Vapor-Liquid Equilibrium Thermodynamics of N 2 + OH 4" Model and Titan Applications
W . R E I D T H O M P S O N !
Laboratory for Planetary Studies, Space Sciences Building, Cornell University, Ithaca, New York 14853
A N D
J O H N A . Z O L L W E G A N D D A V I D H . G A B I S
Laboratory for Chemical Thermodynamics, School of Chemical Engineering, Cornell University, Ithaca, New York 14853
Received October 9, 1990; revised March 12, 1992
Calculations of the vapor-liquid equilibrium thermodynamics of the N 2 + C H 4 system show that the tropospheric clouds of Titan are not pure CH 4, but solutions of CH 4 containing substantial quantities of N 2 . The conditions for saturation, latent heat of condensation, and droplet composition all depend on this equilib- rium. We present a thermodynamic model for vapor-liquid equi- librium in the N 2 + CH 4 system which, by its structure, places strong constraints on the consistency of experimental equilibrium data, and confidently embodies temperature effects by also includ- ing enthalpy (heat of mixing) data. Selected equilibrium and en- thalpy data are used in a maximum likelihood determination of model parameters. The model can be readily evaluated to compute the saturation criteria, composition of condensate, and latent heat in Titan's atmosphere for a given pressure-temperature ( p - T ) profile. For a nominal p - T profile, the partial pressure of CH4 required for formation of C H 4 + N 2 condensate is -20% lower than that required to saturate pure CH4, and -25% higher than that which would be computed by Raoult's law. N 2 constitutes 16-30% of the cloud condensate, and higher altitude clouds are generally more N2-rich. The N 2 content of condensate is i of that computed from Raouit's law and about 30% greater than that computed from Henry's law. Heats of condensation are -10% lower than for pure CH4. Above 14 km altitude, the liquid solution becomes metastable with respect to a solid solution containing less N 2 : freezing of liquid droplets will be accompanied by the exsolu- tion of about 30% of the dissolved N2, probably leading to an underdense, porous texture. The refractive index, single-scattering albedo, and density ofCH 4 + N 2 cloud droplets of the appropriate composition and phase should be used in modeling and spacecraft planning studies for Titan. Cassini investigations with sufficient altitude resolution (primarily Huygens probe experiments) can potentially detect vertical motion of particles by determining whether condensate and gas are in local thermodynamic equ i l i b r i um. ~ 1992 Academic Press, Inc.
187
INTRODUCTION
Ti tan ' s A t m o s p h e r e 2
Prior to the Voyager encounters with Saturn and its satellites, the presence of C H 4 had been known for de- cades (Kuiper 1944). While its abundance was uncertain, the strengths of the observed bands implied an equivalent base pressure of at least 15 mbar (Trafton 1972). Pressure broadening of the 1. l-/zm band indicated the p resence of other undetected gases (Lutz et al. 1976), and models of the possible photolysis of N H 3 early in T i tan ' s history (Atreya et al. 1978, Chang et al. 1979) suggested that N 2 was a likely candidate. Voyager measu remen t s p roved N 2 to be an important const i tuent along with C H 4 : prominent ionospheric emission by N 2 detec ted by the Voyager UVS exper iment (Broadfoot et al. 1981) along with a molecular mass =28 Da implied by the combined constraints of the radio occultat ion (RO) exper iment and IRIS measure- ments (Tyler et al. 1981, Hanel et al. 1981) showed the a tmosphere to be N2-dominated with a few percent of C H 4 . At an altitude z = 1400 km, the gas-phase C H 4 mole fraction Yen4 = 8% (Smith et al. 1982), while the combined constraints of the IRIS-obse rved C H 4 band and RO limit s t ratospheric YCH 4 to 1.0-1.7% for an a tmosphere com- posed only of Nz and C H 4 (Lel louch et al. 1989). For some solutions of Ti tan ' s range of al lowed a tmospher ic thermal s t ructures (Lellouch et al. 1989), a mean molecular mass consis tent with the presence of an additional heavier com- ponent is implied: if present , this consti tuent is expected
To w h o m cor respondence should be addressed.
2 For more background see Hun ten et al. (1984) and T h o m p s o n (1985).
to be Ar. The range of Ycn4 is 0.5-3.4% if Ar is present. (There is no direct evidence for Ar; the UVS upper limit at high altitudes suggests only a mole fraction <40% in the troposphere (ibid.).)
The extreme temperature ranges allowed are 70.5 K < T < 74.5 K for the tropopause (at pressure p = 0. I bar) and 93.1 K < T < 100.6 K for the surface (at pressure p 1.5 bar) (Lellouch et al. 1989). Titan's surface temperature decreases by only - 2 K from the equator to the poles (Flasar et al. 1981). Thus, knowledge of the properties of Titan's atmospheric gases and condensates over a temper- ature range of 70 to -100 K is required.
The atmosphere at the surface is in equilibrium with oceans or lakes of C 2H 6 + C H 4 + N 2 solution (Lunine et al. 1983, Thompson 1985) which contain other dissolved solutes, both organic and inorganic (Raulin 1987, Dubou- loz et al. 1989). The composition of the surface liquid and the surface value of Ycu4 are strictly coupled (Thompson 1985) but neither is well constrained at present. If Ycn4 >~ 2% in Titan's troposphere, condensation of clouds will occur through some altitude range. The presence of C H 4
in condensed form is necessary to produce a satisfactory match to Titan's IRIS-measured thermal emission spec- trum from 200 to 600 cm- ~ (Courtin 1982, Thompson and Sagan 1984, Toon et al. 1988, McKay et al. 1989). While both large column densities of radius r = 0.1 ~m droplets and much lower column densities of r ~ 100 ~m droplets can improve the match to the Voyager IRIS spectra (Toon et al. 1988), radiative balance and microphysical argu- ments suggest that diffuse clouds with large droplets are more likely (Toon et al. 1988, McKay et al. 1989). These clouds are actually not pure CH 4 , but a solution of N 2 in CH 4 (Thompson 1985, Thompson et al. 1990), which will be liquid for T > 80.6 K (z ~< 14 km) in the troposphere but, depending on the degree of supercooling, will be a solid solution at higher altitudes.
Equi l ibr ium T h e r m o d y n a m i c s o f N 2 + CH4
To be able to constrain the saturation profile of Titan's atmosphere and compute the equilibrium between the gas and the cloud condensate, it is necessary to derive a reli- able thermodynamic model for binary equilibrium in the N 2 + C H 4 system. The N 2 + C H 4 system is substantially nonideal (Thompson et al. 1990). A simple model for the C2H 6 + C H 4 + N 2 ternary was obtained by Thompson (1985), but it is not very good for calculations of the N, + CH 4 binary because of the inaccuracies introduced by the very limited ternary data set. The binary was mod- eled using a regular solution formulation by Dubouloz et al. (1989), but shows systematic deviations from experi- mental measurements o f - 10%. An empirical representa- tion fit to mutually consistent data between the C H 4 triple
point of 90.68 K and 105.0 K was derived by Thompson et al. (1990). This model is quite accurate and simple to use, but does not make full use of the constraints imposed by thermodynamic consistency. Kouvaris and Flasar (1991) have used an integration of the Gibbs-Duhem equa- tion to compute the vapor composition along isopleths (lines of constant composition). This method is thermody- namically rigorous but requires accurate experimental data at the temperatures where it is to be applied, and is rather difficult to utilize for general calculations.
Many of the limitations and possible experimental bi- ases of previous models can be overcome by formulating a model in which the temperature dependence is built-in, and in which both the experimental data at a given T and the T-dependence of the model are strongly constrained by thermodynamic theory. The confidence of T-depen- dence in a model can be greatly improved by simultane- ously utilizing vapor-liquid equilibrium (VLE) and heat of mixing (excess enthalpy, H E) measurements. Obtaining a reliable T-dependence in the model is especially im- portant since data that can be rigorously evaluated for consistency only exist above 90.68 K, while most or all of Titan's cloud condenses at lower temperatures.
In this paper, using an accurate nonideal equation of state for N 2 + C H 4 gas, selected VLE data demonstrated to be thermodynamically consistent, and a maximum like- lihood fitting technique which uses both VLE data and H E measurements, we derive a robust model for the N 2 + CH 4 system which can be readily utilized in accurate calculations for low-T (~< 125 K) outer planet applications.
THERMODYNAMICS AND MODEL
The free energy of mixing of n moles of an ideal liquid is n AG l = n R T Z i X ~ In X/, where R is the universal gas constant and Xs is the liquid-phase mole fraction of species i. For real solutions, thermodynamic excess quantities are used to express the deviations from ideality: AGto t =
AG ~ + G E. The excess free energy of mixing for the liquid is defined as
nG E = n R T ~ X i l n y , = n ~ X , l ~ E, (1) i i
where '~i is the activity coefficient and/x~ is the excess chemical potential of component i.
The partitioning of molecules between the liquid and gas phase is determined by nonideal interactions in both phases. For ideal gaseous and liquid states the equilibrium would be expressed by Pi = YiP = Xip~ at (Raoult's law) where Pi is the partial pressure of species i, Yi is the mole fraction of i in the gas, and pSat is the vapor pressure of pure i. The more general expression is
THERMODYNAMICS OF N: + C H 4 ON TITAN 189
~biYiP = ~/;Xip~ at' ,
m
where y; y~®(0, p) and p~at' = sat sat sat = " Pi q~i O ( p i , 0 ) . ¢~i is the fugacity coefficient of the gas__at pressure p, 4~ at is 4~i at pressure psat, and the terms O(0, p) and O(p sat, 0), respectively, correct Yi and ~)sat to standard reference states (see Prausnitz e t al. 1967). (The overall Poynting correct ion O(p TM, p) --~" O(0, p )O(p sat, 0).)
Modeling the thermodynamics of VLE usually consists of adopting a form for G E (motivated by some combination of theory and practical utility) and using a form of the Gibbs -Duhem relation
0 R T In T i = i~ Ei = -~ni [nGE]r.p,,i
to derive activity coefficients. When the Yi are in hand, iter- ative techniques (see Prausnitz e t al. 1967) can be used to find the mole fractions Xi and Yi consistent with a given T and p. For a system in phase equilibrium, the Gibbs phase rule states that for nc components (i) and np phases, the number of degrees of f reedom 0 is
0 = n c -- n p + 2.
A two-component system with a gas phase and one liquid phase is then completely defined by p and T, whereas n c - 2 additional quantities (some of the X i, Yi, or related values) must be known to fully determine the state of two- phase mult icomponent systems.
E q u a t i o n s o f S t a t e a n d V a p o r P r e s s u r e
While other equations of state can be employed for the nonideal gas (see Kidnay e t al. 1985), we use the familiar virial equation of state which, including terms up to sec- ond order, can be written as
Z ~ p v / R T = 1 + Bmix o-! + Cmix 0-2, (5)
where Z is the compressibili ty factor, v is the molar vol- ume, and the gas mixture virial coefficients are
Bmi x = y 2 B l l + 2Y~ Y2BI2 + y2B22,
Cmi x = y3CII I -{-- 3 y 2 y 2 c , 1 2 + 3 y I y 2 c 1 2 2 + Y~C222 •
(2) components Ciii using the method of Orbey and Vera (1983), and employ an extension of the Prausnitz (1969) combining rules to compute the cross-coefficients. See the Appendix for a detailed description of the calculation of virial coefficients.
The vapor pressures of the pure components are com- puted using the equations and constants derived by Igles- ias-Silva et al. (1987). Their method accurately spans the range between triple point and critical point temperatures . At the temperatures relevant here, the computed vapor pressures for N2 agree closely with those computed by the third-order virial expressions of Brown and Ziegler (1980), so the two methods for computing p ~ are nearly
(3) equivalent. The same is true for CH 4 down to its triple point, T,p = 90.68 K, but in Titan 's a tmosphere condensa- tion to the liquid state occurs down to 80.6 K. Computing saturation conditions below the CH4 triple point requires a vapor pressure equation accurate for the supercooled liquid. While the parameter izat ion of Brown and Ziegler has no constraining principle for extrapolat ion through the triple point, the Iglesias-Silva equation is constrained to match a theoretical asymptot ic form at Ttp, and there- fore should more accurately predict the vapor pressure of
(4) the supercooled liquid. Details of the calculation of vapor pressures are given
in the Appendix.
(6)
F u n c t i o n a l F o r m f o r G E
We use a three-term Redl ich-Kis te r expansion (Redlich et al. 1952) to obtain a functional form for G E in terms of the liquid mole fractions Xi;
G E = R T X I X 2 [ a + b ( X l - X 2) + c ( X I - X2)2], (Va)
where we denote N 2 by subscript 1 and CH 4 by sub- script 2. 3
In order to describe the thermodynamics of the system accurately over a wide range of temperatures we allow the three constants a, b, and c to be functions of in- verse T:
a = a o + a l T - l + a2 T - 2 ,
b = b 0 + b 1T-I, (Vb)
C = C O + Cl T - 1 .
We have found that the computat ional strategy of Hay- den and O'Connell (1975) yields second virial coefficients (B;j) that correspond closely to experimental measure- ments. Because of this good agreement and the limitations of experimental data (especially for cross-coefficients at relevant temperatures) , we use that method to calculate all B i f s . We compute the third virial coefficients for pure
The corresponding form for H E is obtained using the Gibbs-Helmhol tz relation
3 The form of Eq. 7a derives from the boundary condition G E = 0 at Xg = 0 (i = 1,2). The simplest equation obeying this condition is G E = aRTXtX 2, a "one-term" equation. To fully account for nonideal effects in the Nz + CH4 system, we find that a three-term expansion with temperature-dependent coefficients is required.
190 T H O M P S O N , Z O L L W E G , A N D G A B I S
H E [ O(GE/T)] =- k Jp,x
= R T X I X 2 [ a ' + b ' ( X l - X 2 ) + c ' ( X ] - X2):], (8)
w h e r e a ' = a l T i + 2 a 2 T - 2 , b, = b t T ~,andc ' = c i T l . The forms for the activity coefficients obtained from Eqs. (7) and (3) are
l n y I = X~[a - (1 - 4Xl)b + (1 - 8X I + 12X~)c]
= X2[(a + 3b + 5c) - 4(b + 4 c ) X 2 + 12cX~],
l n y 2 = X~[a + (1 - 4X2)b + (1 - 8 X 2 + 12X~)c]
= X~[(a - 3b + 5c) + 4(b - 4 c ) X 1 + 12cX~].
(9)
Once the constants are determined, these yg can be used to achieve i terative solutions of the vapor- l iquid equilibrium (Eq. 2).
M a x i m u m L i k e l i h o o d F i t t i n g M e t h o d
A method commonly used to fit V L E data in a thermodynamica l ly consistent way is that of Barker (1953), where the analytical express ion for G E is related to total pressure via Eqs. (3) and (9), or some analogous express ion, and (p , T, X) data are fit in a least-squares sense to obtain the paramete rs in the G E representat ion. For our representa t ion of G E, if we take y~ -~ Yi and p y _ ~ p~at, the Barker method is equivalent to a least- squares fit to the equat ion 4
Ptot =
(Xt /Ol )p~ ~t' exp{X~[(a + 3b + 5c) - 4(b + 4 c ) X 2 + 12cX~]}
Here we replace the simple least-squares approach of the standard Barker method with a maximum likelihood estimation of the parameters which allows for uncertain- ties in all measured quantities. This approach varies the parameters and measured quantities to maximize the probability that the set of observations will be obtained given the model and the measurement uncertainties (Skjold-Jorgensen 1983). As with simpler least-squares methods, systematic errors within the data will bias the
4 Note that the term RT should be deleted from the three-term Red- lich-Kister form given in Eq. (9) of Thompson et al. (1990), making it identical to the middle quantities in Eq. (9) above, while the total pres- sure expression for the case of an ideal gas in their Eq. (10) should read PT = XIP ° exp[X~(a - (I - 4Xi)b + ...)] + X2p ° exp[X~(a + (1 - 4X,.)b + ...)].
parameter est imates , so we examine the deviat ions of pressures and composi t ions f rom the model and retain only data which are thermodynamica l ly consistent .
An additional advantage of this model is its ability to accommoda te H E data s imultaneously with V L E mea- surements . H E data strongly constrain the t empera ture dependence of G E through (8) and facilitate the detect ion of, and/or limit the bias of, V L E data which are inconsis- tent with a smooth t empera tu re dependence of G E (see Kidnay et al. 1985). (For a detailed description of the model contact J. A. Zollweg).
EXPERIMENTAL DATA
An extensive review and evaluation of the thermody- namics of N2 + CH4 in the gas and liquid phases was per formed by Kidnay et al. (1985). In this s tudy thermody- namic consis tency tests were used to evaluate V L E data: departures of the computed quanti ty (GE/T)x_o.5 f rom the curve predicted by excess enthalpy (H E) measu remen t s (see Eq. 8) identify inconsistent data. We have both screened the data with an initial test and examined the point-by-point deviations of candidate exper imenta l mea- surements f rom our model predictions: since our model is by nature thermodynamica l ly consistent , this serves as a similar but more detailed test than that used by Kidnay et al. We find that the data of McClure e t al. (1976) at 90.68 K, of Parrish and Hiza (1974) at 95.0, 100.0, 105.0, and l l0.0 K, of Kidnay et al. (1975) at 112.0 K, and of Stryjek et al. (1974) at 113.71 K provide the most consis- tent collection for a low tempera ture model . With the except ion of the addition of Kidnay et al. (1975), these sources form a subset of those judged consis tent by Kid- nay et al. (1985). In our model , H E data are fit simultane- ously with the V L E data: we use the H E measuremen t s of McClure et al. (1976) at 91.5 and 105.0 K along with the selected V L E data in the m a x i m u m likelihood calcula- tions.
THE N 2 + CH 4 THERMODYNAMIC MODEL
P a r a m e t e r V a l u e s a n d M o d e l - E x p e r i m e n t
C o m p a r i s o n s
The max imum likelihood values of the pa ramete r s and their approximate uncertainties assuming no correlat ions (the square roots of the diagonal e lements of the covari- ance matrix) are shown in Table I. The model predict ions and residuals are compared with selected exper imenta l data relevant to Titan condit ions in Fig. I. The deviat ions of the data used in the analysis are small and/or random, indicating good consis tency. We list the roo t -mean-square deviations for all the data sets utilized in Table II.
THERMODYNAMICS OF N2 + CH4 ON TITAN 191
TABLE I T h e r m o d y n a m i c Model Pa ramete r s for the
N 2 - C H 4 System
Parameter Value Covariance ½
TABLE II Deviations for Individual Data Sets
T 8Prm s K bar ~Xrm s 8 Yrms
a o 0.8096 0.0181 a I - 52.07 3.56 a z 5443. 175. b o -0.0829 0.0111 b I 9.34 1.04 c o 0.0720 0.0268 c~ -6 .27 2.57
FIG. 1. Comparison of experimental data and model predictions. (a) Total pressure versus composition, pressure deviations, and composition deviations. Open symbols represent vapor compositions Yi and filled symbols represent liquid compositions X i at total pressure (p). (b) Excess enthalpy versus composition, and enthalpy deviations. See text for references appropriate to each temperature.
192 T H O M P S O N , Z O L L W E G . A N D GABIS
N , + CH4 Be low the CH+ Triple Poin t
There are no VLE measurements along isotherms (and therefore amenable to the present analysis) below the CH 4 triple point T~p = 90.68 K. Such data are difficult to obtain. 5 Yet much of the altitude range at which condensation may occur in Titan's atmosphere lies below this temperature: for the nominal model of Lellouch et al. (1989) the surface temperature is 93.9 K, and 90.68 K is reached just below z = 3 kin. Condensation to the liquid state occurs up to z = 14 km, T = 80.6 K. Above this altitude the liquid is metastable with respect to a solid solution (Thompson 1985, Thompson et al. 1990).
There are data which span (p, T) regimes relevant to Titan, although their form prevents direct assessment of thermodynamic consistency and inclusion in our model. Omar et al. (1962) and Fuks and Bellemans (1967) plot total pressure Ptot versus T along lines of constant composition (isopleths). The data of Omar et al. are more extensive, with Plot ranging from about I bar down to the vapor-liquid line. Kouvaris and Flasar (1991) have computed vapor compositions along iso- pleths by integration of the Gibbs-Duhem equation, using selected data from Omar et al. along with isopleths estimated from interpolated isothermal data to define the paths.
The strong constraints on temperature dependence imposed by the functional form and the inclusion of H E data in our model should allow it to be used with reasonable confidence at these lower temperatures. In particular, we can compute the model-predicted iso- pleths at the compositions studied by Omar et al. (1962) and assess the differences. The results are shown in Fig. 2. Some of the isopleths agree well with the model, while others show large deviations. The XN+ = 0.037 isopleth deviates greatly from the model; Kouvaris and Flasar also found these data to be suspect, and did not include them in their analysis. Both the position and the implied slope of the sequence of points for XN+ = 0.128 also deviate substantially. For the other isopleths the experimental points match the model lines better. Because several of the isopleths agree very well with the computed lines and the offsets do not seem to be systematic, we feel this comparison validates the model for Titan applications both above and below 90.68 K-- in fact, the model seems reliable down to the N2-CH ~ eutectic at 62.5 K. The systematic offsets of some isopleths of Omar et al. from the model calcula-
One could s tudy part of the composi t ion range by starting with pure N 2 and adding CH+, but for T < T~p solidification would be encountered at a sufficiently high value ofXcH + (at the univariant triple point). Calcula- t ions of equilibria below T~p also require extrapolat ion of the CH4 liquid vapor pressure into the solid field.
FIG. 2. Compar i son of model predict ions with the resul ts of Omar et al. (1962). Plot is total pressure versus tempera ture : lines and symbols show paths of cons tant liquid composi t ion in p T space. Heavy lines are the vapor pressure curves for pure N 2 (left) and CH4 (right). Symbols are the data of Omar et al. Light lines are the total pressure curves computed for the same composi t ions reported by Omar et al. Our model, which does not rely on the data of Omar et al. or any data below 90.68 K, predicts the equil ibrium condit ions well even at very low tempera tures (see text for a d iscuss ion of the residual offsets).
tions suggest that those isopleths actually correspond to different liquid phase compositions than Omar et al. reported. We believe this results from a substantial uncertainty in measurements of the liquid phase compo- sition in their work.
APPLICATIONS OF THE MODEL
Gas Saturat ion and Cloud Compos i t ion in Titan's A t m o s p h e r e
We now use the model to compute the equilibrium between the gas and condensates in Titan's atmosphere. The p - T profile for Titan's atmosphere has been re- ported by Lindal et al. (1983) and reanalyzed with allowances for atmospheric Ar and CH+ content by Lellouch et al. (1989). The nominal profile computed by Lellouch et al. for the Ar-free case is very similar to that of Lindal et al., on which we base the saturation and composition profiles here. Our model describes the N2-CH 4 binary and does not include Ar. Lellouch et a[. included Ar in their equilibrium condensation model, but their regular solution model is relatively inaccurate for all species (Kouvaris and Flasar 1991).
THERMODYNAMICS OF N2 + CH4 ON TITAN
T A B L E I I I
G a s and Liquid Proper t ies for N o m i n a l A t m o s p h e r i c Profile
193
sat 0 A H V ¢bs. at 8 m l z C m i x v Z z T p B i Ci ~bi Pi -,
km K bar cm3/mol cmS/mol bat (plat,p) J/mol cm3/mol cmS/mol cm3/mol
N o t e . For each (z, T, p), quantities for pure N 2 or the gas mixture are given on the first line, followed by quantities for pure CH4 on the second line. AH v (= - AHCi ") are heats of vaporization.
Since there is no direct evidence for Ar and its presence would not produce major changes in the results, we leave its inclusion for further work.
Pure component and gas mixture properties along the nominal p - T profile are shown in Table IlI. We compute the vertical profiles of C H 4 saturation mole fraction
Ycn4 and of condensate composition XN2 by iteration from the surface up to the tropopause. The surface value of YcH4 is probably controlled by the composition of primarily C2H6-CH4-N 2 liquid at Titan's surface (Lunine et al. 1983, Thompson 1985, Dubouloz et al. 1989). A minimum of 700 m of C 2 H 6 would be produced
1 9 4 T H O M P S O N , Z O L L W E G , A N D G A B I S
T A B L E IV C o m p o s i t i o n a n d H e a t o f C o n d e n s a t i o n for N o m i n a l
A t m o s p h e r i c Profi le
z T p Yc°n4 p'c~ 4 Ycn 4 X u 2 7 u 2 7c.q 4 H ~ A H c km K bar bar J / tool J / tool
Note. ~.n4 is the saturation mole fraction for pure CH 4, while YcH~ is the saturation mole fraction above the equilibrium solution. Condensation in rising gas parcels ceases above z = 28 kin.
altitude (below 80.6 K) freezing should occur, so the fur- ther increase Of XN2 to its maximum of 0.29 at z = 20 km applies only to the metastable l iquid- - the solubility of N: in the solid is typically 8-10% less in this region (Omar et al, 1962, see Thompson 1985). (Ar may further reduce the
1.022 61.3 -7860. freezing point, but probably not by large amounts. See 1.024 64.3 -7856. Van't Zelfde et al. (1968) for the A r - C H 4 solid-l iquid 1.026 67.4 -7852. phase diagram.) In a rising gas parcel, condensat ion con- 1.028 71.3 -7839. 1.031 74.8 -7833. tinues to occur up to z = 28 km (T = 73.5 K); above this 1.036 81.9 -7812. level, gaseous C H 4 retains its minimum value YCH 4 = 1.040 87.2 -7803. 1.041 89.5 -7819. 0 . 0 2 0 into the stratosphere. 1.045 95.1 -7807. In Fig. 4 we compare the results of our model with the 1.054 105.3 -7782. 1.063 115.3 -7762. empirical representation of Thompson et al. (1990), with 1.068 121.7 -7771. the results plotted by Kouvaris and Flasar (1991) resulting 1.075 129.1 _ 7 7 6 8 . from their integration along isopleths, and with the simpler 1.081 135.6 -7772.
1.088 142.7 -7768. estimates which result from assuming an ideal solution 1.003 148.7 -7775 (Raoult's law) or (for N, solubility) a gas dissolving spar- 1.084 145.0 -7858. 1.088 150.8 -7869. ingly in a solvent (Henry's law). In Fig. 4a we examine 1083 150.8 -7925 the predictions of Yc~ 4. The saturation values predicted 1.072 145.9 -8014.
by Thompson et al. (1990) are virtually identical to those computed from our detailed model, and the results of Kouvaris and Flasar (1991) for YCH4 are also similar. The calculations of Kouvaris and Fiasar are subject to the accuracy of Ptot and liquid composit ions in individual data sets, while direct parametric modeling of the VLE data without implicit constraints as in Thompson et al. (1990)
over geologic time at current photochemical rates (Yung et al. 1984), but since we do not know the equivalent depth or composit ion of the oceans, YCH 4 is limited only by the constraint that the T profile of the troposphere is less steep than a wet adiabat or pseudoadiabat, which would seem to limit condensat ion to altitudes above - 5 km. ~
The Yen4 and X y 2 profiles are shown in Table IV and Fig. 3; we also show the value l~CH4 that would apply if N~ were not accounted for. Throughout the troposphere the gaseous mole fraction of C H 4 required for condensa- tion is 10-20% less than that required to condense pure CH 4 . The N, mole fraction in the liquid XN~ = 0.16 at the surface and increases to 0.28 at z = 14 km. Above this
For purposes of computation, we start with a value of L,,H~ which exceeds the saturation value at the surface. The equilibrium is computed by an iterative method which starts with XcH, = I, computes activity coefficients Yi (Eq . 9), computes new liquid composit ions [from Eq. 2, Xi = Yi+iP/('Y~PY')], and then iterates until self-consistency is achieved. If ~ X ~ > 1, the gas is supersaturated and a second iteration starts with the computed XN, and iterates %, X N , and Y(,H4 until a stable solution with ~iXi - 5~iY,-= 1 is found. The saturation composit ions Y, and X i are then determined. The computed YcH 4 serves as the starting value for the next higher altitude. Algorithms for the calculation of saturation conditions, liquid and vapor composit ions , and latent heats are available f rom the authors.
40 ¸
35 ¸
30
E 25
-o 20
~ 1 5
10
5
0 0.00
- - - X ( N 2 , 1 i q u i d ) - - Y ( C H 4 , r n o d e l )
- - - Y ( C H 4 , p u r e ) - - - X ( N 2 , m e t a - l i q )
- - - X ( N 2 , s o l i d )
/ /
'\ / /
( /
/ /
/ , /
0.10
Y (CH4)
0 . 2 0 0 . 3 0
X ( N 2 )
FIG. 3. C H 4 saturation mole fraction and condensate composit ion versus altitude for the nominal p - T profile. Gas-phase saturation mole fractions above pure CH4, Y(CH4,pure ) , and above lhe equilibrium C H 4 + N 2 c o n d e n s a t e , Y(CH4,mode l ) , are plotted on the left. On the right the mole fraction of N , in the condensate predicted by our model, X(N2,1iquid), and its extension to altitudes where the liquid is metastable against freezing, X(N2,meta-l iq) , are shown. The equilibrium composi- tion of the solid solution, X(N2,sol id) , from Thompson (1985) is also shown.
THERMODYNAMICS OF N 2 + CH 4 ON TITAN 195
a
30
25
~: 20 "x,
e" -o15-
. - -
< 1 0
.
C ] , .
0.00
Li l t - - - - P u r e C H 4 [ . . . . . I d e a l S o l ' n
- - - T H 9 0 \ ---- KF91 \ - - T h i s Model
\
\ \
' , ~ X X \ \
• , . . . , . . . . . . . , • . . , . . .
0.02 0.04 0.06 0.08 0.10 O.
50 ¸
25
E 20
d . 015 . - I - -
- . I - -
< 1 0
0 2 0.10
, . - .... Ideal $ol'n ,
'X)~ i --z--TH90KF91Henry's Low ',, This Model
/ f ...-- 0.20 0.30 0.40 0.50
Y(CH4) X(N2)
FIG. 4. Comparisons of pure-component properties, simple models, and the results of Thompson et al. (1990), Kouvaris and Flasar (1991) (hereafter referred to as TH90 and KF91, respectively), and the present model. (a) Comparisons of CH 4 saturation mole fractions YCH 4. Profiles for pure CH4, an ideal solution (Raoult's Law), the TH90 parameterization, the KF91 numerical integration, and the present model are shown. TH90 and the present results agree very closely. (b) Comparisons of N 2 mole fractions in the condensate X N . Profiles for an ideal solution (Raoult's Law), Henry's law (see Kidnay et al. 1985), TH90, KF91, and the present model are shown. Deviations i f TH90 and KF91 are more noticeable for liquid-phase compositions.
is more subject to biases caused by possible inaccuracies in the experimentally difficult measurement of equilibrium gas phase composition. While all three models do a rea- sonable job of predicting gas-phase saturation conditions, our present model is more strongly constrained and less prone to biases caused by high experimental YCH4'S or other errors inherent in particular data sets. The actual YCH4 is about 20% lower than that which would be in equilibrium with pure CH 4, and 25% higher than that which would be computed from Raoult's law.
In Fig. 4b we show the vertical profile of XN2 in the condensate. An ideal solution would have about 100% more N2, while Henry's law would predict about 30% less than that computed from the models. The empirical parameterization of Thompson et al. (1990) is close to the line computed from our detailed model, matching it at the surface but progressively underestimating XN2 toward higher altitudes. The results of Kouvaris and Flasar (199 l) are generally close, but deviate by _+5-10% at higher altitudes. This is probably due to the inaccuracies in XN.2 in the data of Omar et al. (1962) seen in Fig. 2. (Kouv- arm and Flasar note that their integration can still estimate Yi well, even with modest errors in Xi.) We believe the temperature dependence and consistency built into our model allows it to predict both X~ and Yi well at low temperatures, without the inaccuracies in one or the other caused, in the other models, by strong dependence on problematic or sparse data.
Entha lpy o f Condensa t i on (La ten t Hea t )
For an ideal solution the enthalpy of condensation A H c = E i X i A H c , where AH/c is the value for pure i (Brown and Ziegler 1980, Table III). The actual value A H c = A/-ff + H E, where H E is given by Eq. 8. In Fig. 5 we show the altitude profile of AH c in Titan's atmosphere. Because IAHC2[ < [AHCH41 , the latent heat of condensa- tion of an ideal CH 4 + N 2 solution is lower in magnitude (here, by about 10%) than that of pure CH4. Since H E > 0 (the computed excess enthalpy is endothermic), the actual value of IAH c] is lower still. AN: increases with altitude sufficiently rapidly to reverse the sense of altitude dependence of AH c compared to the trend for pure CH 4 .
IMPLICATIONS
Our thermodynamically constrained model is well suited to providing a best current estimate of saturation conditions, condensate compositions, and latent heats in Titan's atmosphere; these can be readily computed, or read from Table IV for the nominal atmospheric p - T pro- file. We have also shown the levels of accuracy of the simpler parameterization of Thompson et al. (1990) and the partially parameterized numerical method of Kouvaris and Flasar (1991). (Because the model of Thompson et al. (1990) implicitly includes gas-phase nonideality and standard-state corrections within the parameters, its use may still be expeditious in some calculations.)
FIG. 5. Latent heats of condensation versus altitude for the nominal p - T profile. Results for pure CH4, an ideal solution having the composi- tion of CH~ + N 2 equilibrium condensate, and the present model are shown. The magnitude of &H c is about 10% less than for pure CH 4.
We list several types of studies which, as they reach a given level of sophistication, need to allow for the detailed behavior of the CH 4 + N~_ system.
• The saturation profile is changed, influencing the im- plications of models of CH 4 band structure in the near-infrared, where interband minima sample deep into the troposphere (Griffith e t al . 1992). Limits on the abundance of CH4 in the lower atmosphere placed by the T profile in the lowermost few km (Eshleman e t al . 1983) are affected because of a different wet adiabatic lapse rate. Cloud microphysical and radia- t ive-convective models (Toon e t al. 1988, McKay e t al . 1989) are affected by the reduction of latent heat and of the quantity of C H 4 gas thermal opacity in the troposphere.
• The refractive index and absorption properties of the cloud particles are changed. For a given amount of cloud thermal opacity in the C H 4 far-infrared colli- sion-induced rotational transition (Thompson and Sa- gan 1984, Toon e t al. 1988), the corresponding cloud mass will increase (Thompson 1985). The contribu- tion to N= thermal opacity by dissolved N 2 in the cloud is small, but scattering models will be mildly influenced by the higher real, and by the lower (CH4- dominated) imaginary, index of refraction of the cloud droplets.
• Conditions for nucleation and growth of cloud drop- lets are changed; a new complexity to cloud micro- physics and precipitation is added by the fact that condensate falling through the atmosphere finds itself
in disequilibrium with its surroundings, even if the atmosphere is locally saturated.
• New ramifications of certain measurements planned for the Cassini Huygens probe arise. The composi- tions of liquid tropospheric cloud droplets can be calculated from their refractive index, and compared with measured gaseous compositions to determine whether droplets are in local thermodynamic equilib- rium. Departures will provide at least some con- straints on growth versus sedimentation rates. Also, through the various effects listed above, modeling of the results of other Cassini probe and orbiter investi- gations will be intertwined with the vapor-liquid equilibrium in the atmosphere.
Titan presents a unique environment for study, with complex cloud thermodynamics, substantial expanses of liquid hydrocarbons probable at the surface (Lunine e t al.
1983, Thompson 1985, Dubouloz e t al . 1989), and many organic species, including hydrocarbons and nitriles, present as atmospheric gases (cf. Thompson e t al. 1991), stratospheric condensates (cf. Sagan and Thompson 1984, Frere and Raulin 1990), minor ocean/lake solutes (Dubou- loz e t al. 1989), and surface sediments (Thompson e t al.
1989; Thompson and Sagan 1992). New experimental ef- forts continue to be needed to provide accurate laboratory data of several types, so that physical models can confi- dently advance our understanding of this unique world.
A P P E N D I X
E q u a t i o n o f S t a t e C a l c u l a t i o n s
We use a second-order virial equation of state (Eq. 5l, with the virial coefficients of the mixture Bmi ~ and Cmi~ defined (Eq. 6) in terms of pure- component and cross-coefficients Bit and C~i k . Here we provide a means of calculating those coefficients.
We use the method of Hayden and O'Connell (1975) (HO75) to calcu- late all second virial coefficients and cross-coefficients. HO75 represent the interactions embodied in B in the form
Btot~,t Bfr~, + Bmc~,~,bl~_bouo d = btl{.['(T.) t A exp[~H/(kT/e)]}.
All of the above terms (in addition to Boltzmann's constant k and temperature T) are expressible as functions of four constants: dipole moment /x, radius of gyration R' , critical temperature T~, and critical
pressure Pc- Computationally,
b o 27r/3N~ltr ~
where No is Avogadro 's number, and the molecular interaction potential parameter
o- = (2.44 - oJ')(Tclpc) I'~.
where
o~' = 0,006R' + 0.02087R '2 - 0.00136R '3,
so b 0 = f (R ' , T~, Pc). The function
T H E R M O D Y N A M I C S OF N 2 + CHa ON TITAN 1 9 7
f ( T . ) = 0.94 - 1 .47T, 1 - 0 .85T. 2 + 1.015T. 3,
where
T . t = e / k T - 1.6to'
and the (other) molecular interaction potential parameter
e = kT~(0.748 + 0.91to')
so that f ( T . ) = f ( T ; R ' , T~). For the second term,
A = - 0 . 3 - 0.05/% and AH = 1.99 + 0.2/x2.,
where the reduced dipole momen t
tL , = 1~2/t30 '-3,
so (with the equat ions above) A and AH = f(p. , R ' ,Tc , Pc). The four cons tan t s needed to calculate pure -component second virial
coefficients B, are listed in Table V. The cross-coefficients B~ are com- puted in the same way, except that the intermediate parameters in the computa t ion e, tr, to', and p.. are computed from pure -component values according to the mixing rules:
e,j = 0.7(e,ej) + 0.6(ei -t + e f I) l,
o',j = (o'io)) t/2,
tob = o.5(to; + toj) ,
~**,,j = m#/(%o-~).
[Our equat ion for % corrects Eq. (32) in HO75. Also, note that a l though the virial expans ion used by HO75 is Z = 1 + B ( p / R T ) , the second (but not higher) virial coefficients are identical in the v -l and ( p / R T ) virial expans ions . ]
We compute the third virial coefficients for pure componen ts Cii i using the method of Orbey and Vera (1983). They express these in the form
\ P c /
T A B L E V
Parameters and Constants for N2-CH 4 V L E C a l c u l a t i o n s
Nitrogen Methane
Tc, K 126.20 190.53 Pc, bar 34.002 45.955 V c, cm 3 mo1-1 89.80 99.20 Ttp, K 63.15 90.68 Pry, bar 0.1252 0.1170 co 0.039 0.011 R ' , A 0.55 1.12 tz 0.0 0.0 a 4 3.065972 3.159023 b 0 - 23.52451 - 19.36816 b i 6103.604 8799.140
where Tr ~ T/Tc is the reduced tempera ture and the funct ions F0 and 'Fi are given by
Fo(T,) = 0.01407 + 0.02432/T~ 8 - 0.00313/TI °5
F I ( T r) = -0 .02676 + 0.01770/T~ "8 + (/.040/T~ '°
_ 0.003/T 6.o _ 0.00228/T~ °.5,
so C,~ = f ( T ; to, Tc, p~). The "acent r ic fac tor" to is also tabulated in Table V.
There is no simple analog of the method of Hayden and O 'Connel l for comput ing the third virial cross-coefficients . In o ther modeling, we have found that an extens ion of the Prausni tz combining rules (Prausni tz 1969) can be used to define pseudo-crit ical parameters :
L,iij = (T~J~f L
2toi + toi toi(j - - 3
2Zc. i + Zc,i Zc'iiJ - 3
R Tc,iiaZc,i6 Pc.i(/ -- Vc,i 0
which we employ in the Orbey and Vera (1983) method to compute Cu2 and C m . At the low tempera tures relevant here, the equat ion of state is much more sensi t ive to the second (B) coefficients than to the third (C) coefficients.
Finally, note that the virial equat ion of state can also be writ ten
P u Z = ~ = 1 + B ' ( P / R T ) + C ' ( P / R T ) 2.
As already stated, the second virial coefficients are the same: B ' = B. However , the third virial coefficients in a P / R T e x p a n s i o n are related to those in a 1/v expans ion by C ' = C - B 2.
Vapor Pressure Equation
Here we use a vapor pressure equat ion derived from "ex t ended asymptot ic behav io r" by lglesias-Silva et al. (1987) (IS87). This method is essential ly an interpolation technique between the critical point and the triple point, where the equat ion must satisfy theoretical constraints at these two ex t remes .
The fundamenta l form is
q(t) = (qo(t)~' + q~(t)x) I/x,
where
q(t) =- 1 + p(t) - Pro Pc -- Ptp
T - Ttp t - - - -
L - Lp'
the ~p subscr ipt represents triple point condi t ions , and the ~ subscript represents critical point condit ions, qo(t) is the asympto t ic form of the
198 T H O M P S O N , Z O L L W E G , A N D GABIS
vapor p ressure near the triple point, and q~(t) is the form near the critical point. The parameter ized asympto t ic forms chosen by 1S87 are
qo(t) = a,i + al(al t + l)%'e exp / \ k - a : + _ b _ d R I a f t + 1 /
q~(t) = 2 - a4(1 - t) + as(I - t) 2-~'1
+ a6(1 t) 3 + a7(1 - t) 4.
The identities
a o = I - Pto/(Pc - Pry)
al = - ( a l l - l)e": bo/R
a , = bl/RTtp
a3 = CT¢ - Lp) /Lr
formally reduce the number of parameters to eight: N, 6), bo, bl , a4. a 5 ,
a 6, and a 7. IS87 chose
N = 87Ttp/T ~ and ® - 0.2
and infer the relat ionships
a~ = -0 .11599104 + 0.29506258a4 - 0.00021222a]
a 6 = -0 .01546028 + 0.08978160a] - 0.05322199a~
a 7 = 0.05725757 - 0.06817687a 4 + 0.00047188a],
reducing the number of parameters to three: a4, b0, and b~, so that eventual ly q = f ( t ; a4, b0, b I ; Pc, Ptp, To, Ttp). The latter seven parame- ters and cons tan t s are listed in Table V.
A C K N O W L E D G M E N T S
We thank F. M. Flasar and J. C. G. Calado for helpful d iscuss ions and the reviewers for useful suggest ions . This work was supported by the N A S A Planetary A tmosphe re s Program through Grant N A G W - 1444.
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