NON-TABULAR RAPIDEVALUATION OF THERMODYNAMIC DERIVATIVES
June, 1952
TABLE I11
TABLE IV
x
x
Gas
Ns A Os Air
COLLISION DI.4METERS (U 10’ CW.)AT 0’ Chapinan and XenHandCowling‘ nard” Loeb’2 Strong15 book13
3.756 3.664 3.620
3.75 3.64 3.61 3.72
3.16 2.86 2.98
790
3.50 3.36 3.39
3.15 2.88 2.98
This
work
3.773 3.665 3.632 3.741
(14) E. H. Kennard, “Kinetic Theory of Gases,” McOraw-Hill Book Co., lnc., New York, N . Y.,1938, p. 140. (15) J. Strong, “Procedures in Experimental Physics,” PrenticeHall Publishers, Inc., New York, N . Y.,1938, p. 96.
Gas
N2 0 2
A
108 Ckr.
AT
0’
van der Wads
Gas i’sotherms
Molecular refraction
Present VIScosity
2.94 2.98 3.15
2.98 2.65 2.42
2.40 2.34 2.96
3.773 3,632 3,665
reported are higher than other values but it must be realized that the assumptions made in each particular method of calculation make an analytical comparison difficult.
A JACOBIAN METHOD FOR THE RAPID EVALUATION OF THERMODYNAMIC DERIVATIVES, WITHOUTTHE USE OF TABLES BY RICHSRD c. PINKERTON Conlribution No. 180 f r o m the Institute for Atomic Research and Department of Chemislry,‘ Iowa Stale College, Anzes, Iowa Received February 7, 1961
The method of Shaw, which makes use of Jacobians to calculate easily the thermodynamic derivatives, is presented in such a way that no tables are necessary. With it, any of the first partial derivatives may be set down rapidly from memory.
Many students of thermodynamics are familiar with the Bridgman method2 for finding the first partial derivatives of the thermodynamic functions, because it is the Bridgman table of Jacobians that appears in most textbooks. Fern have learned how to use the method given by S h ~ , usually 3 called the “Method of Jacobians.” This scheme has the advantage of being more general and hence more flexible than Bridgman’s system, although it requires learning a few simple properties of Jacobians. Using the Bridgman table, one obtains an expression in which T and P are always the independent variables. This feature is inherent in the structure of the table. In Shaw’s method, alternate expressions containing other pairs of variables are readily derived. However, in the method as it is usually presented, one must get accustomed to five extra symbols, which are used as abbreviations for the six Jacobians formed from the state variables, P, V , T and S. In the following outline, the additional symbols of Shaw will not be necessary. The table used in his system is usually reprinted in the form of a tenby-ten array. It can be condensed to a five-byfive form which is extremely easy to duplicate from memory. However, with the aid of a few simple rules, any of the Jacobians may be set down immediately without the aid of a table. Only a knowledge of the four fundamental equations is required. This gives ready access to some 720 first partial derivatives which arise from the ten thermodynamic variables used in the method, with many possibilities for alternate expressions. Those not already familiar with the Jacobian method will find it helpful to refer to the more (1) Work was performed in the Ames Laboratory of the Atomic Energy Commission. (2) P. W. Bridgman, “Condensed Collection of Thermodynamic I~’orniulas,”Harvard University Press, Cambridge, Mass., 1926. (3) A. N. Shaw, Phil. Trans. E o y . SOC.(London), 8234, 299 (1935).
complete tpeatments given b e l o ~ v . ~Otherwise the advantages of this method may not at first appear to be worth its consideration. The brief summary of the properties of Jacobians is given here so that anyone may take advantage of the device after a few practice calculations. The Jacobian is defined as
The independent variables 1‘ and s may be any desired, and for the purposes of a single problem are usually taken as implied. That is, one usually writes down only J(z,y). The following relations come from the definition and should be recognized on sight, after use J(zJ) = 0
(2) (3)
(4)
The last relation (4) is used to evaluate any desired derivative. Skaw proceeded by defining five new symbols for the basic Jacobians J(V,T), J(P,V) = J(T,S), J(P,S), J(P,T) and J(V,S). He then calculaled the Jacobians for 100 pairs of variables in terms of P, V , T, S and the five symbols listed above. These were collected in a ten-by-ten array which has been reprinted in the references ~ i t e d .The ~~~ (4) (a) 0. A. Hougen and IC. R.1. Watson, “Chemical Process Prinoiplea,” Part 11, John Wiley and Sons, Inc., New York, N . Y., 1947, PP. 465-471. (b) T. K. Sherwood and C. E, Reed, “Applied Mathematics iu Chemical Engineering,” McCraw-Hill Bodk Co., Inc., New York, N. Y., 1939, pp. 170-182. ( 0 ) H. Margenau and G. M . Murphy, “The Mathematics of Physics and Chemistry,” D. Van Nostrand Co., Inc., New York, N. S., 1043, pp. 17-24.
RICHARD C.PINICERTON
800
Vof. 6G
Jacobians listed are for the pairs of variables formed example will be given to assist in the formulation from the quantities P, V , T , S , E , H , A , F , &(rev) of the rule. and W(rev). Actually only 45 independent relaSuppose it is required to find J(H,A). First tions are present because of equations (2) and (3). recall that T and V occur with H , and that -S The purpose of the table was to give access to the and -P occur with A . Then mentally multiply more complex Jacobians which contain either one these two pairs together as though they were two energy variable and one state variable, or two binomials, only making sure to put down the energy variables. This made it unnecessary to variables associated with H first. That is, write recalculate them when using the system, but the down table is clumsy and not always available. -TS -T P -V S -V P With the aid of the following rules, any of the leaving a space. Then after each pair write down Jacobians may be set down in a matter of seconds. (I).-Those Jacobians containing P, V , T or S the Jacobian containing the conjugate pair, keeping are left untouched or are transformed by one of the the same order and leaving the sign intact. Thus, - T S is followed by J(S,T). Next, -TP is folfollowing relations lowed by J(S,V) etc. The complete expression is J(P,V) = J(T,S)
(5)
This relation arises from the cross-differentiation of any of the four fundamental equationss and is very useful. It can be remembered by noting that the intensive variables are paired with the extensive, and that the two intensive variables occupy the same relative positions (ie., if P appears first, T appears first). J(T,V) J(P,S)
J(H,A) = -2'8 J(S,T)
- TP J(S,V)
-
V S J(P,T)
-
c
I c
T'I'J(P,1')
Now suppose one wants to evaluate (bFflbS),\. The expression just obtained is used, along Jvith J(S,A) = - J(A,S) = S J(T,S) + P J( T',S)
+ J(P,T) J(V,S) +
J(V,P) J(T,S) = 0 ( 6 )
This relation is sometimes conveniently used to eliminate any one of the simple Jacobians, but is not necessary. It is equivalent to the definition (l).s (11).-Those Jacobians containing one each of an energy variable (E, H , F , A , Q(rev) or W(rev)) and a state variable (P,V , T or 8 ) : One needs to remember the two undifferentiated state variables occurring with each energy differential in the fundamental equations. These two are set down, and each is followed by a Jacobian containing first, the conjugate of the state variable in that term (that is, the variable appearing as a differential) and second, the state variable in the desired Jacobian. For example, to find J(A,S): from the relation dA = -SdT - P d V write down -SJ(T, - PJ)V, and then add S to obtain J(A,S) = - SJ(T,S) - PJ(V,S). (111).-Those Jacobians containing two energy variables: these are the most complex. As a rule, they contain from three to four terms, and each term consists of two state variables and one Jacobian. Because it is irksome to derive them when needed, Shaw felt that a table was desirable. An
-
.
(5) The four fundamental equations of thermodynamics referred t o TdS VdP, d F = - S d T 4here are d E = T d S - P d V , dH V d P a n d d A = - S d T - PdV. (6) T o p u t down (6) easily from memory, write any one of 'the variables (in this case S) in the same relative position in each of the three terms (here the last position). Then permute the remaining three among the rest of the positions 80 that each appears b u t once in any term and none occupies the same relative position in the three terms.
+
Dividing through by J(S,T) or - J(P,V) results in
Then
This rather complicated derivative, which has been derived here in a short space, would take some time to obtain accurately by ordinary megns. It is true that derivatives of such complexity are seldom required. But it is also true that many refrain from adopting short cuts and learning easier methods because they are often of limited utility. It is hoped that this simplification will encourage more students and teachers of thermodynamics t o adopt the Shalv system. To anyone wishing a proof or justification of the above rules, it is suggested that a condensation of the original table be attempted.
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