ) = - (g) dT v (Cs) bV
("1
a(s V ) = J(s,V ) -A b(T, V )
- T J(S,) -p
(slgebraic simplification)
J ( V ,H ) =
J(H, S ) - p J(H, V )
=
-
(??) bV (%) aT v T
Cv [mT] dT T =
-
v
Therefore,
(3)
++
T J (S, S ) V J(P, S ) T(4) T J(S, V ) V J(P, V ) TV J(P, S ) - P (using the null theorem) T J(S, V ) V J(P, V ) =
+
This is the general solution. We shall now find the particular solution for the independent variables T and P. Thus, letting (u, y) = ( T , P), we obtain
-w)
J ( p , s)
aT v
T
=
XT, P )
- (5)= aT
P
CP ( V ) =
r w
[-1
- (CITPT
aT
=
- FCF
P
m)(mIp
J ( P V ) bV T
+
The partial derivative having the most complicated expression for it, with the independent variables T and P, is (bS/bH),. The reader can test his skill by verifying the following expression:
Substituting these partial-derivative expressions found for the primary Jacobians in the general solution, the particular solution for the independent variables T and P is found to be as follows:
The same expression is found using Bridgman's table and subsequently simplifying algebraically. Although this expression is somewhat complicated, the derivation was entirely straightfoward and relatively short,
Despite its great complexity, no more effort is required to find this expression than in the previous examples. Our last mle is a generalization of Bridgman's: T o find a particular derivative in terms of three others which involve altogether more than two independent variables, Jind expressions for each of these four derivatives in terms of the same (no matter which) pair of independent variables, following the above rules, and solve these four equations simultaneously to eliminate the unwanted derivatives. Employing this rule, the reader may wish to show that
In an alternative procedure due to S h a ~the , ~primary Jacobians are eliminated instead. Sometimes, as in the above example, the application of Shaw's procedure requires knowledge of one more property of Jacobians. For this property and the application of Shaw's method to the above example, the reader is referred to Reid's "Principles of Chemical Thermodynamics," pp. 258 and 259.8
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