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Statistical Rate Theory Insight into Evaporation and Condensation in Multicomponent Systems Atam Kapoor and Janet A. W. Elliott* Department of Chemical and Materials Engineering, UniVersity of Alberta, Edmonton AB, Canada T6G 2V4 ReceiVed: July 19, 2010; ReVised Manuscript ReceiVed: September 22, 2010
Current approaches to mathematically modeling liquid-vapor mass transport (e.g., film theory, penetration theory, boundary layer theory) treat bulk phase transport accurately with diffusion models, but leave the transport across the interface to be described by empirically determined mass transfer coefficients. In multicomponent systems, this requires empirical mixing rules for the single-component mass transfer coefficients. Such approaches can only give estimates of net rates at the interface but cannot examine the movement of individual components. Here we use statistical rate theory to provide new physical insight into evaporation and condensation at interfaces in systems containing multiple volatile components. In contrast to the traditional multicomponent mass transfer approach, we show ranges where one component evaporates while the other condenses even when the net transport is unidirectional. Introduction Current approaches to mathematically modeling liquid-vapor mass transport1,2 (e.g. film theory, penetration theory, boundary layer theory), treat bulk phase transport accurately with diffusion models, but leave the transport across the interface to be described by empirically determined mass transfer coefficients. Here we use Statistical Rate Theory3–26 to provide new physical insight into evaporation and condensation at interfaces in systems containing multiple volatile components. Multi-component evaporation and condensation are central to many processes on earth such as: climate, biological mass transfer (i.e., respiration, perspiration or transpiration), soil water management, and separations in the chemical process industries. Current theories used to predict multicomponent evaporation or condensation require effective multicomponent diffusivities, empirical film coefficients, and overall mass transfer coefficients. Single-component diffusivities are predicted using the elementary kinetic theory, Chapman-Enskog theory,27,28 or by mere correlation of experimental data using empirical expressions given by Fuller, Schettler, and Giddings.29 These singlecomponent diffusivities are further used to approximate effective multicomponent mass transfer coefficients using analytical techniques (mixing rules) laid out by Gilliland,30 Toor (binary mixtures31), and Hsu and Bird (ternary mixtures32). Deviations up to 25% in the correlated mass transfer coefficients are not uncommon. As well, such approaches can only give estimates of net rates at the interface but cannot examine the movement of individual components. In this paper, we propose to use statistical rate theory to investigate multicomponent evaporation and derive expressions for prediction of evaporation flux of individual components which can be made as precise as desired and are free from any fitting parameters. Statistical rate theory is a theory of interfacial transport that has had remarkable success in accurately predicting evaporation and condensation for pure systems.3–6 The assumption that every quantum state is equally probable leads * Corresponding author. E-mail:
[email protected]. Phone: 1-780492-7963. Fax: 1-780-492-2881.
to the equilibrium statistical thermodynamics of Boltzmann and Gibbs. A comparably general assumptionsthat each available quantum transition is equally likelysleads to statistical rate theory, a general theory of nonequilibrium statistical thermodynamics. Our results yield statistical rate theory equations that are as broadly and easily applicable to a wide number of circumstances as multicomponent phase equilibrium calculations. As well, we anticipate that this work will be a catalyst for more refined statistical rate theory models for multicomponent interfacial transport, describing particular circumstances in specific fields. Statistical rate theory (SRT) was proposed by Ward in 19777 and published in its more current form in 19828 and 1983.9 To date, more than 95 journal papers have been written on applications of statistical rate theory and its validity has been experimentally confirmedsonly a few references are given here. Key improvements allowed SRT to be applied to nonisolated systems,10,11 and to systems where the two bulk phases involved in the transport had different temperatures.3 SRT fundamentals are described in detail in a book chapter.12 SRT has been applied to gas absorption by liquids13,14 and solids,15 gas adsorption/desorption,10–12,16–19 adsorption at the solid/solution interface,20–22 electronexchange between isotopes in solution,9,12 evaporation/condensation,3–6 crystal growth from solution,23 surface diffusion,24 and biological cell membrane transport.25,26
Governing Equations Using the statistical rate theory expression for predicting evaporation flux for a single component3,4 and assuming that single molecular events occurring during the evaporation of individual components from a multicomponent liquid phase are identically distributed independent events, we can generalize the total solution evaporation flux for a C-component mixture as a summation of individual component evaporation fluxes
10.1021/jp106715v 2010 American Chemical Society Published on Web 10/15/2010
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C
J LV )
that corresponding to the instantaneous liquid composition. Thus, we write the equilibrium exchange rate as
∑ JiLV i)1
where JiLV
{ [ [
µiL
µiV
)] )]}
( (
hiV 1 1 ) Ke,i exp + - L L V V k kT kT T T µiL µiV hiV 1 1 - L exp - L + k TV kT kT V T
Ke,i )
-
(1)
where L and V refer to the liquid and the vapor phase, respectively, JLV is the total evaporation flux from the liquid to the vapor phase, subscript “i” refers to the component i, µLi and µVi are the chemical potentials of component i in the liquid and vapor phase respectively, hVi is the molecular enthalpy of component i in the vapor phase, k is the Boltzmann constant, TL and TV are the liquid and vapor temperatures, and Ke,i is the equilibrium molecular exchange rate of component i. To evaluate the above expressions (eq 1) for each component in the liquid phase, we need expressions for the equilibrium molecular exchange rate (Ke,i), and enthalpies and chemical potentials for each component as functions of liquid and vapor phase compositions. For simplicity, we will only consider the case of an ideal liquid mixture with incompressible liquid components and an ideal vapor mixture with each component vapor treated as an ideal gas. The approach can be extended to include nonideal effects;5 however, such extension would not qualitatively affect the results emphasized in this paper. For an ideal incompressible liquid mixture and an ideal vapor mixture, the expressions for chemical potentials of each component in the liquid and the vapor phases are given by33
µiL(T L, P L)
)
µiL(T L, P∞,i(T L))
+
L V∞,i [P L
- P∞,i(T )] + L
kT L ln xi
(2)
( )
µiV(T V, P V) ) µiV(T V, P∞,i(T V)) + kT V ln
(
ye,iPeV ) xiP∞,i(T L) exp
JiLV )
(3)
where xi and yi refer to mole fractions of component i in the L and P∞,i(T) refer liquid and the vapor phases respectively, V∞,i to the pure species i liquid molecular volume and saturated vapor pressure at temperature T, and PL and PV refer to the pressure in the liquid and the vapor phase, respectively. The reference chemical potential in the saturated vapor phase can be approximated using the ideal gas chemical potential which can be obtained using Boltzmann statistics and the Born-Oppenheimer approximation in terms of the translational, vibrational, rotational, and electronic partition functions.3,4 Using classical kinetic theory for the equilibrium state, the equilibrium exchange rate is determined as the rate at which molecules from the vapor phase collide with the liquid-vapor interface. As has been done before,3,10,11 a hypothetical equilibrium state is defined as that which the system would come to if it were isolated at a particular instant. Assuming the mass of the liquid dominates the system, the equilibrium temperature will be the liquid temperature, whereas the equilibrium vapor phase pressure and vapor phase composition must be found as
ye,iPeV
√2πmikT L
where ∆Si TV ) ln k TL
[( ) (
L V∞,i
kT L
l)1
P∞,i(T V)
L V∞,i
kT
L
)
[PeV - P∞,i(T L)]
(5)
Equation 5 can be solved iteratively to obtain PVe . Substituting eqs 2-4 in eq 1, we obtain the final expression for predicting evaporation flux for each component in a C-component mixture in terms of mathematically computable and experimentally measurable quantities as
∑
yiP V
(4)
√2πmikT L
where ye,i and mi are the equilibrium vapor concentration and molecular mass of component i, and TVe ) TL and PVe are the equilibrium temperature the equilibrium pressure in the vapor phase of the instantaneously isolated system. PVe can be evaluated by equating the chemical potentials in the liquid and the vapor phases at equilibrium resulting in
ni
and
ye,iPeV
4
[ ( ) exp
P∞,i(T L) PV
[P L - P∞,i(T L)] +
[
( )]
∆Si ∆Si - exp k k
)] [
(
+ ln
qvib,i(T V) qvib,i(T L)
)
1 1 - L × TV T
] (
]
+
) ()
θl,i xi θl,i TV + 4 1 - L + ln + V 2 yi exp(θl,i /T ) - 1 T
(6)
where for species i, qvib,i(T) is the vibrational partition function at temperature T, θl,i is the characteristic temperature of vibration, and ni is the number of vibrational degrees of freedom of a molecule. Equation 6 can be evaluated to predict evaporation rates for multicomponent liquid mixtures if the vibrational partition functions and the characteristic temperatures of vibration are known for all evaporating components. This data is not readily available for most substances and the need for this data arises only when the liquid-phase temperature is appreciably different from the vapor-phase temperature. The liquid- and vapor-phase temperatures are not in general equal when phase change is taking place. For example, temperature jumps (∼5-8 K) across the liquid-vapor interface have been experimentally observed6,34,35 for evaporation of pure water and hydrocarbons. However, it was shown that the terms involving the vibrational partition functions and characteristic temperatures of vibration can be neglected for such small temperature jumps without altering the qualitative behavior for evaporation systems.5 In this work, we assume that the liquid and vapor temperatures are the same, eliminating the need to know the vibrational partition functions and the characteristic temperatures of vibration. A detailed analysis on how to evaluate SRT expressions
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Kapoor and Elliott
when temperature jumps are involved and vibrational data for components is known has been laid out by Ward et al.3,4,6 Results Calculations were performed to predict the interfacial evaporation rates for an equimolar (starting composition) liquid mixture of water and ethanol with an initial mass of 200 kg coexisting with a vapor phase at 100 kPa for different constant vapor mole fractions of water and ethanol. The flat interfacial area was taken to be 100 cm2, and the liquid and vapor phases were assumed to be at the same temperature (27 °C). Figure 1 shows the results for one such case in which the vapor mole fractions of water and ethanol were held at 0.01 and 0.05, respectively. It is interesting to see that ethanol is found to condense initially but evaporates at long times. A high mole fraction of ethanol in the vapor phase results in initial condensation while water is evaporating at all times due to its low vapor mole fraction. This leads to an increased liquid mole fraction of ethanol in time resulting in switching of the flux direction for ethanol from condensation to evaporation after some time. It must also be noted that the mole-fraction curves monotonically reach their equilibrium values even though the mass curves are not monotonic. This nonmonotonicity of mass curves (switch in the direction of flux for a component in an evaporating system) for evaporation of a multicomponent mixture is mathematically similar to a competitive process. Although Azizian et al. studied a similar kind of competitive behavior in multicomponent adsorption from a solution onto a solid surface,21 where the ad-species are competing to occupy the lattice sites available on the solid surface, it is counterintuitive to obtain the same type of behavior in an ideal liquid-vapor mixture as the components are not competing with each other and there are no lattice sites (or anything physically analogous). We expect such nonmonotonicity of mass curves to be seen only for cases where a solution for coexistence of the liquid and vapor phase can be found for eq 5 (and even then, such nonmonotonicity is not assured). A slight modification of eq 5 leads us to the following bounds in which nonmonotonicity of mass curves might be observed
∑ i
ye,iPeV ≈
∑ i
xiP∞,i(T L) ⇒
P∞min PVe
e
∑ i
ye,i e
P∞max PVe
(7) where P∞min and P∞max are the minimum (water in this case) and the maximum (ethanol in this case) of the saturation pressures of evaporating components respectively. Figure 2 shows the competitive evaporation/condensation results for vapor mole fractions lying within the bounds given by eq 7. Figure 2a shows the case of low vapor mole fractions of both water and ethanol. Water condenses initially but is found to evaporate at long times. Figure 2b,c shows similar phenomenon in which ethanol evaporates initially but is found to condense at long times. It is interesting to note that although ethanol mole fraction in the vapor phase was not changed from case (a) to case (b), ethanol evaporates at long times for case (a) while it condenses at long times for case (b). Figure 2d shows that, for high mole fraction of ethanol and low mole fraction of water in the vapor phase, ethanol condenses initially but evaporates at long times. Figure 3a,b shows that, for very small vapor mole fractions of water and ethanol, the total liquid solution mass and the masses of the individual components monotonically decrease
Figure 1. SRT calculations for an initially equimolar liquid of water and ethanol having a total initial mass of 200 kg at a liquid temperature of 27 °C and 100 kPa pressure in the vapor. The interface is considered to be flat and having 100 cm2 interfacial surface area. The subscripts 1 and 2 refer to water and ethanol, respectively. The vapor mole fractions of water and ethanol are held constant at 0.01 and 0.05, respectively. Plot a shows the total liquid solution mass and individual component (water and ethanol) mass as functions of time. Plot b shows component mole fraction in the liquid phase as functions of time starting from an equimolar composition at time t ) 0. Plot c shows evaporation flux of water and ethanol as functions of time.
(evaporation taking place) while they monotonically increase (Figure 3c,d, condensation taking place) for very large vapor mole fractions. This is in accordance with our expectations because the vapor mole fractions (cases a, b, d) lie outside the bounds given by eq 7. Discussion and Conclusions This Raoult’s law level of mass transport description is illuminating and clearly demonstrates competitive-like behavior
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Figure 2. SRT calculations for an initially equimolar liquid of water and ethanol having a total initial mass of 200 kg at a liquid temperature of 27 °C and 100 kPa pressure in the vapor. The interface is considered to be flat and having 100 cm2 interfacial surface area. The subscripts 1 and 2 refer to water and ethanol, respectively. Plots a, b, c, and d show the total liquid solution mass and individual component (water and ethanol) mass as functions of time for different constant mole fractions of water and ethanol in the vapor: (case a) y1 ) 0.03 and y2 ) 0.01; (case b) y1 ) 0.05 and y2 ) 0.01; (case c) y1 ) 0.03 and y2 ) 0.03; and (case d) y1 ) 0.01 and y2 ) 0.05, respectively. Competitive evaporation is observed for all cases as the total liquid solution mass curve and the individual mass curves are not all monotonic and in every case at least one component switches flux direction from condensation to evaporation or vice versa at long times. All cases (a-d) refer to vapor mole fractions which lie within the bounds given by eq 7.
Figure 3. SRT calculations for an initially equimolar liquid composition of water and ethanol having a total initial mass of 200 kg at a liquid temperature of 27 °C and 100 kPa pressure in the vapor. The interface is considered to be flat and having 100 cm2 interfacial surface area. The subscripts 1 and 2 refer to water and ethanol, respectively. Plots a, b, c, and d show the total liquid solution mass and individual component (water and ethanol) mass as functions of time for different constant mole fractions of water and ethanol in the vapor: (case a) y1 ) 0.005 and y2 ) 0.005; (case b) y1 ) 0.01 and y2 ) 0.01; (case c) y1 ) 0.01 and y2 ) 0.03; and (case d) y1 ) 0.05 and y2 ) 0.05, respectively. The total liquid solution mass curve and the individual mass curves are monotonic in behavior. Cases a, b, and d refer to vapor mole fractions which do not lie within the bounds given by eq 7.
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of evaporating and condensing multicomponent systems. Thus, application of statistical rate theory for predicting multicomponent evaporation/condensation is a significant improvement over the previous theories as the expressions can be made as precise as desired and are free from any fitting parameters. These expressions can be easily applied to mixtures with any number of volatile components (and not just binary or ternary mixtures, as has been the case before). Further, we also get physical insight into the pathways followed by the system during multicomponent evaporation or condensation depending on the initial conditions imposed on the systemsthis insight was lacking before. These pathways are very sensitive to the vapor pressure and the vapor mole fractions next to the interface and this can be captured by using a quantum probabilistic model (as is statistical rate theory). This method can be extended to include nonidealities using equation-of-state models for nonideal mixtures to extend this work for nonideal liquid-vapor mixtures. This extension has previously been investigated in much detail for pure components.5 The approach presented herein is anticipated to lead to more refined statistical rate theory models for multicomponent interfacial transport in specific circumstances. Acknowledgment. This research was supported financially by the Natural Sciences and Engineering Research Council (NSERC) of Canada. A.K. holds an NSERC Vanier Ph.D. Scholarship. J.A.W.E. holds a Canada Research Chair in Interfacial Thermodynamics. Note Added after ASAP Publication. This paper was published on the Web on October 15, 2010. Changes were made to all figures and the corrected version was reposted on November 2, 2010. References and Notes (1) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena, 2nd ed.; John Wiley & Sons, Inc.: New York, 2006; pp 514-538, 858860. (2) Geankoplis, C. J. Mass Trasport Phenomena; Holt, Rinehart and Winston, Inc.: Austin, TX, 1972; pp 23-33, 300-304. (3) Ward, C. A.; Fang, G. Expression for predicting liquid evaporation flux: Statistical rate theory approach. Phys. ReV. E 1999, 59, 429–440. (4) Fang, G.; Ward, C. A. Examination of the statistical rate theory expression for liquid evaporation rates. Phys. ReV. E 1999, 59, 441–453. (5) Kapoor, A.; Elliott, J. A. W. Nonideal statistical rate theory formulation to predict evaporation rates from equations of state. J. Phys. Chem. B 2008, 112, 15005–15013. (6) Persad, A. H.; Ward, C. A. Statistical rate theory Examination of Ethanol Evaporation. J. Phys. Chem. B 2010, 114, 6107–6116. (7) Ward, C. A. Rate of gas absorption at a liquid interface. J. Chem. Phys. 1977, 67, 229–235. (8) Ward, C. A.; Findlay, R. D.; Rizk, M. Statistical rate theory of interfacial transport. I. Theoretical development. J. Chem. Phys. 1982, 76, 5599–5605. (9) Ward, C. A. Effect of concentration on the rate of chemical reactions. J. Chem. Phys. 1983, 79, 5605–5615. (10) Elliott, J. A. W.; Ward, C. A. Statistical rate theory description of beam-dosing adsorption kinetics. J. Chem. Phys. 1997, 106, 5667–5676. (11) Elliott, J. A. W.; Ward, C. A. Temperature programmed desorption: A statistical rate theory approach. J. Chem. Phys. 1997, 106, 5677–5684.
Kapoor and Elliott (12) Elliott, J. A. W.; Ward, C. A. Statistical rate theory and the material properties controlling adsorption kinetics on well defined surfaces. In Studies in Surface Science and Catalysis. Vol. 104: Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, 1997; pp 285-333. (13) Tikuisis, P.; Ward, C. A. Rate of gas absorption by liquids and surface resistance. in Transport Processes in Bubbles, Drops and Particles; Chhabra, R., Dekee, D., Eds.; Hemisphere: New York, 1992; pp 114-132. (14) Ward, C. A.; Tikuisis, P.; Tucker, A. S. Bubble evolution in solutions with gas concentrations near the saturation value. J. Colloid Interface Sci. 1986, 113, 388–398. (15) Ward, C. A.; Farahbakhsh, B.; Venter, R. D. Absorption rate at the hydrogen-metal interphase and its relation to the inferred value of the diffusion coefficient. Z. Phys. Chem. Neue Folge 1986, 147, 89–101. (16) Ward, C. A.; Findlay, R. D. Statistical rate theory of interfacial transport. III. Predicted rate of non-dissociative adsorption. J. Chem. Phys. 1982, 76, 5615–5623. (17) Findlay, R. D.; Ward, C. A. Statistical rate theory of interfacial transport. IV. Predicted rate of dissociative adsorption. J. Chem. Phys. 1982, 76, 5624–5631. (18) Ward, C. A.; Elmoselhi, M. Molecular adsorption at well-defined gas-solid interfaces: Statistical rate theory approach. Surf. Sci. 1986, 176, 457–475. (19) Rudzinski, W.; Panczyk, T. Kinetics of isothermal adsorption on energetically heterogeneous solid surfaces: A new theoretical description based on the statistical rate theory of interfacial transport. J. Phys. Chem. B 2000, 104, 9149–9162. (20) Rudzinski, W.; Plazinski, W. Kinetics of solute adsorption at solid/ solution interfaces: A theoretical development of the empirical pseudofirst and pseudo-second order kinetic rate equations, based on applying the statistical rate theory of interfacial transport. J. Phys. Chem. B 2006, 110, 16514–16525. (21) Azizian, S.; Bashiri, H.; Iloukhani, H. Statistical rate theory approach to kinetics of competitive adsorption at the solid/solution interface. J. Phys. Chem. C 2008, 112, 10251–10255. (22) Azizian, S.; Bashiri, H. Description of desorption kinetics at the solid/solution interface based on the statistical rate theory. Langmuir 2008, 24, 13013–13018. (23) Dejmek, M.; Ward, C. A. A statistical rate theory study of interface concentration during crystal growth or dissolution. J. Chem. Phys. 1998, 108, 8698–8704. (24) Torri, M.; Elliott, J. A. W. A statistical rate theory description of CO diffusion on a stepped Pt(111) surface. J. Chem. Phys. 1999, 111, 1686– 1698. (25) Skinner, F. K.; Ward, C. A.; Bardakjian, B. L. Permeation in ionic channels: a statistical rate theory approach. Biophys. J. 1993, 65, 618–629. (26) Elliott, J. A. W.; Elmoazzen, H. Y.; McGann, L. E. A method whereby Onsager coefficients may be evaluated. J. Chem. Phys. 2000, 113, 6573–6578. (27) Chapman, S.; Cowling, T. G. The Mathematical Theory of NonUniform Gases, 3rd ed.; Cambridge University Press: Cambridge, UK, 1970; pp 312-319. (28) Cercignani, C. Mathematical Models in Kinetic Theory, 1st ed.; Plenum Press: New York, 1969; pp 101-124. (29) Fuller, E. N.; Schettler, P. D.; Giddings, J. C. A new method for prediction of binary gas-phase diffusion coefficients. Ind. Eng. Chem. 1966, 58, 18–27. (30) Gilliland, E. R. Diffusion coefficients in gaseous systems. Ind. Eng. Chem. 1934, 26, 681–685. (31) Toor, H. L. Diffusion in 3-component gas mixtures. AIChE J. 1957, 3, 198–207. (32) Hsu, H. W.; Bird, R. B. Multi-component diffusion problems. AIChE J. 1960, 6, 516–524. (33) Elliott, J. R.; Lira, C. T. Introductory Chemical Engineering Thermodynamics; Prentice Hall: Upper Saddle River, NJ, 1999; pp 264272, 292. (34) Fang, G.; Ward, C. A. Temperature measured close to the interface of an evaporating liquid. Phys. ReV. E 1999, 59, 417–428. (35) McGaughey, A. J. H.; Ward, C. A. Temperature discontinuity at the surface of an evaporating droplet. J. Appl. Phys. 2002, 91, 6406–6415.
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