OF THE THERMODYNAMIC FUNCTIONS BENJAMIN CARROLL and ALEXANDER LEHRMAN Newark Colleges, Rutgers University, N. J. College of the City of New York
IN
SEVERAL publications in THIS JOURNAL there is an emphasis on exact mathematical treatment of thermodynamics.' Recently it has been shown how the Maxwell relations may be presented in the ~lassroom.~It is our purpose to demonstrate that these Maxwell relations (equations l b to 4b), along with equations ( l a to 4a) from which they are easily ~ b t a i n e dare , ~ sufficient to derive a vast number of the thermodynamic formulas. The present methods used to obtain relations between the derivatives of the thermodynamic functions are unsatisfactory. The fault lies in the lack of a general systematic method. Textbooks derive these relations in various ways, but there is no indication of how the author was led to choose each step or operation in the derivation. To the student each relationship is obtained in an arbitrary manner. He sees the.desired result and that each step is mathematically correct, but unless he memorizes each particular derivation that happens to be used, he cannot repeat the steps. Also if a relationship between derivatives is needed but not given in the text, he is, in many cases, unable to obtain it for lack of a general and systematic method. The student may have recourse to Bridgman's tables.' Here 90 quantities of the type (bX), are defined, where X and Y represent any of the ten thermodynamic functions, so that the quotients of the pairs of them form all possible h s t partial derivatives. Thus the partial derivative (bX/bZ), nyly be denoted a s ( ( Z ) . By imposing the further restriction that (ax), = - (b Y ) , Bridgman was able to reduce the number of required quantities to 45. Nonetheless these quantities are arbitrarily defined and to the student the use of the tables appears quite artificial. Lennan presented a method' based on quantities similar to those devised by Bridgman but redefined in such a manner as to permit the resolution of the 45 forms into fewer, more basic forms. In this method the quan-
tity d X , is the "vired" of X with Y constant, whereX and Yare any two of the ten thermodynamic functions, and is defined by the relation
.
See for example, 5. E. WOOD,J. CHEMEDUC.,20, 80 (1943). a COPFIN,C. c.,&id., 23, 584 (1946). 8 See for example F. Darum~s,"Mathematical Preparation for Physical Chemistry," McGraw-Hill Book Company, Inc., New York, 1928, p. 193. ' BRWGMAN, P. W., Phys. Rw. (2) 3, 273 (1914); also see "A Condensed Collection of Thermodynamic Formulas," Harvard University Press, Cambridge, Mass., 1925, and G. N. LEWIS AND M. Rlwnnm, "Thermodynamics," McGraw-Hill Book Company, Inc., New York, 1923, pp. 163-5. LERMAN, F., J. Chem. Phys., 5, 792 (1937). 389
However, in using Lerman's method, one must acquire the operating rules for vireds, and although these can be derived from the equation just given, they appear as a new and arbitrary set of mathematical operations. Were it not for the method subsequently presented, we would perhaps consider Lerman's method as the one most suitable for student use. The more general method of obtaining thermodynamic relations by the method of Jacobians as described by ShawEhas many desirable features, but the understanding of functional determinants places this method outside the realm of an undergraduate course in physical chemistry or thermodynamics. The recommended method described in this article is, with some minor modifications, the method devised by Tobolsky7 which appears to have been overlooked by textbook writers. As given here the method is broken down into its simplest steps and is applicable to the derivatives for a one-phase system of constant mass. There are used the four fundamental equations following and the four Maxwell rehtions readily obtained from them. The student will find it convenient to add equations (5) and (6) to the set of equations to be kept in front of him during the derivation of a formula. TABLE 1
. d~
=
T ~ S P ~ V(la)
( )
dB
=
Tds
( b ~ )= ~
+
VdP (2a)
dA = -SdT - PdV =
(3a)
-SdT + VdP (4a)
as (-1 a~
= -
=
as
a~ -
)
=
) )
(~b)
rz)
P
aT v
- )a~
P
(2b)
(3b) (4b) (5)
as 6
(6)
Smw, A. N., Trans. Rcy. SOC.London, A234,299 (1935).
' TOBOLSKY, A,, J. Chem. Phys., 10, 644 (1942).
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JOURNAL OF CHEMICAL EDUCATION
THE METHOD
The procedure consists of five simple steps as explained and illustrated below. Consider the problem of while those of dP yield evaluating the Joule-Thomson coefficient, ( b T / b P ) , with T and P chosen as the independent variables. Step I . Set up the differential equation containing the variables appearing in the partial derivative. For It follows that the case of (bT/bP), the differential equation is as follows: dT
=
XdP
+ YdH
(7)
which is the solution to the problem. Occasionally the equation for X does not contain the term Y and thus the value for X can be read directly without solving the simultaneous equatio?~. To illustrate the directness and simplicity of this method let us take the problem of evaluating (bE/bV), in terms of the independent variables T and V . This is a widely used thermodynamic relation. Following the procedure described above we obtain the following:
so that the sought quantity is X, since it can be seen that
In general to evaluate the partial @B/bC),, theequation to be set up would be dB
=
XdC
+ YdD
Step 2. If dE, dH, dA, or dF appear in the differential equation in step 1 and further if A, E, H, and/or F is not one of the selected independent variables, the differential should be replaced by its value as given in equations ( l a to 4a). Thus in equation (7) substitute for dH its value in equation (2a). Hence dT
=
XdP
dy
=
(g)"dz + E) ,?d
dE = XdV + YdT
Step 8.
TdS
Step
+ Y ( T a + VdP).
Step 3. Express all differentials in terms of the two selected independent variables by using the fundamental equation for a total differential
Step
.
s.
- PdV
T($)"~T
=
CdT
+ T(&)"
Step 6.
X
T (aP~ ) " P
+ Y [T (%Jp
GIT
+Y
aE
= (m)T
($IT
(9)
Since PV
=
= .O. = R -
V
7
.
.
. ..
d~
+ T (%)= dp + ~
d p ](10)
Applied to a van der Waals gas, equation (13) yields .. -. a
Step 4. It usually is desirable to replace partial differentials involving the entropy with their equivalent quantities in the Maxwell relations (lb to 4b) and in equations (5) and (6). Thus in equation (10) we replace (bS/bP). by - (bV/bT), [from equation (4b)land (bS/bT), by CJT [from equation (611, obtaining dT
= XdP
(13)
RT, it will be seen that
("1a
+ Y [c,~T- T($)
dP
+V ~ P ]
(av"> = =- -vz
since
rand
(p++)(Vpb)=RT
Substituting in (13),
(11)
Step 5. To find the value of X equate the coefficients As of the differentials of each of the independent variables. This gives rise to two simultaneous equations containing the two unknowns, X and Y, and leads directly to . lt follows that the value of X. For the case in hand [equation ( l l ) ]the coefficients of dT yield
p = - - -RT V - b
("Ia
a Vz
a = -
V2
( ab pT ) v = m R
~ T
~ T
C
It then follows that XdP
- P ~ V= XdV
dV
Substituting in (13)
=
=
Applied to an ideal gas this yields
where u, x, and y represent any of the thermodynamic functions. In this case, as P and Tare the independent variables. we have
dT
X
+ T ( % ) = ~ v - P ~ V= X ~ V+ Y
Step 4.
=
so that
+ YdT
XdV
AUGUST, 1947 RELATION BETWEEN PARTIAL DERIVATIVES
Substituting in equation (16) The problem of h d m g the relation between partial RaT derivatives is clearly the problem of solving each partial V' RT = R C, -C.= ---in terms of two independent variables which are usually -Va the same for the partials and then by simple algebraic manipulation arriving at the desired expression. This Because of the general relationship of the partial de- . procedure isessentially that employed when one uses the rivatives tables of Bridgman. Thus to find a value for (bS/bP),(bT/i3~S')),when T (17) and P are the independent variables, first find the value of (bS/bP), by the' method recommended, obtaining
v
-
(14)
T
ap
(18)
where X, y, z, and u may be any of the ten thermodynamic functions, final equations may be transformed and sometimes simplified. For example, with the aid of equation (17) it can be seen that (15)
and then evaluate (bT/bS), where again T and P are the independent variables, obtaining T
Multiplying (14) by (16) we obtain
(19) Using this equation in equation (16) we get
An intekesting case is that of finding an expression for C , - C , in terms of T and V . First 6nd the value of C , in terms of T and V. Since C, = T(bS/bT), we have to 6nd the value of T(bS/bT), in terms of T and V. Following the method, we have
+ YdP
l'dS = XdT
where X is the desired quantity. T($)"
dT
+ T ( g ) = dV = XdT
C.dT
+ T(%)"
d~
=
XdT
+
dT
Y[(%)v
+ Y[(&)"
dT
.
"= x +
aP Y(@)"
+(g)T
+ (g)? dv]
(W As for the general thermodynamic problem of 6nding one partial derivative in terms of three others, we take the following problem from Margenau and Murphy, "The Mathematics of Physics and Cherni~try."~ These authors use the method of Jacobians to obtain a solution identical with the one given here. Develop (bP/bT),as a function of C,, C,, and (bT/bP),. .This can be solved by finding expressiqns for the partials'in terms of T and V. The previous value for (bT/bP), as a function of T and P [see equation (12)] can be changed by means of equation (17)to yield
Z
or
($),c. + V = 7 X=C,---
C, - C,
=
(21)
Taking -the value for heat capacities appearing in equation (16),
C,
TG):
C,
= - ----
- C, = - T-
(16)
and dividing this by equation (21) it follows that 1t happens to he unneccessary to solve separately for
c,.
Applied to an ideal gas, PV = RT,equation (16)leads to the well-known expression C, - C, = R since ap aT v
(-)
R =
ap
and
(av-
=
-
RT V2
(22) MAEGENAU, H., AND G. M. MWPHY, "The Mathematics of Physics and Chemistry," D. Van &strand Company, h e . , New York, 1943, p. 23.
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JOURNAL OF CHEMICAL EDUCATION
The method presented here has the property of being applicable to second partial derivative^.^ It i s readily applied to systems where the m d ~ a n i c a lwork Pd?' is other forms of work. Further One may apply the method to systems with more than two using the general es~ressionof the pendent variables, first and second laws, dE = TdS - &F,dei ~
where the parameter r, is associated with the generalized force F, and the work in the change ds, is F&,.
CONCLUSION
will be seen that the method recommended is general and makes possible the straightforward derivation of the relations between derivatives of thermodynamic functions without unnecessa~lyburdening the ~ t manv h nrocedures or without the necessitv of haviw an extensive table of eauations. Each steo by itself-uses a mathematical operation with which students of physical chemistry are familiar and the method directs the student along a simple path to the desired relationship.
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