1, 2
i, j , k r
R
(9) Pitzer, K. S., Hultgren, G. O., Zbid., 80, 4793 (1958). (10) Prausnitz. J. M.. A.I.Ch.E. J. 5.3 - , -11959). - , (11) Prausnitz: J. M.: BensonTP. R., Zbtd., 5, 161 (1959). (12) Prausnitz, J. M., Gunn, R. D., Zbzd., 4, 430 (1958). (13) Reamer, H. H., Olds, R. H.,. Sage, B. H., Lacev, ,. W. N.. Ind. Eng. C h m . 36,88 (1944). (14) Reamer, H. H., Sage, B. H., Lacey, W. N., Zbid., 43, 2515 (1951). Redlich, O., Kister, A. T., Zbid., 40, 345 (1948). Redlich, O.,Kwong, J. N. S., Chem. Revs. 44,233 (1949). Robinson, R. L., Jr., Jacob R. H., Hydrocarbon Process. Petrol. Rejiner 44,No. 4,141 (19657: (18) Rowlinson, J. S., “Liquids and Liquid Mixtures,” pp. 111-2, Academic Press. New York. 1959. (19) Sage, B. H.,’ Budenholzer, R. A., Lacey, W. N., Znd. Eng. Chem. 32,1262 (1940). (20) Sage, B. H., Lacey, W. N., Zbid., 31,1497 (1939). (21) Sa e, B. H., Lacey, W. N., Hicks, B. L., Zbid., 32, 1085
component number one, two component number i, j , k = refers to rth component = reduced property = =
\ -
-_
literature Cited
(1) Chao, K.C., Seader, J. D., A.Z.Ch.E. J . 7,598 (1961). (2) Chueh, P. L.,Muirbrook, N. K., Prausnitz, J. M., Zbid., 11, 1097 (1965). (3) ~, Haselden, G. G., Newitt, D. M., Shah, S. M., Proc. Roy. Sac. 209A, 1 (1951). f4) Lewis. G. N.. Randall. M.. “Thermodvnamics.” revised bv ‘ K.S. Pitzer and L. Brewer, 2nd ed., Appendix I, McGraw-Hili, New York, 1961. (5) Mazzei, L. D.,Jr., “Fugacity Coefficients of Gases in Binary Mixtures,” M.S. thesis, Newark College of Engineering, Newark, N. J., 1961. ( 1 W.”,. IT (6) Myers, A. L., Prausnitz, J. M.. IND.ENG. CHEM.FUNDAMEN- (22),Sage, B. H., Lacey, W. N., Reamer, H. H., Olds, R. H., TALS 4.209 f 19651. Zbzd., 34, 1108 (1942). (7) Olds: R. H., Reamer, H. H., Sage, B. H., Lacey, W. N., (23) Wilson, G.M., Advan. Cryog. Eng. 9,168 (1964). Znd. Eng. Chem. 41, 475 (1949). (8) Pitzer, K. S., Curl, R. F., Jr., J . Am. Chem. Sac. 79, 2369 RECEIVED for review January 14, 1966 (1957). ACCEPTEDMay 17, 1966 ’
\-,
T H E SECOND LAW ACCORDING T O KELVIN STANLEYKATZ Dejartment of Chemical Engineering, T h e City College, City University of New York, New York, N . Y.
A recent axiomatic analysis by Arens has put the classical approach to the foundations of thermodynamics, via the second law formulations of Kelvin and Planck, on as sound a mathematical footing as the approach of Caratheodory based on adiabatic inaccessibility. The present paper gives an engineering exposition of the central portion of Arens’s work, with particular attention to the question, often overlooked, of the existence of adiabatic surfaces in complex systems.
HE classical approaches to the foundations of thermoTdynamics, via the second law formulations of Kelvin and Planck, have long been criticized for lack of rigor. And indeed, a modern mathematical treatment of the subject, such as that of Landsberg (4), may be expected to follow the economical elegant axiomatics of Caratheodory (2, 3 ) . A great deal in the way of clarity and physical insight is of course gained from Caratheodory’s analysis. But something is lost on the way, and that is the primary role of temperature in the analysis of heat. Now a recent axiomatic analysis by Arens (7) has put the classical development of the second law on as secure a foundation as one could want, and we propose here to restate the central portion of his argument in somewhat less severely mathematical terms. Arens’s analysis leads to the thermodynamic identities which make up the formal core of the subject, but it works entirely with reversible processes, and so has nothing to say about the inequalities which are such an important part of the applications of thermodynamics. Some comment may be in order on what seems to us to be the central mathematical difficulty in establishing the notions of entropy and absolute temperature from the classical formulations of the second law. This is the question of the existence of adiabatic surfaces. I t does not arise for a simple fluid, where two variables fix the thermodynamic state, and the differential expression representing the reversible heat transfer appears in the form
(Such expressions are, technically, linear differential forms, 462
I&EC F U N D A M E N T A L S
and we here distinguish the symbols representing them with a tilde.) For a differential form in two variables, such as this , the differential equation
(r=O is simply an ordinary differential equation, and so has in general a family of solutions q ( x , y ) = constant
which are the null surfaces (here, curves) of &ht-at is, the adiabatics of the thermodynamic system in question. T h e situation is quite different when more than two variables are involved. Here the corresponding differential equation for the null surfaces f(x,y,z)dx
+ g(x,y,z)dy + h(x,y,z)dz
= 0
is not an ordinary differential equation, and it will have the desired family of solutions q ( x , y , z ) = constant
only if certain compatibility conditions among the coefficientsf, g , and h are satisfied. Indeed, the harmless-looking differential equation ydx f dy
+ dz = 0
has no such solutions. Unless these compatibility conditions are satisfied for the heat form, there will be no adiabatic surfaces. A criJcia1 point of Arens’ argument is the demonstration that no matter how many variables are involved initially, the
heat form can always be reduced to an expression in just two variables, so that adiabatic surfaces will always exist. I t will appear further that one of these two variables will be the empirical temperature, so that a natural analytical machinery will exist for defining the absolute thermodynamic temperature. The order of presentation here is second law before first, but this order is adopted only to bring us more quickly to the central ideas of entropy and absolute temperature. The Heat Form and the Second l a w
We consider here closed systems (bodies) in various equilibrium states, characterized by the values of certain thermodynamic variables. Alj one of these variables, we may always take an empirical temperature, t , so that we have always families of isothermal surfaces
= fdt.
which
;* O
i
Y
= ydx
+ djj + dz
has passing through each point, x = a, y = b, z = c, a family of null arcs (indexed on p;i
12
= c
(11
4
that is, the coefficients in the differential form d o not all vanish simultaneously. With these formal axioms out of the way, we introduce the second law in the form:
A3.
fi
=
0 along every isothermal arc whose ends can be connected by adiabatic arcs.
This is simply a compact way of making the classical statement that no heat can be transferred to (and hence no work done by) a body executing a semi-carnot cycle made up of adiabatic pieces and a single isothermal piece. We consider a point in the state space (no change in the body at all) as representing an adiabatic process, and it accordingly follows from A3 that
$i
t = constant
spanning the thermodynamic state space. Thermal equilibrium between two bodies is taken, in the usual way, to reside in the equality of their empirical temperatures. With each body, we associate a linear differential form, i, in its state variables, representing the differential increment of heat transferred to the body under reversible differential changes of the specified amounts in the appropriate state variables. We adopt further the usual understanding that when two bodies at the same temperature are lumped together as one, they form a new ihermodynamic system at this common temperature (zeroth law), whose heat form is the sum of the heat forms of the component systems. i\ny arc in the state space may be regarded as a finite reversible process, and we may assess the heat transferred to the body under such a process by f along the arc. If f 4 = 0 along every portion of some given arc, then this arc is a (reversible) adiabatic and represents a (reversible) adiabatic process. Every differential form generates such null arcs, whether it generates null surfaces or not. Thus the form noted earlier
As a side matter, it follows that
=
0 around every isothermal loop.
(2)
Existence of Adiabatic Surfaces
We turn now to a purely mathematical argument establishing the existence of adiabatic surfaces. We note first from Equation 2 that is exact on every isothermal surface, and so may be written
4
= udt
+ dv
(3) where u and u are functions of thermodynamic state, the empirical temperature, t , being taken as one of the state variables. We note next that u cannot be a function o f t alone since, if it were, would be an exact differential = d [ f'udt
+
u]
in contradiction of A l . We may thus take u as another state variable, along with t, and in Equation 3 regard u as a function oft, u, and the remaining state variables. We now proceed to show that u is actually a function o f t and u only, no matter how many other state variables there may be. For if it were not, it could be taken as another state variable, along with t and u, and a cycle constructed to violate A3. This construction is very straightforward (see Figure 1). We set up the adiabatic arcs a and b,
-t (1 - p)X
r
along every piece of which f = 0, but these arcs cannot be pieced together to form a null surface, q ( x , y , t) = constant, on which = 0. We note further that as a topological matter, we take the state space of a body to be simply connected (no holes), so that Stokes' theorem appliec; in its simplest form, and we can argue freely from the exactness of a differential form to the vanishing of integrals of this form around closed loops, and vice versa. \Yith these introductory matters in hand, we introduce now two formal axioms in the heat form, 4:
1
AI. The heat form, i , is not exact. A2. T h e heat form, i , is not proportional to dt. The first of these axioms is, of course, familiar. The second says, geometrically, thai: adiabatic surfaces (once their existence has been established) will not be tangent to isothermals. In algebraic terms, it asserts that there is no function f of state for
I
I I I I I
I I
I
I I
I
Figure 1 .
I
c
a,
u o
44
Construction of a cycle violating the second law VOL. 5
NO. 4 N O V E M B E R 1 9 6 6
463
I I
t =
a.
u =
b.
tl
u = uo u1
-
+
(tl
- to)X
(tl
- to)uoX
with u1 = uo
t = to u = uo u = uo
+ +
1
(r
f ( t , P)d* (6) Now f must really depend on t as well as P, since otherwise would be exact, in contradiction of Al-that is, bf/bt may not vanish. Further, by Equation 1, f may not vanish either. (This is in effect another application of A2.) We conclude, the usual smoothness assumptions being implicit, that in Equation 6 neither f nor bf/bt changes sign. To come to the absolute temperature, we lump together two bodies in thermal equilibrium at the common empirical temperature, t . For the first, we have the heat form
O < X < l
- to)X - uo)X - ( t l - to) [UOX
-
(tl
- to)uo
(tl
( ~ 1
+
O < h < l 1/z(u1
- UO)X2]
connect them with the isothermal arc c,
G.
i
t =
tl
u = u =
u1 Ul
-
+
(Ul (0,
- U0)X - v1)X
with u? = uo
1
0
5x5
1
*o
in contradiction of the second law axiom, A3. I t accordingly follows that in Equation 3, u is a function of t and u only, no matter how many other state variables there may be. The heat form, may accordingly be written
;=
Q1
+
;2
u)dt
+
P(t, u ) = constant
at
c;,
the vector (dt, du) can be orthogonal to the vector (f,g) only
follows that the vectors (f,p ) and which is to say that
E)..
It
(E,2)
- - are proportional,
4 is proportional to dP-that
4 = $4,u)d'k(t, u )
is,
+ f d t , *2)d*?
(7)
= f ( t , *)dP
(8)
for a suitable Comparing Equations 7 and 8, we see that 9 is a function of and *Z with
i'(
1 t , P l ) = f ( t , *) . b*/bPl f d t , *2) = f ( t , *) * bP/d*?
from which it follows that
Now, apart from dependence on t , the first member of Equation 9 depends only on P I , and the last member only on \E2. They must accordingly both be equal to the same quantity independent of PI and Pz-that is, they must both be equal to the same function of t . And since the two bodies under consideration might have been any two, it follows that there is a universal function, k ( t ) , such that for any body with a heat form given by Equation 6,
3
= k(t)
And since neither f nor bf/bt changes sign, it follows that k ( t ) must be of one sign. Now Equation 10, considered as a differential equation in f, can be integrated to give
f(t,
=
. exp iS k(e)d@/
and we may define the absolute temperature, T , as
Entropy and Absolute Temperature
With the heat form given by Equation 5 in terms of two state variables, one of them the empirical temperature, we can readily establish the existence of entropy and absolute temperature by the usual Carnot cycle construction. I t may be of interest, however, to proceed in a more formal way, putting in evidence the special role of the axiom (A2), which has not yet been used. We note first that in Equation 5, may or may not depend on u alone, but it certainly cannot depend on t alone, since this would violate A2. We may accordingly choose t and P as the convenient state variables in which to express (r, and rewrite Equation 5 as
*
l&EC FUNDAMENTALS
(r
(5)
In Equation 5, the function p appears as an integrating denominator for the heat form i, and will of course be closely related to the absolute temperature. The adiabatic surfaces are defined by Equation 4, and accordingly appear as cylinderlike surfaces in the thermodynamic state space whose projections on the t, u plane are just the curves of Equation 4.
464
f z ( t , Pz)dPz
(9)
+ b*du dU
if it is also orthogonal to the vector
=
But the composite body being itself a thermodynamic system at temperature t (zeroth law), it follows that
(4)
Since the vanishing of the heat form, (r, is equivalent to the vanishing of
bP
42
*.
g ( t , u)du
so that the differential equation (r = 0 defining the adiabatic surfaces is simply an ordinary differential equation in t and u which may be solved in the form
d P = - dt
f i ( t , Pi)dPi
= fl0, *l)dPl
4,
= f(t,
=
where P I and 'k?are appropriate state variables for the two bodies. For the composite body,
and simply verify that along c
i
(ri
and for the second
- ' / ? ( t l - to)(ul + uo)
" I =; u?-u1
=
D1.
T
=
exp{JL k(e)dO]
and the entropy, S, by D2.
dS = g ( P ) d P
so that
Since k is of one sign, T varies strictly monotonically with t , and so really defines a new temperature scale. Since k is a universal function of t , the same for any body, T is properly an absolute temperature. By DI, T is positive, and the
absence of the lower liimit in the integral there reflects the customary freedom in expanding or contracting the scale of T. There is of course no freedom in setting the origin of the scale of T. The First l a w and Woriking Formulas
With the analysis of the heat form in hand, the development of the working parts of thermodynamics offers no surprises. With each body, we associate a linear differential form, representing the differ(entia1 increment of work done by the body under reversible changes of the specified amounts in the appropriate state variables. As for the heat form, we understand that the work done by a body under a finite reversible along the corresponding arc, process may be assessed as and that when two bodies are lumped together as one, their work forms add. We may axiomatize the first law as
and apply Equation 14 to give the Gibbs-Duhem equation. T o get a t the thermodynamic relationships describing chemical equilibrium, we recognize that the general differential expression (Equation 14) for the energy may be taken to apply to closed systems, subject to suitable restrictions on the N,. We note first on comparing Equation 14 with Equation 13 that
w,
fw
A4.
,uadNa = 0 for bodies, a
but this identity may not be taken, in the manner of our analysis of the heat form, as implying that the pa all vanish, since the A’, are not allowed independent variation. Indeed, for nonreacting systems, the dNa all vanish. If we allow chemical reaction (including phase change), we remove some of the restrictions on the N,. We choose an independent set of reactions, and write them in the form
va,ca= 0 ;
4 - & is exact
w
D3. dU
= 1, 2,
Y
.. .
a
from which, together with A l , it follows that is of course not exact. With A4, we may define the internal energy, U, by
- - w-
(16)
where Ca is the chemical symbol for the a t h substance, and the ua, are stoichiometric coefficients. Introducing the progress variables E,
= q
dN, = va,dE, for the rth reaction
Now, to apply Arens’ development to the analysis of chemical systems, we must say something about the form of &. We write
(1 5 )
(17)
we see that we may account for the total change in N , by
dNa =
vardErfor bodies r
PtdVI
=
so that
i
where we describe the iVi as configuration variables (for simple fluid systems, volumes) and the Pi as their conjugate reactions (for simple fluid systems, pressures). Assembling D3 and Equations 11 and 12 ihen gives the fundamental differential equation for closed systems in the form:
dU = TdS
-
dE, r
va,pa for bodies a
Bringing Equation 18 to 15, and noting that the E, are allowed independent variation, we recover Yar,ua
= 0; r = 1, 2,
...
a
PtdVi for bodies
(13)
I
We now generalize Elquation 13 in the usual way for (possibly) open systems :
dU = TdS --
HadNa = Q
PZdVi I
+
padNa
(14)
a
where the Na are the amounts of the different substances in the system (the same chemical substance in different phases being counted separately), and the p a their chemical potentials. We may then make the usual definitions of the enthalpy and the free energies, and apply them to Equation 14 to find the corresponding differential expressions for them. The Maxwell equations and similar relations can be found from Equation 14 and its analogs for the enthalpy and the free energies, essentially by applying criteria for exactness. T o develop such relations as the Gibbs-Duhem equation requires that we introduce the notion of intensive and extensive properties of the system. Accordingly, we recognize U, S, the Vi,and the hraas extensive so that T , the Pi,and the p a are intensive. (The enthalpy and the free energies become, of course, extensive.) We may then integrate Equation 14 in the usual way to give
u = TS - C PiVt I
+C Q
paNa
(1 9 )
as the equilibrium conditions associated with the independent set of Reactions 16. Equations 19 are of course the usual form of reaction (and phase) equilibrium conditions. However, they have been developed without recourse to the customary maximum-minimum analysis, and indeed our whole presentation has been devoted to the characterization of a merely stationary equilibrium. Acknowledgment
The writer is greatly indebted to Robert A. Graff, City University of New York, for his stimulating help with this work, literature Cited (1) Arens, R., J.Math. Anal. Appl. 6 , 207 (1963). (2) Caratheodory, C., Math. Ann. 67, 355 (1909). (3) Caratheodory, C., Sitzber. Preuss. Akad. Wiss. Physik-Math. KI. 1925, p. 39. (4) Landsberg, P. T., “Thermodynamics,” Interscience, New
York, 1961. for review Novembe; 12, 1965 RECEIVED ACCEPTEDJune 10, 1966
National Meeting, American Institute of Chemical Engineers, San Francisco, Calif., May 1965.
VOL. 5
NO. 4
NOVEMBER 1 9 6 6
465
