ICARUS 97, 187--199 (1992)
Vapor-Liquid Equilibrium Thermodynamics of
N 2 + OH 4"
Model and Titan Applications
W. REID THOMPSON !
Laboratory for Planetary Studies, Space Sciences Building, Cornell University, Ithaca, New York 14853
AND
JOHN A. ZOLLWEG AND DAVID H. GABIS
Laboratory for Chemical Thermodynamics, School of Chemical Engineering, Cornell University, Ithaca, New York 14853
Received October 9, 1990; revised M a r c h 12, 1992
INTRODUCTION
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 equilibrium. We present a thermodynamic model for vapor-liquid equilibrium 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 including enthalpy (heat of mixing) data. Selected equilibrium and enthalpy 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 exsolution 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
equilibrium.
Titan's Atmosphere 2
Prior to the V o y a g e r e n c o u n t e r s with Saturn and its
satellites, the p r e s e n c e of C H 4 had b e e n k n o w n for decades (Kuiper 1944). While its a b u n d a n c e was uncertain,
the strengths of the o b s e r v e d bands implied an equivalent
base pressure of at least 15 m b a r (Trafton 1972). Pressure
broadening of the 1. l-/zm band indicated the p r e s e n c e of
other undetected gases (Lutz et al. 1976), and models of
the possible photolysis of N H 3 early in T i t a n ' s history
(Atreya et al. 1978, Chang et al. 1979) suggested that N 2
was a likely candidate. V o y a g e r m e a s u r e m e n t s p r o v e d N 2
to be an important constituent along with C H 4 : p r o m i n e n t
ionospheric emission by N 2 detected by the V o y a g e r UVS
e x p e r i m e n t (Broadfoot et al. 1981) along with a molecular
mass =28 Da implied by the c o m b i n e d constraints of the
radio occultation (RO) e x p e r i m e n t and I R I S measurements (Tyler et al. 1981, H a n e l e t al. 1981) s h o w e d the
a t m o s p h e r e to be N2-dominated with a few p e r c e n t of
C H 4 . At an altitude z = 1400 km, the g a s - p h a s e C H 4 mole
fraction Yen4 = 8% (Smith et al. 1982), while the c o m b i n e d
constraints of the I R I S - o b s e r v e d C H 4 band and R O limit
stratospheric YCH4 to 1.0-1.7% for an a t m o s p h e r e composed only of Nz and C H 4 (Lellouch et al. 1989). F o r some
solutions of T i t a n ' s range of allowed a t m o s p h e r i c thermal
structures (Lellouch et al. 1989), a mean molecular mass
consistent with the p r e s e n c e of an additional heavier component is implied: if present, this constituent is expected
To w h o m c o r r e s p o n d e n c e should be addressed.
2 F o r more background see H u n t e n et al. (1984) and T h o m p s o n (1985).
~ 1992AcademicPress, Inc.
187
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Copyright © 1992 by Academic Pressl Inc.
All rights of reproduction in any form reserved.
188
THOMPSON, ZOLLWEG, AND GABIS
to be Ar. The range of Ycn4 is 0.5-3.4% if Ar is present. point of 90.68 K and 105.0 K was derived by Thompson
(There is no direct evidence for Ar; the UVS upper limit et al. (1990). This model is quite accurate and simple to
at high altitudes suggests only a mole fraction <40% in use, but does not make full use of the constraints imposed
the troposphere (ibid.).)
by thermodynamic consistency. Kouvaris and Flasar
The extreme temperature ranges allowed are 70.5 K <
(1991) have used an integration of the Gibbs-Duhem equaT < 74.5 K for the tropopause (at pressure p = 0. I bar) tion to compute the vapor composition along isopleths
and 93.1 K < T < 100.6 K for the surface (at pressure p
(lines of constant composition). This method is thermody1.5 bar) (Lellouch et al. 1989). Titan's surface temperature namically rigorous but requires accurate experimental
decreases by only - 2 K from the equator to the poles data at the temperatures where it is to be applied, and is
(Flasar et al. 1981). Thus, knowledge of the properties of rather difficult to utilize for general calculations.
Titan's atmospheric gases and condensates over a temperMany of the limitations and possible experimental biature range of 70 to - 1 0 0 K is required.
ases of previous models can be overcome by formulating
The atmosphere at the surface is in equilibrium with a model in which the temperature dependence is built-in,
oceans or lakes of C 2 H 6 + C H 4 + N 2 solution (Lunine et and in which both the experimental data at a given T and
al. 1983, Thompson 1985) which contain other dissolved the T-dependence of the model are strongly constrained
solutes, both organic and inorganic (Raulin 1987, Dubou- by thermodynamic theory. The confidence of T-depenloz et al. 1989). The composition of the surface liquid and dence in a model can be greatly improved by simultanethe surface value of Ycu4 are strictly coupled (Thompson ously utilizing vapor-liquid equilibrium (VLE) and heat
1985) but neither is well constrained at present. If Ycn4 >~ of mixing (excess enthalpy, H E) measurements. Obtaining
2% in Titan's troposphere, condensation of clouds will a reliable T-dependence in the model is especially imoccur through some altitude range. The presence of C H 4 portant since data that can be rigorously evaluated for
in condensed form is necessary to produce a satisfactory consistency only exist above 90.68 K, while most or all
match to Titan's IRIS-measured thermal emission spec- of Titan's cloud condenses at lower temperatures.
trum from 200 to 600 c m - ~(Courtin 1982, Thompson and
In this paper, using an accurate nonideal equation of
Sagan 1984, Toon et al. 1988, McKay et al. 1989). While state for N 2 + C H 4 gas, selected VLE data demonstrated
both large column densities of radius r = 0.1 ~m droplets to be thermodynamically consistent, and a maximum likeand much lower column densities of r ~ 100 ~m droplets lihood fitting technique which uses both V LE data and
can improve the match to the Voyager IRIS spectra (Toon H E measurements, we derive a robust model for the N 2
et al. 1988), radiative balance and microphysical argu+ CH 4 system which can be readily utilized in accurate
ments suggest that diffuse clouds with large droplets are calculations for low-T (~< 125 K) outer planet applications.
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
THERMODYNAMICS AND MODEL
CH 4 (Thompson 1985, Thompson et al. 1990), which will
be liquid for T > 80.6 K (z ~< 14 km) in the troposphere
The free energy of mixing of n moles of an ideal liquid
but, depending on the degree of supercooling, will be a
is
n AG l = n R T Z i X ~ In X/, where R is the universal gas
solid solution at higher altitudes.
constant and Xs is the liquid-phase mole fraction of species
i. For real solutions, thermodynamic excess quantities
E q u i l i b r i u m T h e r m o d y n a m i c s o f N 2 + CH4
are used to express the deviations from ideality: AGto t =
To be able to constrain the saturation profile of Titan's AG ~ + G E. The excess free energy of mixing for the liquid
atmosphere and compute the equilibrium between the gas is defined as
and the cloud condensate, it is necessary to derive a reliable 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
C 2 H 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 modeled using a regular solution formulation by Dubouloz et
al. (1989), but shows systematic deviations from experimental measurements o f - 10%. An empirical representation fit to mutually consistent data between the C H 4 triple
nG E = nRT~Xilny,
i
= n~X,l~
E,
(1)
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:
~biYiP = ~/;Xip~ at' ,
(2)
m
where y; = y~®(0, p) and p~at'
" = Pisat q~isat O ( p i sat , 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
correction O(p TM, p) --~"O(0, p ) O ( p sat, 0).)
Modeling the thermodynamics of V L E usually consists
of adopting a form for G E (motivated by some combination
of theory and practical utility) and using a form of the
G i b b s - D u h e m relation
0
R T In T i = i~ Ei = -~ni [nGE]r.p,,i
(3)
to derive activity coefficients. When the Yi are in hand, iterative techniques (see Prausnitz e t al. 1967) can be used to
find the mole fractions X i 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 freedom 0 is
0 = n c -- n p + 2.
(4)
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 twophase multicomponent systems.
Equations
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 second order, can be written as
= 1 + Bmixo-! + Cmix0-2,
(5)
where Z is the compressibility factor, v is the molar volume, and the gas mixture virial coefficients are
Bmix = y 2 B l l + 2Y~ Y2BI2 + y2B22,
CH 4
ON TITAN
189
components Ciii using the m e t h o d of O r b e y and Vera
(1983), and employ an extension of the Prausnitz (1969)
combining rules to c o m p u t e the cross-coefficients. See
the Appendix for a detailed description of the calculation
of virial coefficients.
The vapor pressures of the pure c o m p o n e n t s are computed using the equations and constants derived by Iglesias-Silva et al. (1987). Their m e t h o d accurately spans the
range between triple point and critical point temperatures.
At the temperatures relevant here, the c o m p u t e d vapor
pressures for N2 agree closely with those c o m p u t e d by
the third-order virial expressions of B r o w n and Ziegler
(1980), so the two methods for computing p ~ are nearly
equivalent. The same is true for C H 4 down to its triple
point, T,p = 90.68 K, but in Titan's atmosphere condensation 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 parameterization o f B r o w n and Ziegler
has no constraining principle for extrapolation through
the triple point, the Iglesias-Silva equation is constrained
to match a theoretical asymptotic form at Ttp, and therefore should more accurately predict the vapor pressure o f
the supercooled liquid.
Details of the calculation o f vapor pressures are given
in the Appendix.
Functional Form for G E
We use a three-term R e d l i c h - K i s t e r expansion (Redlich
et al. 1952) to obtain a functional form for G E in terms of
the liquid mole fractions Xi;
of State and Vapor Pressure
Z ~ pv/RT
+
(6)
Cmix = y3CII I -{-- 3 y 2 y 2 c , 1 2 + 3 y I y 2 c 1 2 2 + Y~C222•
We have found that the computational strategy of Hayden and O'Connell (1975) yields second virial coefficients
(B;j) that correspond closely to experimental measurements. 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 c o m p u t e the third virial coefficients for pure
GE
=
RTXIX2[a
+ b ( X l - X 2) + c ( X I - X2)2],
(Va)
where we denote N 2 by subscript 1 and C H 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 inverse T:
a = a o + a l T - l + a2 T - 2 ,
b = b 0 + b 1T-I,
(Vb)
C = CO + Cl T - 1 .
The corresponding form for H E is obtained using the
G i b b s - H e l m h o l t z 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 GE =
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
THOMPSON, ZOLLWEG, AND GABIS
HE
[ O(GE/T)]
=- k
Jp,x
= RTXIX2[a'
+ b'(X l
-
X2)
+
c'(X]
-
X2):], (8)
w h e r e a ' = a l T i + 2 a 2 T - 2 , b, = b t T ~ , a n d c ' = c i T l .
The f o r m s 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]
= X 2 [ ( a + 3b + 5c) - 4(b + 4 c ) X 2 + 12cX~],
(9)
l n y 2 = X~[a + (1 - 4 X 2 ) b + (1 - 8 X 2 + 12X~)c]
p a r a m e t e r estimates, so we examine the deviations of
pressures and c o m p o s i t i o n s f r o m the model and retain
only data which are t h e r m o d y n a m i c a l l y consistent.
An additional advantage of this model is its ability to
a c c o m m o d a t e H E data simultaneously with V L E measurements. H E data strongly constrain the t e m p e r a t u r e
d e p e n d e n c e of G E through (8) and facilitate the detection
of, and/or limit the bias of, V L E data which are inconsistent with a smooth t e m p e r a t u r e d e p e n d e n c e of G E (see
Kidnay et al. 1985). (For a detailed description of the
model contact J. A. Zollweg).
EXPERIMENTAL DATA
= X~[(a - 3b + 5c) + 4(b - 4 c ) X 1 + 12cX~].
Once the constants are determined, these yg can be used to
achieve iterative solutions of the v a p o r - l i q u i d equilibrium
(Eq. 2).
Maximum
Likelihood Fitting Method
A m e t h o d c o m m o n l y used to fit V L E data in a
t h e r m o d y n a m i c a l l y consistent w a y is that of Barker
(1953), where the analytical expression for G E is related
to total pressure via Eqs. (3) and (9), or some analogous
expression, and (p, T, X) data are fit in a least-squares
sense to obtain the p a r a m e t e r s in the G E representation.
F o r our representation of G E, if we take y~ -~ Yi and
p y _ ~ p~at, the B a r k e r m e t h o d is equivalent to a leastsquares fit to the equation 4
Ptot =
( X t / O l ) p ~ ~t' exp{X~[(a + 3b + 5c) - 4(b + 4 c ) X 2 + 12cX~]}
+ (XJ4~z)p) ac exp{X~[(a - 3b + 5¢.) + 4(b - 4c)X~ + 12cX~]}.
(10)
Here we replace the simple least-squares approach of
the standard Barker method with a maximum likelihood
estimation of the parameters which allows for uncertainties 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 Redlich-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 pressure 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 + ...)].
An extensive review and evaluation of the thermodynamics of N2 + CH4 in the gas and liquid phases was
p e r f o r m e d by Kidnay et al. (1985). In this study t h e r m o d y namic consistency tests were used to evaluate V L E data:
departures of the c o m p u t e d quantity (GE/T)x_o.5 from the
curve predicted by excess enthalpy ( H E) m e a s u r e m e n t s
(see Eq. 8) identify inconsistent data. We have both
screened the data with an initial test and e x a m i n e d the
point-by-point deviations of candidate e x p e r i m e n t a l measurements f r o m our model predictions: since our model
is by nature t h e r m o d y n a m i c a l l y consistent, this serves as
a similar but m o r e detailed test than that used by K i d n a y
et al. We find that the data of McClure e t al. (1976) at
90.68 K, of Parrish and H i z a (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 m o s t consistent collection for a low t e m p e r a t u r e model. With the
exception of the addition of Kidnay et al. (1975), these
sources form a subset of those judged consistent by Kidnay et al. (1985). In our model, H E data are fit simultaneously with the V L E data: we use the H E m e a s u r e m e n 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 calculations.
THE N 2 + CH 4 THERMODYNAMIC MODEL
Parameter Values and Model-Experiment
Comparisons
The m a x i m u m likelihood values of the p a r a m e t e r s and
their a p p r o x i m a t e uncertainties assuming no correlations
(the square roots of the diagonal elements of the covariance matrix) are s h o w n in Table I. The model predictions
and residuals are c o m p a r e d with selected experimental
data relevant to Titan conditions in Fig. I. The deviations
of the data used in the analysis are small and/or random,
indicating good consistency. We list the r o o t - m e a n - s q u a r e
deviations for all the data sets utilized in Table II.
191
THERMODYNAMICS OF N2 + CH4 ON TITAN
TABLE II
Deviations for Individual Data Sets
TABLE I
T h e r m o d y n a m i c Model P a r a m e t e r s for the
N 2 - C H 4 System
Parameter
Value
Covariance ½
T
K
8Prms
bar
~Xrms
8 Yrms
ao
aI
0.8096
- 52.07
5443.
-0.0829
9.34
0.0720
-6.27
0.0181
3.56
175.
0.0111
1.04
0.0268
2.57
90.68
95.00
100.00
105.00
110.00
112.00
113.71
0.0042
0.0041
0.0033
0.0200
0.0200
0.0202
0.0642
0.0025
0.0233
0.0174
0.0296
0.0251
0.0543
0.0114
-0.0134
0.0112
0.0046
0.0071
0.0265
0.0225
az
bo
bI
co
c~
Pressure vs. Composition
a
0.8
0.6
•
•
•
I
'
I
'
I
90.68 K
• look
"
b
I
'
..-."~
9sK
150
•
t
~ -
•
@
0.4
I
91.5 K
"
I
f--
'
i
"
I
"
~
100
E
0.2
O.0~
0.0
E x c e s s E n t h a l p y vs. C o m p o s i t i o n
r~
,
I
0.2
I
J
I
0.4
0.6
so
I
0.8
0
0.0
Mole f r a c t i o n n i t r o g e n
!
I
I
I
I
I
0.2
0.4
0.8
o.8
Mole f r a c t i o n n i t r o g e n
Pressure deviations
0 - 0 0 0 8 I~"
=
I
Excess Entalpy deviations
|
0.0004~"
• •a
eA&
•
- 0 . 0 0 0 8 ~-
I
0.0
i~
n
I
0,2
i
n
•
i
I
0.4
i
.
--
A
n
I
0.6
0.8
"°
Mole f r a c t i o n n i t r o g e n
'0- . 0
i
02
i
04
'
0.6
"=
' 0.8'
1.0
Mole f r a c t i o n n i t r o g e n
Composition deviations
i
0.004
0.002~
o
I
•
I
i
•
•
[] m
•
[]
Q
-~' OO(~
-c)
~
-o.oo2r
-0.004 "
0.0
i.
0 2
l,
I
0.4
i
I
0.6
I
I
0.8
~
I
1.0
Mole f r a c t i o n n i t r o g e n
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
THOMPSON, ZOLLWEG. AND GABIS
N , + CH4 B e l o w the CH+ Triple P o i n 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 isopleths 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 isopleths 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 - - i n 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 calculaOne could study part of the composition 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). Calculations of equilibria below T~p also require extrapolation of the CH4 liquid
vapor pressure into the solid field.
Pressure vs. temperature
'
0.12
0.10
0.08
mole
[]
•
0
•
•
z~
•
fraction
0.037
0.128
0.197
0.262
0.500
0.604
0.750
I
N2
~0.06
z
0.04
0.02
0.00
60
I
I
70
I
I
80
t
I ~
90
100
T/K
FIG. 2.
C o m p a r i s o n of model predictions with the results of O m a r
et al. (1962). Plot is total pressure v e r s u s t e m p e r a t u r e : lines and s y m b o l s
show paths of constant liquid composition in p
T space. H e a v y lines
are the vapor pressure curves for pure N 2 (left) and CH4 (right). S y m b o l s
are the data of O m a r et al. Light lines are the total pressure c u r v e s
c o m p u t e d for the same compositions reported by O m a r et al. Our model,
which does not rely on the data of O m a r et al. or any data below
90.68 K, predicts the equilibrium conditions well even at very low
temperatures (see text for a discussion 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 composition in their work.
APPLICATIONS OF THE MODEL
Gas Saturation a n d C l o u d C o m p o s i t i o n 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 reported 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).
193
T H E R M O D Y N A M I C S OF N2 + CH4 ON T I T A N
TABLE III
G a s a n d Liquid P r o p e r t i e s for N o m i n a l A t m o s p h e r i c Profile
z
T
p
Bi
Ci
km
K
bar
cm3/mol
cmS/mol
0.0 94.0 1.500
0.5 93.3 1.460
1.0 92.6 1.420
1.5 91.9 1.390
2.0 91.2 1.350
3.0 89.9 1.280
4.0 88.9 1.220
5.0 88.1 1.150
6.0 87.1 1.090
8.0 85.3 0.981
10.0 83.6 0.879
12.0 82.2 0.785
14.0 80.8 0.701
16.0 79.5 0.625
18.0 78.2 0.556
20.0 77.0 0.494
22.0 76.2 0:438
24.0 75.1 0.389
26.0 74.2 0.344
28.0 73.5 0.304
30.0 72.9 0.269
32.0 72.3 0.238
34.0 71.9 0.210
36.0 71.7 0.185
38.0 71.4 0.163
".
40.0 71.2 0.144
-179. 2.110E+02
-485. -6.019E+05
-181. -2.200E+02
-495. -6.525E+05
-184. -6.980E+02
-505.-7.076E+05
-186. -1.227E+03
-515.-7.678E+05
-189. -1.813E+03
-525. -8.335E+05
-194. -3.070E+03
-546. -9.722E+05
-198. -4.211E+03
-563. -1.096E+06
-201. -5.250E+03
-577. -1.207E+06
-206. -6.730E+03
-596. -1.363E+06
-214. -9.998E+03
-634. -1.703E+06
-222. -1.398E+04
-673. -2.109E+06
-229. -1.809E+04
-709. -2.523E+06
-236. -2.315E+04
-748. -3.026E+06
-244. -2.890E+04
-789. -3.593E+06
-252. -3.590E+04
-833. -4.277E+06
-259. -4.371E+04
-878. -5.036E+06
-265. -4.978E+04
-911. -5.623E+06
-273. -5.947E+04
-959. -6.556E+06
-279. -6.874E+04
-1002. -7.445E+06
-285. -7.693E+04
-1037. -8.227E+06
-289. -8.471E+04
-1069. -8.970E+06
-294. -9.328E+04
-1103. -9.786E+06
-298. -9.948E+04
-1127.-1.037E+07
-300. -1.027E+05
-1139. -1.068E+07
-302. -1.078E+05
-1158. -1.117E+07
-304. -1.114E+05
-1170. -1.150E+07
~bi
Pisat
bat
0.885
0.989
0.888
0.989
0.892
0.990
0.896
0.991
0.900
0.991
0.906
0.992
0.911
0.993
0.915
0.993
0.920
0.994
0.928
0.995
0.936
0.996
0.941
0.996
0.947
0.997
0.952
0.997
0.956
0.998
0.960
0.998
0.963
0.998
0.966
0.998
0.968
0.999
0.970
0.999
0.972
0.999
0.973
0.999
0.974
0.999
0.975
0.999
0.976
0.999
0.976
0.999
0
AH V
¢bs.at
-,
(plat,p) J/mol
5.0066
0.1772
4.7373
0.1629
4.4788
0.1495
4.2306
0.1371
3.9926
0.1255
3.5766
0.1061
3.2790
0.0929
3.0543
0.0833
2.7896
0.0725
2.3566
0.0560
1.9956
0.0433
1.7308
0.0348
1.4934
0.0277
1.2958
0.0223
1.1188
0.0177
0.9723
0.0143
0.8831
0.0123
0.7709
0.0100
0.6875
0.0084
0.6276
0.0073
0.5796
0.0064
0.5345
0.0057
0.5059
0.0052
0.4921
0.0050
0.4719
0.0047
0.4588
0.0045
0.983
1.006
0.984
1.006
0.985
1.006
0.986
1.006
0.987
1.006
0.988
1.005
0.990
1.005
0.990
1.005
0.991
1.005
0.993
1.005
0.994
1.004
0.995
1.004
0.996
1.003
0.996
1.003
0.997
1.003
0.997
1.003
0.998
1.002
0.998
1.002
0.998
1.002
0.998
1.002
0.998
1.002
0.998
1.001
0.998
1.001
0.998
1.001
0.998
1.001
0.998
1.001
4853.
8637.
4890.
8655.
4927.
8673.
4963.
8691.
4999.
8708.
5061.
8735.
5109.
8763.
5147.
8784.
5191.
8806.
5266.
8842.
5334.
8882.
5389.
8917.
5441.
8949.
5488.
8978.
5534.
9007.
5577.
9034.
5606.
9055.
5642.
9076.
5674.
9097.
5699.
9113.
5721.
9127.
5741.
9139.
5756.
9149.
5764.
9155.
5774.
9161.
5781.
9166.
0.966
0.922
0.966
0.922
0.966
0.923
0.966
0.923
0.966
0.923
0.967
0.924
0.967
0.925
0.968
0.927
0.969
0.928
0.970
0.931
0.972
0.934
0.974
0.938
0.975
0.941
0.977
0.944
0.978
0.947
0.980
0.951
0.982
0.955
0.983
0.957
0.984
0.961
0.986
0.964
0.987
0.967
0.988
0.970
0.989
0.973
0.991
0.976
0.992
0.978
0.993
0.981
8ml z
Cmi x
v
cm3/mol
cmS/mol
cm3/mol
Z
-202.1 -9.860E+03
4998. 0.9592
-203.4 -1.030E+04
5099. 0.9597
-204.9 -1.078E+04
5206. 0.9602
-206.3 -1.122E+04
5280. 0.9605
-207.9 -1.182E+04
5398. 0.9611
-211.1 -1.306E+04
5618. 0.9620
-214.0 -1.428E+04
5834. 0.9629
-216.8-1.567E+04
6142. 0.9643
-220.0 -1.727E+04
6413. 0.9653
-226.4 -2.082E+04
6993. 0.9672
-233.2 -2.514E+04
7663. 0.9691
-239.5 -2.984E+04
8456. 0.9713
-246.3 -3.544E+04
9327. 0.9732
-253.1 -4.180E+04
10312. 0.9751
-260.3 -4.939E+04
11423. 0.9768
-267.5 -5.790E+04
12682. 0.9785
-273.0 -6.533E+04
14182. 0.9804
-280.4 -7.584E+04
15761. 0.9819
-287.0 -8.640E+04
17637. 0.9835
-292.6 -9.635E+04
19800. 0.9850
-297.8 -1.062E+05
22226. 0.9864
-303.1 -1.170E+05
24946. 0.9877
-307.2 -1.262E+05
28152. 0.9889
-310.1 -1.336E+05
31908. 0.9902
-313.8 -1A29E+05
36100. 0.9912
-316.9 -1.514E+05
40788. 0.9921
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
THOMPSON,
194
ZOLLWEG,
TABLE IV
C o m p o s i t i o n a n d H e a t of 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 Profile
z
T
p
km
K
bar
0.0
0.5
1.0
1.5
2.0
3.0
4.0
5.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
22.0
24.0
26.0
28.0
30.0
32.0
34.0
36,0
38,0
40,0
94.0
93.3
92.6
91.9
91.2
89.9
88.9
88.1
87.1
85.3
83.6
82.2
80.8
79.5
78.2
77.0
76.2
75.1
74.2
73.5
72.9
72.3
71.9
71.7
71.4
71.2
1.500
1.460
1.420
1.390
1.350
1.280
1.220
1.150
1.090
0.981
0.879
0.785
0.701
0.625
0.556
0.494
0.438
0.389
0.344
0.304
0.269
0.238
0.210
0.185
0.163
0.144
Yc°n4
p'c~4
Ycn 4
Xu 2
7u 2
0.109
0.103
0.096
0.090
0.084
0.074
0.067
0.064
0.058
0.049
0.042
0.037
0.033
0.029
0.026
0.023
0.023
0.021
0.020
0.020
0.020
0.020
0.020
0.020
0.020
0.020
0.161
0.167
0.173
0.181
0.188
0.202
0.212
0.214
0.224
0.242
0.260
0.268
0.278
0.287
0.297
0.303
0.287
0.291
0.283
0.264
1.839
1.830
1.821
1.807
1.797
1.774
1.759
1.762
1.748
1.723
1.700
1.696
1.687
1.683
1.676
1.675
1.724
1.730
1.766
1.828
7c.q4
bar
0.118
0.112
0.105
0.099
0.093
0.083
0.076
0.072
0.067
0.057
0.049
0.044
0.040
0.036
0.032
0.029
0.028
0.026
0.024
0.024
0.024
0.024
0.025
0.027
0.029
0.031
0.164
0.150
0.137
0.125
0.113
0.095
0.082
0.073
0.063
0.048
0.037
0.029
0.023
0.018
0.014
0.011
0.010
0.008
0.007
0.006
H ~
J/tool
AH c
J/tool
1.022
1.024
61.3 -7860.
64.3 -7856.
1.026
1.028
1.031
67.4
71.3
74.8
-7852.
-7839.
-7833.
1.036
1.040
1.041
81.9
87.2
89.5
-7812.
-7803.
-7819.
1.045
1.054
1.063
95.1
105.3
115.3
-7807.
-7782.
-7762.
1.068 121.7 -7771.
1.075
1.081
1.088
129.1
135.6
142.7
_7768.
-7772.
-7768.
1.003
1.084
1.088
148.7
145.0
150.8
-7775
-7858.
-7869.
1083
1.072
150.8
145.9
-7925
-8014.
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.
AND GABIS
altitude (below 80.6 K) freezing should occur, so the further increase Of XN2 to its maximum of 0.29 at z = 20 km
applies only to the metastable liquid--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
freezing point, but probably not by large amounts. See
Van't Zelfde et al. (1968) for the A r - C H 4 solid-liquid
phase diagram.) In a rising gas parcel, condensation continues to occur up to z = 28 km (T = 73.5 K); above this
level, gaseous C H 4 retains its minimum value YCH4 =
0 . 0 2 0 into the stratosphere.
In Fig. 4 we compare the results of our model with the
empirical representation of T h o m p s o n et al. (1990), with
the results plotted by Kouvaris and Flasar (1991) resulting
from their integration along isopleths, and with the simpler
estimates which result from assuming an ideal solution
(Raoult's law) or (for N, solubility) a gas dissolving sparingly in a solvent (Henry's law). In Fig. 4a we examine
the predictions of Yc~ 4. The saturation values predicted
by Thompson et al. (1990) are virtually identical to those
computed from our detailed model, and the results of
Kouvaris and Flasar (1991) for YCH4are also similar. The
calculations of Kouvaris and Fiasar are subject to the
accuracy of Ptot and liquid compositions in individual data
sets, while direct parametric modeling of the VLE data
without implicit constraints as in T h o m p s o n et al. (1990)
40 ¸
over geologic time at current photochemical rates (Yung
et al. 1984), but since we do not know the equivalent
depth or composition of the oceans, YCH4 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 condensation 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 condensation 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
35 ¸
--- -
X(N2,1iquid)
Y(CH4,rnodel)
- -- X(N2,meta-liq)
--
- Y(CH4,pure)
---
30
X(N2,solid)
E 25
(
/
-o 20
'\
//
/
/
/
/
/
~15
,/
10
5
0
For purposes of computation, we start with a value of L,,H~ which
e x c e e d s the saturation value at the surface. The equilibrium is computed
by an iterative method which starts with XcH, = I, computes activity
coefficients Yi ( E q . 9), computes new liquid compositions [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 , a n d Y(,H4 until a stable solution
with ~iXi - 5~iY,-= 1 is found. The saturation compositions Y, and X i
are then determined. The computed YcH4 serves as the starting value for
the next higher altitude. Algorithms for the calculation of saturation
conditions, liquid and vapor compositions, and latent heats are available
f r o m the authors.
0.00
0.10
Y(CH4)
0.20
0.30
X(N2)
F I G . 3. C H 4 saturation mole fraction and condensate composition
versus altitude for the nominal p - T profile. Gas-phase saturation mole
fractions above pure CH4, Y ( C H 4 , p u r e ) , and above lhe equilibrium
C H 4 + N 2 c o n d e n s a t e , Y ( C H 4 , m o d e 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-liq), are shown. The equilibrium composition of the solid solution, X(N2,solid), from Thompson (1985) is also
shown.
195
THERMODYNAMICS OF N 2 + CH 4 ON TITAN
a
30
50 ¸
l
Li
t
[
25
---.....
- - -
\
----
\
~: 20
"x,
- -
,
Pure CH4
Ideal Sol'n
TH90
KF91
T h i s Model
.
- ....
Ideal
$ol'n
,
'X)~i z-T
-H90KF91LHoew
nyrs',,
This
25
Model
E 20
\
\\
e"
-o15-
d
.015
',~XX\\
. - -
.-I--
-.I--
<10
<10
.
C]
,
.
0.00
•
,
.
0.02
.
.
,
.
0.04
.
.
.
.
.
.
0.06
,
•
0.08
.
.
,
.
0.10
.
.
O. 2
Y(CH4)
0
0.10
/ f
...-0.20
0.30
0.40
0.50
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 YCH4. 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 reasonable 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.2in the data of Omar et al. (1962) seen in Fig. 2. (Kouvarm 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.
Enthalpy of Condensation (Latent Heat)
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 condensation 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 IAHc] 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 profile. 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.)
196
T H O M P S O N , Z O L L W E G , A N D GABIS
30-
~
~
25
E
.f/
/'
-----
-
-
Pure
-
CH4
Ideal
Sol'n
This
Model
I
I
]
/
2O
/
,¢
/
/
<1
/
/
/
/
/
/
/
/
/
/
5
o ..........
,,
//
i
",
/
//
". . . . . . . . . .
( ...........
-7.6 -7.8 -8.0 -8.2 -8.4 -8.6 -8.8 -9.0 -9.2
AHc, kJ/mol
FIG. 5. Latent heats of condensation versus altitude for the nominal
profile. Results for pure CH4, an ideal solution having the composition 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.
p-T
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 implications 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 radiative-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 collision-induced rotational transition (Thompson and Sagan 1984, Toon e t al. 1988), the corresponding cloud
mass will increase (Thompson 1985). The contribution 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 (CH4dominated) imaginary, index of refraction of the
cloud droplets.
• Conditions for nucleation and growth of cloud droplets are changed; a new complexity to cloud microphysics 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 compositions 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 equilibrium. Departures will provide at least some constraints on growth versus sedimentation rates. Also,
through the various effects listed above, modeling of
the results of other Cassini probe and orbiter investigations 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 (Dubouloz e t al. 1989), and surface sediments (Thompson e t al.
1989; Thompson and Sagan 1992). New experimental efforts continue to be needed to provide accurate laboratory
data of several types, so that physical models can confidently advance our understanding of this unique world.
APPENDIX
Equation
of State Calculations
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 purecomponent and cross-coefficients Bit and C~ik . Here we provide a means
of calculating those coefficients.
We use the method of Hayden and O'Connell (1975) (HO75) to calculate all second virial coefficients and cross-coefficients. HO75 represent
the interactions embodied in B in the form
Btot~,t
Bfr~, + Bmc~,~,bl~_bouod =
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,
bo
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 O F N 2 + CHa ON T I T A N
f(T.)
= 0.94 -
1.47T, 1 - 0 . 8 5 T . 2 + 1.015T. 3,
197
w h e r e Tr ~ T/Tc is the reduced t e m p e r a t u r e and the functions F0 and'Fi
are given by
where
Fo(T,) = 0.01407 + 0.02432/T~ 8
T.t = e/kT-
-
0.00313/TI °5
1.6to'
F I ( T r) = - 0 . 0 2 6 7 6 + 0.01770/T~ "8 + (/.040/T~ '°
and the (other) molecular interaction potential p a r a m e t e r
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.,
_ 0.003/T 6.o _ 0.00228/T~ °.5,
so C,~ = f ( T ; to, Tc, p~). The " a c e n t r i c f a c t o r " to is also tabulated in
Table V.
There is no simple analog of the m e t h o d of H a y d e n and O ' C o n n e l l for
computing the third virial cross-coefficients. In other modeling, we have
found that an extension of the Prausnitz c o m b i n i n g rules (Prausnitz 1969)
can be used to define pseudo-critical parameters:
where the reduced dipole m o m e n t
L,iij (T~J~f L
=
tL, = 1~2/t30'-3,
2toi
toi(j
so (with the equations above) A and AH = f(p., R',Tc, Pc).
The four c o n s t a n t s needed to calculate p u r e - c o m p o n e n t second virial
coefficients B, are listed in Table V. The cross-coefficients B~ are computed in the same way, except that the intermediate parameters in the
c o m p u t a t i o n e, tr, to', and p.. are c o m p u t e d from p u r e - c o m p o n e n t values
according to the mixing rules:
e,j = 0.7(e,ej) + 0.6(ei -t + e f I)
tob = o.5(to; + toj),
~**,,j =
Zc'iiJ -
2Zc. i + Zc,i
3
Pc.i(/ --
R Tc,iiaZc,i6
Vc,i 0
l,
o',j = (o'io)) t/2,
m#/(%o-~).
[Our equation for % corrects Eq. (32) in HO75. Also, note that although
the virial e x p a n s i o n 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
expansions.]
W e c o m p u t e the third virial coefficients for pure c o m p o n e n t s Cii i using
the m e t h o d of Orbey and Vera (1983). T h e y e x p r e s s these in the form
toi
+
3
--
which we employ in the Orbey and Vera (1983) m e t h o d to c o m p u t e Cu2
and C m . At the low t e m p e r a t u r e s relevant here, the equation of state is
m u c h more sensitive to the second (B) coefficients than to the third (C)
coefficients.
Finally, note that the virial equation of state can also be written
Pu
Z = ~ = 1 + B ' ( P / R T ) + C ' ( P / R T ) 2.
As already stated, the second virial coefficients are the same: B ' = B.
H o w e v e r , 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 e x p a n s i o n by C ' = C - B 2.
\Pc /
Vapor Pressure Equation
TABLE V
Parameters and Constants for N2-CH 4
VLE Calculations
Tc, K
Pc, bar
Vc, cm 3 mo1-1
Ttp, K
Pry, bar
co
R', A
tz
a4
b0
bi
Nitrogen
Methane
126.20
34.002
89.80
63.15
0.1252
0.039
0.55
0.0
3.065972
- 23.52451
6103.604
190.53
45.955
99.20
90.68
0.1170
0.011
1.12
0.0
3.159023
- 19.36816
8799.140
Here we u s e a vapor pressure equation derived from " e x t e n d e d
asymptotic b e h a v i o r " by lglesias-Silva et al. (1987) (IS87). This method
is essentially an interpolation technique between the critical point and
the triple point, where the equation must satisfy theoretical constraints
at these two e x t r e m e s .
The f u n d a m e n t a l form is
q(t) = (qo(t)~' + q~(t)x) I/x,
where
q(t) =- 1 + p(t) - Pro
Pc -- Ptp
Tt--
-
L
Ttp
-
Lp'
the ~p subscript represents triple point conditions, and the ~ subscript
represents critical point conditions, qo(t) is the a s y m p t o t i c form o f the
198
THOMPSON, ZOLLWEG, AND GABIS
vapor p r e s s u r e near the triple point, and q~(t) is the form near the critical
point. T h e p a r a m e t e r i z e d a s y m p t o t i c forms c h o s e n by 1S87 are
qo(t) = a,i + al(alt + l)%'e exp /\ k - a : + _ b _ d R I
aft+ 1 /
q~(t) = 2 - a4(1 - t) + as(I - t) 2-~'1
+ a6(1
t) 3 + a7(1 - t) 4.
DUROULOZ, N., F. RAULIN, E. LELLOUCH, AND D. GAUTIER 1989.
T i t a n ' s h y p o t h e s i z e d ocean properties: The influence of surface temperature and a t m o s p h e r i c composition uncertainties. Icarus 82,
81-96.
ESHLEMAN, V. R., G. F. LINDAL, AND G. L. TYLER 1983. lS Titan wet
or dry'? Science 221, 53-55.
FRERE, C., AND E. RAULIN 1992. Microphysical modeling of Titan's
aerosols, submitted for publication.
Ft, ASAR, F. M., R. E. SAMUELSON, AND B. J. CONRATH 1981. T i t a n ' s
atmosphere: T e m p e r a t u r e and d y n a m i c s . N a t u r e 292, 693-698.
The identities
a o = I - Pto/(Pc - Pry)
al = - ( a l l a,
=
l)e": bo/R
bl/RTtp
a3 = CT¢ - L p ) / L r
formally reduce the n u m b e r of p a r a m e t e r s to eight: N, 6), bo, bl , a4. a 5 ,
a 6, and a 7. IS87 c h o s e
N = 87Ttp/T ~ and
® - 0.2
and infer the relationships
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 n u m b e r of p a r a m e t e r s to three: a4, b0, and b~, so that
eventually q = f ( t ; a4, b0, b I ; Pc, Ptp, To, Ttp). The latter s e v e n parameters and c o n s t a n t s are listed in Table V.
ACKNOWLEDGMENTS
W e t h a n k F. M. Flasar and J. C. G. Calado for helpful discussions
and the reviewers for useful suggestions. This work was supported by
the N A S A Planetary A t m o s p h e r e s Program through Grant N A G W - 1444.
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