Stokes' Theorem and the Geometric Basis for the Second Law of Thermodynamics Lawrence H. Bowen North Carolina State University. Raleigh. NC 27695 The second law of thermodynamics is a difficult concept to present in physical chemistry, and the manner of its presentation has been the subject of much discussion. Many physical chemistry texts take the traditional approach, referring to observations on heat engines. A few texts start with the supposition of astate function S,which increases in irreversible processes. The entropy Sis sometimes immediately related to "disorder", leading to difficulties in qualitative interpretation of spontaneous ordering processes such as precipitation ( I ) . Honig (2) recommended that more attention he eiven to the CarathBodorv formulation of the second law,'& being more general thin the usual ones and more (lirrctlv relarcd to chemical vhenomena. He has developed this approach more recentliin a book (3).There is no evidence that this mathematical and seemingly abstract approach has achieved any favor among undergraduate teachers-it is not used in any undergraduate physical chemistry text to date. However, the relationship between geometry and the CarathBodory version of the second law seems to deserve more attention because geometry is a visual mathematics and a visual representation of entropy should be useful, especially for chemists who find difficulties in relating heat engines to chemical reactions. In the following discussion, the approach is closely related to that of Honig (2) but is simplified to emphasize the geometric interpretation. Any thermodynamic system can be represented by a state characterized by the values of a series of independent variables, a minimum of three: for example, V , T, and n for a pure substance with no external fields. This suggests an analogous relationship between thermodynamic variables A(V,T,n) and geometric variables A(x,y,z). In geometry, dependent variables may he vectors and have a directional character as well as magnitude. In usual notation. A(r) = A,i
+ A,j + A,k
(1)
where The vector character of certain thermodynamic variables is not a property usually discussed hut is one which is important to the CarathBodory formulation. Three basic ideas of vector algebra are necessary for the following discussion. These are the definitions of (1) the gradient vector operator,
a
a ay
V=i-+j-+kdr
a ar
where it is noted that V points in the same direction as r; (2) the scalar product of two vectors,
(3) the vector product, A X B = i(A$, -A$,,)
+ j(A,B, - A,B,) + k(A,B,
- A$,)
(5)
where it is noted that the vector A X B is perpendicular to 50
Journal of Chemical Education
both A and B. The magnitudes of these two products depend on the magnitudes of A and B and on the angle between. Thus, A.B = AB cos 0 and IA X BI = AB sin 0, where 0 is the angle between A and B. Two parallel vectors will have maximum scalar product and zero vector product while perpendicular vectors will have zero scalar product and maximum vector product. The magnitude of a vector is, of course,
Stokes's Theorem The geometric hasis for CarathBodory's formulation of the second law is a standard theorem of vector analysis, Stokes's theorem (4):
where $ denotes the cyclic line integral of A taken around the line hounding asurface, S, and SsJ is the integral of V X A taken over the surface vector: dS = idydz
+ jdxdr + kdxdy
(8)
which always points perpendicular to the surface, S, of a three-dimensional figure. This theorem can be proved by vector analysis ( 4 ) , but its consequences are more easily discussed. According to Stokes's theorem, the necessary and sufficient condition that $A.dr = 0 is that V X A = 0 for all values of A(x,y,z). This means that
+
and similarly for the other derivatives. Since A.dr = A,dx Aydy A&, the fact that $A.dr = 0, independent of path, impl~esthat A,dx Aydy A,dz = dF,the exact differential of a function F(x,y,z). Only then will
+
+
+
and thus
+
+
A,dy A,dz is an exact differential, Now, if dF = A,dx then A, = (aFlax),,, etc., and A is the gradient vector of the scalar function F. Equation 9 can be rewritten:
The order of differentiation does not matter for a scalar function. The Flrst Law The above discussion can be applied directly to the first law of thermodynamics. Thus, the condition that a dependent variable, the internal energy U(V,T,n), he a state function is that V X A = 0, where
with al,az,aa heing scalars required t o give A consistent dimension (dr = ial-'dV + jaz-'dT + ka-'dn). Thus,
and similar equalities can he written for the other cross derivatives. By their definitions, Q and W depend on path. The first law states that the sum d U = dQ + d Wgives a path indepeudent function U for which $dU=O (15) Geometrically, Uisa function that hasagradient vectorA pointing in the direction of r, and
= VUalways
The Second Law
I t is easily shown that dQ(reversible) does not correspond to a state function Q (even with reversibility heing specified so that dQ can he expressed in terms of state variables V, T, n). There is no gradient vector VQ! Thus for constant n, d U = C,dT (aU/aV)dV = dQ dW. If the path is reversible, d W = -pdVand
+
+
dQ(reversih1e) = CUdT+
The geometry of converting the path-dependent dQ(reversihle) to the state variable d S deserves closer scrutiny. If A is a gradient of a function, A = VF, then A always points along r. If a path is chosen, such that irl is constant, then A must he perpendicular to d r such that A.dr = 0 (Fig. 1).Note that a t every point on the surface of this sphere there are inaccessible points characterized by an infinitesimal change in r to another sphere off the surface. This surface is not a common one in thermodynamics, since it places severe restrictions on the variation of independent variables. A more general surface will involve variations in r , hut although A.dr is non-zero when d r is not perpendicular to r, the positive and negative contributions exactly cancel, according to Stokes's theorem, if a cyclic path is chosen as illustrated in Figure 2. Even though A.dr is not always zero, V X A will he, as A points in the direction of r and V. There are no inaccessible points in this case, as d r can change arbitrarily. Now suppose that an arbitrary vector B exists, which does not point in the direction of r. As d r lies in the surface mapped out by the variations of r, the vector B will always be perpendicular to the surface element B.dr = 0 (Fig. 3). There may he many such surfaces crossing a given point, as the magnitude and orientation of B will in general depend on the path taken. The vector B is not required to have a fixed orientation a t a given point. However, if B does have a fixed orientation a t a given point, the surface element B.dr = 0 will he fixed (Fig. 4)
(17)
or for an ideal gas with (BUI~V)T = 0,
+
Since (aC,laV)~+ nRIV, dQ(reversihle) is not the differential of a state function. However, for an ideal gas, as made clear by Wall (5),T1is an integratingfactor that does give a state variable, d S = dQ(reversih1e)lT.
Figure 3. A surface Bdr = 0. B always points perpendicularto the surface, but ns wientstlon to r is arbltrarv. Figure 1. me surface Adr = 0.
Figure 2. A general surface wilh S A d r = 0. A must always point in the direction of r.
Flgure 4. ma fixed surface Bdr = 0. A = B/A (see taxi)
Volume 65
Number 1 January 1988
51
rather than diffuse. In this case, all ooints off the surface are inaccessible by the path B.dr = b. ~dseemsclear that B could he reoriented to a new vector A which points in the direction of r by dividing by ascalar factor X(x,y,z) such that A = BIX. The vector A is the gradient of a scalar function F(x,y,z), since V X A = 0, and thus $A.dr = 0. I t seems unlikely that avector B such as in Figure 3 would also have an integrating factor X if its orientation to r at a given point is arbitrary. The Caratb6odory theorem (2, 3) concerns the case where B mav have manv orientations a t a given r. This theorem asserts that if inaccessible states exist infinitesimallv close to the surface R.dr = Oat evers ooint. B/Xis then an integrating factor X(x y,z) exists such that A ; the gradient of a function. Thus, a fixed surface B.dr = 0. suchas Figure 4, is sufficient forthe existence of A, hut not necessary. I t is only necessary that some values of d r be forbidden at every point. Now, examine the thermodynamic path denoted by dQ = 0. Such a path is always followed by system and surroundings together. That is, heat flow into a system is always balanced by heat flow from the surroundings. Chemistry students quickly confront the fact that any specific mixture of chemical compounds at a specific initial state of this mixture and its surroundings (usually fixed temperature and pressure) will react in one direction only. They soon learn that phase changes for a single species are also unidirectional a t given temperature and pressure. Thus, for any state of thesystem andsurroundings-there exist inaccessit~fe states (those prior in time). Buchdahl (6) states the second law as:"in the neighborhood of any state 2 o f an adiahatically isolated system here are states inaccessible from 2".If dQ = R.dr. that is. dQ(reversihle), . . which is a function of state variables V, 'l',n, ;hen according to Carathhodory the exiitenceof inaccessible points ti.e., thedirectional oath of natural processes) implies the existence of an integrating factor X
Figue 5. The thermodynamic rutace dS = A* = 0 A I reversibe adiaoat c pdhs lie an mls surfaceand no adiabatic parn can result in s surface of lower
for dQ(reversih1e). Thus, a state function must exist, called the entropy, which is defined by dS =
dQ(reversib1e)
(19)
X
and $dS = 0. There exists a gradient vector A = VS which points along r and is calculated by BIA. Honig (2, 3) gives a detailed discussion of the equality between X and the absolute temperature T. This discussion can he summarized by noting that the only commonparameter among systems that can exchange heat reversibly is their temperature. Thus X can he a function of T only, and if X = T, the absolute temperature so defined is identical to the ideal gas temperature. The second law is not only a definition of entropy, hut also an inequality, The sign of the inequality is arbitrary but results in the usual identification of A with T. Inaccessible points, required by Carath6odory for the existence of an entropy function, are chosen to have lower S. Of course, AS in eq 20 has two parts, AS for the system and AS for the surroundings. Clearly, there may he a decrease in S i n one as long as the other comoensates. This is the basis for anv, chemical orderine phenomenon. It should be noted that only positive values the variables V. T.n are allowed. Also. the scalar f a c t m a , . a 2 , a 3 (eq 13) ark different for d U and d ~Paths . of constan; energy and paths of constant entropy lie on the positive quadrant of a sphere (Fig. 5), although the coordinates are different. Thus, for an ideal gas, PVIn is constant for constant energy, while P(Vln)r is constant for constant entropy (y being the heat capacity ratio, C,,/C,). There is a fundamental difference between U and S. If A.dr = d U for system and surroundings, the first law allows no change in radius: energy is conserved. However, if A.dr = dS, the second law allows expansion of the sphere in the direction of increased entropy (Fig. 6). For example, the path dQ = 0 allows a number of surfaces B.dr = 0 for adiahatic changes of an ideal gas. The volume and entropy may increase indefinitely if the gas performs no work on expansion. The temperature and entroov mav increase indefinitely if an excess pressure is applied on-compression. However, not all points are accessible. The temperature can never decrease below the surface P(Vln)r = constant. Any cyclic adiahatic path must he reversible. I t seems pertinent to note that temperature is the key variable in determining the inaccessible region (both P and Vln have allowed range from zero to infinitv) and temnerature is also the kev in converting d~(reversib1e)to a state variable dS. Since, from the definition, a reversible heat flow affects the entropy more a t low Tthan high, the natural flow of heat from high T to low T is simple to describe (only that direction of heat flow will produce a net positive AS). Chemical reactions are more difficult to discuss. However, without resorting to molecular models or heat engines the geometric view of the directional process of events leads directly to a state function, the entropy, and an inequality for changes in that function, AS 2 0. ~
~~
~~~
~~~~~
Acknowledgment
The constructive comments of J. M. Honig, E. F. Carter, and A. F. Coots on a draft version were very helpful in improving this paper. 1. McGlashan,M.L. Chemieol Thermdynamica;A~demie:NmYork,1979;pplll-115. 2. Honig, J. M . J . Chem. Educ. 197P.52.416. 3. Honk. J. M . Thormdvumics: Princiole* ChorocWzim - Phvaieoi . and Chamicol &mas; Elsevim ~ k York, k 1982;dp 4-2. 4. Amen. G. Mothamtical Methods for Physicisls, 2nd e6.:Acadsmic: New York. 1970: pp 51-59. 5. Wa1l.F. T . Chemical TkrmodywMcs,3rd ed.: W . H . Freeman: Sao Franciam,1974:p
Figure 6. One of the allowed adiabatic surfaces with dS = A& positive, but dqreversible)= Bdr = 0.Both dSand do referto system +surroundings. 52
Journal of Chemical Education
44.
6. Buehdahl, H. A. Tha Concept$ of Clossieoi T h e r m d y ~ m i e sThe ; Univmity Pmss: Cmbridpe, 1966: p68.
