J. Phys. Chem. 1989, 93, 6939-6943 set of parameters the mean field steady-state rate constant is always larger than the Smoluchowski one. This is probably fortuitous since the mean field theory does not appear to contain any additional physical ingredients. We have seen that there is a close formal correspondence between the mean field theory and the results predicted by the statistical nonequilibrium thermodynamic theory of Keizer.I3-l5 For the trapping problem, all three approaches agree in the low-concentration limit in two and higher dimensions. This is, however, not the case for the steady-state fluorescence intensity. The difference is particularly apparent when the relative diffusion coefficient is small, and it appears that the statistical nonequilibrium thermodynamic approach needs to
6939
be reexamined at least in this limit. We would not be surprised if this approach would turn out to be operationally identical with the mean field one for the simple reaction schemes considered in this paper. Finally, we mention that the mean field theory, which at present is restricted to the steady state, can be generalized to describe frequency domain fluorometry, as will be shown elsewhere.43 Acknowledgment. This paper is dedicated to Robert Zwanzig on the occasion of his 60th birthday. I am really enjoying having Bob around! I have benefitted from discussions with N. Agmon, R. Cukier, J. Keizer, J. Lakowicz, X. Zhou, and R. Zwanzig.
Thermodynamics at Isothermal, Isobaric Steady States: Vapor Pressure, Colligative Properties, and the Electromotive Forcet Joel Keizer Institute of Theoretical Dynamics and Department of Chemistry, University of California, Davis, California 95616 (Received: February 17, 1989; In Final Form: May 9, 1989) Previously, it has been demonstrated that a consistent thermodynamic theory of matter at nonequilibrium steady states can be developed based or) the properties of molecular fluctuations. Here we use that theory to treat the vapor pressure of solutions at steady states as well as the three classical colligative properties of freezing point depression, boiling point elevation, and osmotic pressure. The osmotic pressure appears to be especially sensitive to nonequilibrium modifications in the chemical potential, and it may provide a direct means of measuring the nonequilibrium component of the chemical potential of a solvent. The related effect of nonequilibrium chemical potentials on the electromotive force is calculated by using theoretical methods for three types of redox reactions in a continuously stirred tank reactor. These reactions model possible oxidation mechanisms by peroxydisulfate, which has proven to be a useful tool in experimental measurement of the nonequilibrium electromotive force. The dependence of this effect on the bimolecular rate constant for oxidation by peroxydisulfate should help delineate in what systems the nonequilibrium electromotive force is easy to measure.
I. Introduction Over the past two decades it has become well understood that away from equilibrium the properties of matter can be modified in significant ways. While the visually most compelling modifications are probably the spontaneous oscillations, pattern formation, and chaotic behavior that are exhibited by the Belousov-Zhabotinski reaction,I fundamental changes in the properties of matter occur even at stable steady states. For example, when liquids are subjected to a steady temperature gradient, one finds modifications in the spectrum of scattered light as a result of the flux of heat through the liquid.2 Similar effects have been observed for fluids under steady shear,3 and in recent years it has been possible to model this behavior using molecular dynamic^.^ Recent measurements of the electromotive force in a continuously stirred tank reactor5 suggest that even the electrochemical properties of solutions can be modified at steady state. Some time ago it was proposed that these, and other properties of matter at steady state, could be explained through a knowledge of the molecular statistical properties at steady state.6-8 The statistical theory of nonequilibrium thermodynamics has provided a useful framework for thinking about these question^,^ and in several cases it has provided successful quantitative comparisons with experiment.’*’* That theory also has suggested a generalization of classical equilibrium thermodynamics that is applicable at steady state and that can be used to analyze the physical and chemical properties of matter in a fashion analogous to the equilibrium theory.I3 At steady state the thermodynamic theory is truly “dynamic” in that it depends on the transport coefficients as well as the usual local equilibrium thermodynamic functions. This paper is dedicated to Robert Zwanzig on the occasion of his 60th birthday. His contributions to nonequilibrium statistical mechanics are enormous and served to inspire many in my generation, including myself.
0022-3654/89/2093-6939$01.50/0
Just as in equilibrium thermodynamics, in the steady-state theory there is an entropy function which is related to the macroscopic statistical properties of the system.I3 Because this function was originally defined through its relationship to the covariance matrix of the extensive variables, ni, it is written as 2 and has been called the “sigma function”. Here we will often refer to it as the generalized entropy or just the entropy. According to ideas developed previously, the generalized entropy can be separated into a local equilibrium part, S, and a nonequilibrium part Z(n;f,R) = S(n)
+ ujuj(n;f;R) I
where the column vector f consists of fluxes of the extensive variables (e.g., mass, momentum, or energy) that maintain the steady state and R consists of variables that may be needed to characterize reservoirs attached to the system. The functions vi (1) Field, R. J.; Burger, M. Oscillations and Traueling Waues in Chemical Systems; Wiley: New York, 1985. (2) Beysens, D.; Garrabos, Y.; Zalczer, G. Phys. Reu. Lett. 1980.45. 403. (3) Beysens, D. Physica 1983, 118A, 250. (4) Evans, D. J. Physica 1983,118A, 51. Hoover, W. G. Physica 1983,
118A, 111. ( 5 ) Keizer, J.; Chang, 0.-K. J . Chem. Phys. 1987, 87, 4064.
(6) Keizer, J. J . Chem. Phys. 1976, 65, 4431. (7) Keizer, J. J. Chem. Phys. 1978, 69, 2609. (8) Keizer, J. Arc. Chem. Res. 1979, 12, 243. ( 9 ) Keizer, J. Statistical Thermodynamics of Nonequilibrium Processes; Springer-Verlag: New York, 1987. (10) Tremblay, A. M. S.; Siggia, E.; Arai, M. Phys. Reo. A 1981.23, 1451. Kirkpatrick, T. R.; Cohen, E. G. D.; Dorfman, J. R. Phys. Reu. A 1982, 26, 972. Ronis, D.; Procaccia, I. Phys. Reu. A 1982, 26, 1812. (11) Keizer, J. Chem. Rev. 1987, 87, 167. (12) Keizer, J. J . Chem. Phys. 1987, 87, 4074. (1 3) Reference 9, Chapter 8.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Keizer
depend explicitly on n, f, and R. The local equilibrium entropy, S(n), is the customary equilibrium entropy function evaluated at nonequilibrium values of the extensive variables, n. A stirred tank reactor provides a simple example of the flux and reservoir variables, which in that case are proportional to the flow rate and the concentrations of reactants in the inflow lines, respecti~ely.'~*'~ Using statistical nonequilibrium thermodynamics we have shown that the form of eq 1 can be obtained from the underlying statistical relationship13
energy of reaction for an electrochemical cell and the electromotive force t of a namely,
6940
where kB is Boltzmann's constant and the covariance matrix, mu, is determined by the equal time covariance of fluctuations in the extensive variables at steady state, i.e. cij = ( (ni -
np)(nj - ny))
(3)
Here nss represents the average value of the extensive variables at the steady state. The generalized entropy, Z, allows one to define a reversible process at steady state as one in which the parameters f and R are fixed and for which the generalized entropy production v a n i ~ h e s . ' ~ While the thermodynamic theory of steady states has rather broad applicability, we will be concerned here only with the properties of dilute solutions that are isothermal and isobaric. One can show, in general, that the thermodynamic pressure, $ 1 -(8E/dV)x,N, is identical with the mechanical pressure, p.14 Furthermore, for dilute, isothermal solutions it has previously k e n shownI5 that the generalized thermodynamic temperature, T = (dE/dZ),,, where E is the internal energy, is identical with the local equilibrium kelvin temperature, T = ( d E / a S ) , Moreover, in dilute solution one finds that to a good approximation eq 2 can be written (4)
where f i i is the generalized chemical potential of species i and the prime on "refers to the remaining mole (or particle) numbers. It should be remembered that in all of these derivatives the variables f and R are held fixed although for ease in writing we will not explicitly indicate this fact. In an isothermal, isobaric ensemble at steady state the fundamental thermodynamic function is the generalized Gibb's energy,I4 G, whose differential gives the reversible work, exclusive of pressure-volume work; i.e. (5)
just as the Gibb's energy does for such an ensemble at equilibrium. In addition to eq 5 one has the generalized Gibbs-Duhem relationship14
where p is the pressure. Equation 6 can be useful in analyzing changes in the generalized chemical potentials. For isothermal, isobaric changes the left-hand side of eq 5 takes the form
AGaI1= -en€
t
All of the identities satisfied by the generalized thermodynamic variables are the same as those satisfied by their equilibrium counterparts. This means that to obtain a valid relationship at steady state one need only add "hats" to the familiar equilibrium id en ti tie^.'^ Thus, for example, the partial molecular volume of species i is given by Vi =
(dfii/aP)T,N
ele -
AP$,,/en
(1 1)
11. Steady-State Thermodynamic Criterion for Solvent-Vapor
Equilibrium One of the simple predictions of equilibrium thermodynamics is that the properties of dilute solutions are dominated by the entropy of mixing associated with the dissolution of solute.I6 These properties are collectively called colligative properties and include lowering of the vapor pressure by nonvolatile solutes, boiling point elevation, freezing point depression, and osmotic pressure differences across semipermeable membranes. It is an important achievement of thermodynamics that the magnitude of these effects can be explained in terms of other measurable quantities, such as the heat of fusion and molar volume of the pure solvent. The derivation of these properties depends on the fact that at equilibrium the chemical potential of a component must be the same in phases between which that component can be exchanged, which in turn depends on the fact that at constant temperature and pressure Recalling eq 5 , which is the analogue of eq 12 for the generalized Gibbs energy at steady state, it should not be surprising that equality of the generalized chemical potentials, pi, can be envoked as a criterion for steady state between phases that communicate with one another. Indeed, consider a dilute solution with several nonvolatile solutes that is maintained at a stable steady state at fmed temperature and pressure. This can be accomplished, for example, in a continuously stirred tank r e a c t ~ r . ~ *If' ~the solvent is in contact with its vapor phase, then in the absence of non-pVwork eq 5,7,and 8 imply for any reversible process (recall that f and R are fixed) that m
i=2
where component one is the solvent and the superscripts s and g stand for the solution and gas phases. Now imagine a specific reversible process of this sort in which the only change is a transfer of solvent molecules from solution to the gas phase, Le., -dN,(') = dN,(g) = d t a n d dNi = 0, 2 Ii Im. In this case eq 13 implies that (p,(%)- fil(s)) d{ = 0
(9)
since $ = p and F = T , as noted above. There is another useful thermodynamic relationship between the generalized Gibbs free
If the vapor is in an equilibrium state, there will be no corrections ~
(14) Keizer, J. J . Chem. Phys. 1985, 82, 2 7 5 1 . (1 5) Keizer, J. J . Chem. Phys. 1984, 80, 41 85.
=
where AGEll Aecell- AGaII,where G is the local equilibrium free energy. The second term in eq 11 represents a correction to the local equilibrium prediction of the electromotive force, as given by the Nernst potential. The existence of corrections to local equilibrium thermodynamic functions at steady state is rather general, and in the remainder of this article we examine the influence of these corrections for some of the familiar properties of solutions, including vapor pressure, colligative properties, and the electromotive force.
(7)
The generalized Gibbs energy is an extensive variable, so for a composite system that is made up of two subsystems one has d 6 = d e , + dG2 (8)
(10)
where e is the electron chargc, n is the number of electrons involved is the generalized Gibbs energy in the cell reaction, and AGCeII change (per molecule) for the reaction. Because Z can be broken up intoits local equilibrium and nonequilibrium contributions, so can G = E - TZ + pVand the generalized chemical potentials, pi. Thus eq 10 can be written
~
(16) Rock, P. A. Chemical Thermodynamics; University Science: Mill Valley, 1983.
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6941
Properties of Matter at Steady State to its chemical potential and, thus, eq 14 takes the even simpler form P1(8)
= PI(')
(15)
Equation 15 should be contrasted with the local equilibrium result that would be predicted on the basis of the irreversible thermodynamic^.^^^^ According to that theory, the transfer rate of molecules across the gas-liquid surface is determined by the local equilibrium chemical potentials. If the coupling to other elementary processes (e.g., heat transport across the surface) is neglected, the mass-transfer rate can be written9 as d t / d t = wxP(Pl(S)/kBT) - eXP(Pl(g)/kBT)l
= ( n / ~ B n ( P l ( s )- PI(g))
(16)
where the linearization is appropriate when differences in the local equilibrium chemical are small. At steady state the left-hand side of (16) is zero and, therefore,
=
(17)
Pl(s)
The difference between the local equilibrium thermodynamic prediction in (17) and the steady-state prediction in (1 5) is the statistical correction to the chemical potential, pi, as determined by eq 4. Indeed, using eq 1, which gives the decomposition of the generalized entropy, Z, into its local equilibrium and nonequilibrium parts, we can rewrite eq 15 as
= Pl(s) + plnc
(18)
At steady states the first term on the right-hand side of (18) is, in fact, the largest term (except near critical pointsI3) and corresponds to the local equilibrium approximation given in eq 17. This correction has the same origin as the predicted correction to the Nernst expression for the electromotive force in eq 11. Indeed, using a calculation based on irreversible thermodynamics,l* patterned after the calculation in eq 16 and 17, one easily sees that irreversible thermodynamics predicts that the electromotive force is determined by AGccl,= -en€
(19)
with AGeII the usual local equilibrium Gibbs energy change for the cell reaction. These results highlight the fact that the origin of these nonequilibrium corrections is the statistical correlations among intensive variables that cannot be predicted by equilibrium statistical mechanics. Indeed, the usual theory of irreversible processes, either in its linear form due to Onsager or its nonlinear generalization, is based on the idea of local equilibrium functions and, therefore, does not contain information about these nonequilibrium statistical effects. As we have shown earlier, however, this defect can be removed within the context of statistical nonequilibrium thermodynamics, which provides explicit expressions for the statistical correlations required for the definition of the entropy in eq 2 and 3. 111. Generalized Colligative Properties at Steady State The thermodynamic criterion in eq 18 provides a simple way to measure the nonequilibrium component of the chemical potential of a volatile solvent (or solute, for that matter). Neglecting fugacity coefficients, one can write the local equilibrium chemical potential of the gas pI(g) = poI(7')
+ kBTIn p 1
(20)
where we have chosen molecule number-based chemical potentials. The generalized chemical potential of the solvent in solution, on the other hand, is
~~
(17) deGroot, S .
R.; Mazur, P. Nonequilibrium Thermodynamics;
North-Holland: Amsterdam, 1982. (18) Reference 9, Section 5.6.
where k * , is the pure solvent chemical potential and al is its activity,I6 taken as unity for the pure solvent at 1 atm pressure. Substituting these expressions in eq 18 gives
Pine = kBT In
(Pl/P1lC) =
kBT In (PI/P'lal)
(22)
with p o l the pure solvent equilibrium vapor pressure determined by kBT In p'l = boI- I . L O ~ (23) and
pile = p'lal
(24)
the local equilibrium prediction of the vapor pressure.16 According to eq 22, the nonequilibrium component of the solvent chemical potential can be determined by measuring deviations of the vapor pressure from its local equilibrium value given in eq 24. In dilute solution at equilibrium, Raoult's law holds, so that plIe = p',X1, where XIis the mole fraction of solvent. Any deviations from this value reflect a different "escaping tendency" of the solvent from solution as determined by a change in entropy from its local equilibrium value. Writing p 1 = pile plneand
