J . Phys. Chem. 1985,89, 3279-3285
0.030
I
I
10
20
30
1
I
40
60
I
60
TIME (MSEC)
Figure 3. 1(2P1,2)decay profiles following addition of IC1 to 310 mtorr of 02(lA) with 10-70 mtorr of H20. The circles, triangles, squares, and hexagons refer to 9.8, 6.5, 3.2, and 1.6 mtorr of ICl, respectively.
profile moved forward in the flow tube as the IC1 flow was increased. In this respect, the behavior of IC1 is similar to 12,which also dissociates more rapidly in larger concentrations. This result is expected whenever I(2P1/2)plays a rate-limiting role in the dissociation process. Upon adding up to 10 mtorr of IC1 to 310 mtorr of chemically generated 02('A), the I(2PI,2)profiles decayed
3279
significantly within the flow tube and the peak I(2P1/2)concentrations did not scale linearly with the initial concentration of IC1. These results implied that a substantial fraction of the initial 02(lA) was consumed in the process of dissociating the ICI. To quantify this effect, the I(zPl/2)concentrations were normalized to the IC1 flows and plotted as a logarithmic function of time in the flow tube. As shown in Figure 3, the I(2Pl/2)decays following the dissociation of IC1 were found to be exponential in character, with decay rates that were proportional to the IC1 concentration with a slope of approximately 9000 torr-' s-I. This loss of I(2P,/2) reflects a much greater loss of OZ('A), since these species are maintained in equilibrium by the rapid energy-transfer reaction. The loss of I(2P1/2)and O,(lA) at long times in the flow tube can be explained by the quenching13of I(zP1/2)by H20. There is also a substantial loss of I(2P1/2)and 0 2 ( l A ) at early tiimes in the flow tube, as can be seen in Figure 3 by the extrapolations of the exponential decays back to t = 0. The reduction of the t = 0 intercept with increasing IC1 concentrations corresponds to the deactivation of approximately 20 02('A) molecules for each IC1 molecule added to the flow. Under similar conditions, the dissociation of I2 is known to occur with a much smaller loss of 02(1A).6The inefficiency of the IC1 dissociation process may be explained either by a low branching ratio in one of the energytransfer steps or by significant deactivation or quenching of 0 2 ( l Z ) or one of the intermediate states of IC1. In summary, the dissociation of IC1 by O,(lA) appears to proceed primarily through O,('Z) as an intermediate and is markedly slower and less efficient than the dissociation of I2 by 02(lA).
Acknowledgment. This work was supported by the Air Force Weapons Laboratory under Contract AFWL-29610-82-C-0082. Registry No. ICI, 7790-99-0;02,7782-44-7; H 2 0 , 7732-18-5. (13) Grimely, A. J.; Houston, P. L. J . Chem. Phys. 1978, 69, 2339.
Langmuir Constants for Spherical and Linear Molecules In Clathrate Hydrates. Validity of the Cell Theory Vijay T. John* Department of Chemical Engineering, Tulane University, New Orleans, Louisiana 70118
and Gerald D. Holder Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261 (Received: August 13, 1984)
The Lennard-Jones and Devonshire (LJD) cell theory as applied to the calculation of Langmuir constants for spherical and linear guest molecules in hydrate cavities is critically examined. Using the crystallographic locations of the host water molecules, and modeling binary guest-host interactions by Kihara type. potentials, we carried out more precise calculations for Langmuir constants, and the results are compared to Langmuir constants obtained by smearing the host water molecules over the surface of a sphere (the LJD approach). The disparity between the two methods of calculation is especially pronounced for the large cavity of structure I hydrate, which is the most asymmetric of hydrate cavities. Further, the variation of Langmuir constants obtained from the two methods is dependent on the Kihara effective size and energy parameters used. The results may be significant to the precise determination of gas hydrate phase equilibria.
Introduction Gas Hydrates are clathrate inclusion compounds formed by water (the host) and low molecular weight gases (the guest). Thermodynamic stability of the inclusion complex is due to dispersion interactions betwen the enclathrated guest molecule and the host lattice made up of hydrogen-bonded water molecules. 0022-3654/85/2089-3279$01.50/0
Hydrates of nonpolar gases crystallize in one of two structures commonly referred to as structures 1 and 11. Lattice properties of these structures are shown in Table I which indicates that both structures are characterized by small and large cavities. The formation of either structure I or structure 11is dependent on the relative stabilities of the structures when a given gas (or gas 0 1985 American Chemical Society
3280 The Journal of Physical Chemistry, Vol. 89, No. 15, 1985
John and Holder
TABLE I: Lattice Properties of Structure I and Structure 11 Hydrates structure I structure I1
no. of large cavities per unit cell no. of small cavities per unit cell no. of water molecules per unit cell coordination no., small cavities coordination no., large cavities
6 2
46 20 24
A
Cavity Surface
8 16
136 20 28
mixture) is enclathrated at the thermodynamic conditions in effect. The theory of hydrate equilibrium was originally developed by van der Waals and Platteeuw' subject to the following assumptions: (1) The free energy of the water molecules of the crystal lattice is independent of the nature of the enclathrated gas molecule. Thii assumption is valid as long as the enclathrated molecule does not distort the lattice structure. (2) The encaged molecule is localized in its cavity and not more than one gas molecule can occupy a given cavity. (3) At the thermodynamic conditions of interest, classical statistical mechanics can be used. The fundamental equation derived from the theory is
where 1 . 1 ~is the chemical potential of water in the hydrate phase, pfi is the chemical potential potential of water in the metastable phase of empty hydrate lattices, Y, is the number of cavities of type m per water molecule,A is the fugacity of gas species i, T is the temperature, k is Boltzmann's constant, and Cim is the Langmuir constant of molecular species i in a cavity of type m. The Langmuir constant is defined as
where h is the molecular partition of the encaged species i in cavity type m, and q is the molecular partition of species i in the gas phase with the volume term excluded. The inverse relationship between C and f in eq 1 implies that accurate calculation of the Langmuir constant is necessary for precise determination of the equilibrium fugacity of species i at a given temperature. Earlier researchers1q2have used the LJD cell theory to derive analytic expressions for the Langmuir constants using the inherent assumption of the cell theory that the host molecules are "smeared" over the surface of a perfect sphere. This work examines the smoothed cell assumption and presents results of more accurate calculations for the Langmuir constants of spherical and linear guest molecules in hydrate cavities.
Theory and Discussion Langmuir Constants for Spherical Guest Molecules. For monotonic (perfectly spherical) guest molecules the Langmuir constant can be rigorously expressed (omitting subscripts) as
where W = W(rJ,r$)is the potential energy of interaction between the guest molecule and the host lattice (the cell potential) when the guest is located at positional coordinates r,O,r$. In evaluating this cell potential, we have assumed that the binary interaction between a guest molecule and a host molecule can be represented by the spherical core potential r(x) = m x 5 2a
where x is the distance between molecular centers, a is the core radius of interaction, is the distance between molecular cores
Smoothed-Cell Potential
/
F I I 1. Schematic representation of the interaction between a spherical guest molecule and a perfectly spherical hydrate cavity.
TABLE U Optimal Cell Radii and Coordination Numbers Used in Smoothed-Cell Potential Calculationso
cell radius, %, optimal maximum minimum optimal coordination no. 3.803 3.875 20 3.972
structure I, small cavityb structure I, large cavity structure 11, small cavitye structure 11, large cavity
4.664
4.017
4.30
21
3.926
3.748
3.870
20
4.727
4.679
4.703
28
"The table also lists the maximum and minimum distances of the water molecules from the center of the hydrate cells, a measure of cell cBased on a cell asymmetry. bBased on a cell constant of 12.0 constant of 17.31 at which r ( x ) = 0, and e is the depth of the intermolecular potential well. Calculation of the cell potential W(r,O,r$)requires that the dependence on the angular coordinates O and 4 be emphasized. If the LJD cell theory assumption of smearing the water molecules over the cavity surface is used, the angular dependence is lost, and eq 3 reduces to
where R is the cavity radius and ( W(r))is the angle-independent smoothed cell potential which can be expressed analytically as
with
6N = ((1 - r / R - u / R ) - -~ (1
+ r/R - a/R)-q/N
Figure 1 is a two-dimensional idealized representation of the interaction between a guest molecule and the host molecules of a perfectly spherical hydrate cavity. As represented, the true potential V(r,B,r$)is a function of the angular coordinates 0 and 6. Smearing the water molecules over the surface would lead to the smoothed cell potential ( W ( r ) ) .If the hydrate cavities were perfectly spherical and the water molecules evenly distributed over the surface, the smoothed cell potential ( W ( r ) )would be the average of the angle-dependent potential W(r,O,+)at all radial distances. In the real case, however, the hydrate host lattice is not spherically symmetric. Application of the LJD approach requires spherical symmetry, and for this purpose previous researchers'~~ have assigned cell radii R to be used in calculating the smoothed cell potential and the Langmuir constant from eq 5 and 6 . In earlier research: we have obtained optimal values of the cell radii R and coordination number z by fitting the average of the true potential W(r,O,d)obtained by discrete summation of ~~~
(1) van der Waals, J. H.; Platteeuw, J. C. Adu. Chem. Phys. 1959, 2, 1. (2) McKoy, V.;Sinanoglu, 0. J . Chem. Phys. 1963, 38, 2946.
&
True Potential
~~
~~
~
~
(3) John, V. T.; Holder, G. D. J . Phys. Chem. 1981,85, 1811. (4) Pamsh, W. R.; Prausnitz, J. M. Ind. Eng. Chem. Process. Des. Deu. 19172, 11, 26.
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3281
Cell Theory of Langmuir Constants 3.0 (a)
9
c
3.0
I
Q*
2.0
-
II
, , I
I
2.0
Q* ' 1.0
1.0 0.0 2.5
3.0 a
3.5
2.5
(A)
3.0 a
3.0
3.0
2.0
2.0
1"0
1.0
Q"
3.5
(A)
tt
0.0
2.5
3.0
3.5
2.5
3.0
3.5
4.0
0 (A) 0 (A) Figure 2. The effect of intermolecular size and energy parameters on Q* = C / C for spherical molecules: (a) structure I, small cavity; (b) structure I, large cavity; (c) structure 11, small cavity; (d) structure 11, large cavity. (-) e / k = 130 K. (---) c / k = 300 K. (V) a = 0.2 A, ( 0 )a = 0.3 A, ( 0 ) a = 0.4 A, ( 0 ) a = 0.5 A, (A) a = 0.88 A.
binary guest-host interactions to the smoothed cell potential ( W(r)),at different radial distances. These optimal cell radii vary shghtly with the molecular parameters (u,a,e) used; averaged values are shown in Table 11. Table I1 also lists the minimum and maximum distances of the water molecules from the cavity center, obtained from crystallographic data;sv6 this measure of cavity asphericity indicates that the large cavity of structure I is the most asymmetric of all hydrate cavities. Table 111 lists results of a parametric study on the accuracy of using the smoothed cell potential with the radius values R shown in Table I1 to calculate the Langmuir constant. Thus Langmuir constants C were calculated exactly from the configurational integral of eq 3 and compared to values obtained from the smoothed cell approach (C* in eq 5 ) . In carrying out the exact computations, we evaluated the cell potential W(r,O,$) using a discrete summation of binary interactions between the guest molecules and the host molecules located at their crystallographic positions. A Gaussian ten-point integration routine' was modified to calculate the triple integral of eq 3; to improve accuracy, the integration limits were halved, the total integrqj being the sum of the subintegrals. Thus, the total number of integration points for the triple integral is 2000. It is of interest to mention that the upper limit R in eq 3 is inconsequential to the calculations; (5) von Stackelberg, M.; Muller, H. R. Z.Electrochem. 1954, 58, 215. (6) Mak, T. C. W.; McMullan, R. K. J . Chem. Phys. 1965, 42, 2732. (7) Carnahan, B.; Luther, H. A.; Wilkes, J. 0. 'Applied Numerical
Methods"; Wiley: New York, 1969.
in a discrete summation scheme based on the crystallographic locations of the host molecules, the cell potential W(r,O,$) is independent of the cell radius value. Further, W(r,O,$)becomes highly repulsive a t radial distances approaching the cavity wall, thus leading to a negligible contribution of the integrand in eq 3 at these radial distances. An alternative Monte Carlo method of evaluating the configurational integral has been used by Tester et a1.* to study the hydrate equilibrium of small gas molecules that form structure I hydrates. Results of that work which is theoretically correct, are invalid since incorrect structural information on gas hydrates were used. The present study was directed to examining the deviations between the Langmuir constants obtained from the discrete summation approach and from the smoothed cell approximation as a function of molecular size and energy parameters. Figure 2a-d and Table 111show the values of Q* = C / C , the ratio of the true Langmuir constant to the smoothed cell Langmuir constant, as a function of the molecular parameters u, a, and e . It is observed that Q*deviates only slightly from unity for a range of molecular size parameters, indicating the validity of the smoothed cell assumption. Deviations from unity appear to be most significant for molecules in the large cavity of structure I which is the most aspherical of hydrate cavities. Figure 2a-d also indicates that Q*values show a minimum and increase to values greater than unity as the molecular dimension approaches cavity (8) Tester, J. W.; Bivins, R. L.;Herrick, C. C. AZChE J . 1972, 18, 1220.
3282
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985
John and Holder
TABLE I I I :Langmuir Constants of Spherical Guest Molecules as a Function of Molecular Parameters’ u, 8, a, A ilk, K C, atm-I C,atm-l Q* Structure I, Small Cavity 0.968 2.5 0.2 130.0 1.11 X loT2 1:14 X 5.94 X 6.95 X 0.855 3.0 1.52 x 10-3 1.227 3.5 1.87 x 10-3 2.5 0.3 130.0 1.25 X 1.33 X 0.939 6.53 X 7.70 X 0.848 3.0 1.583 5.13 X 3.24 X lo-’ 3.5 300.0 2.28 2.87 0.792 2.5 0.3 3.58 x 103 0.624 1.61 x 103 3.0 1.82 x 10-3 2.583 4.70 x 10-3 3.5 Figure 3. Schematic representation of a linear molecule in a hydrate
Structure I, Large Cavity 1.40 X 7.22 X 1.41 X IO-’ 2.68 X lo-’
1.05 X 5.98 X 9.22 X 8.32 X lo4
1.326 1.208 1.524 3.215
130.0
1.62 X 9.67 X 2.55 X 0.0
1.25 X 8.50 X 1.35 X 0.0
1.293 1.138 1.89
0.4
300.0
2.08 1.05 x 103 1.95 x 104 1.74 x 103 0.0
1.40 1.07 x 103 1.68 x 104 5.35 x io2 0.0
1.491 0.98 1 1.161 3.245
2.5 2.75 3.0 3.25
0.88
300.0
8.04 X 10’ 3.25 x 103 2.03 x 103 2.13 X
1.02 X 5.43 x 2.33 x 3.10 X
IO2 103 103 10”
0.790 0.599 0.871 6.87
2.5 2.75 3.0 3.25
0.88
130.0
3.41 X 7.91 X 2.56 X 3.60 X 10”
3.36 X 8.76 X 2.44 X 1.40 X 10“
1.015 0.908 1.049 2.57
2.5 3.0 3.5 3.75 2.5 3.0 3.5 3.75
0.2
130.0
1.19 X 6.65 X 1.57 x 10-3 0
1.15 X 7.00 X 1.31 x 10-3 0
1.036 0.949 1.2
0.3
130.0
1.36 X 7.31 X 3.57 x 10-5 0.0
1.34 X 7.71 X 2.52 x 10-5 0.0
1.012 0.948 1.417
2.5 3.0 3.25 3.5 3.75
0.3
300.0
2.78 2.16 x 103 1.80 x 103 2.14 x 10-3 0.0
2.98 2.69 x 103 1.90 x 103 1.07 x 10-3 0.0
0.933 0.805 0.948 2.0
2.5 3.0 3.5 4.0 4.25
0.3
130.0
1.81 X 8.51 X 1.21 1.64 4.41 x 10-3
1.61 X 8.63 X 1.29 1.62 3.47 x 10-3
2.5 3.0
0.5
130.0
2.10 X 1.47 X lo-’
1.95 X 1.52 X lo-’
1.123 0.992 0.939 1.016 1.272 1.078 0.967
3.5
2.38
2.55
0.935
4.0 4.25 2.5 3.0 3.5 4.0 4.25
3.28
2.5 3.0 3.5 3.75
0.25
2.5 3.0 3.5 3.75
0.4
2.5 3.0 3.25 3.5 3.75
130.0
cavity. in cell potential W(r,O,qi)with angular position, due to cavity asymmetry and the discrete positions of the water molecules. These fluctuations increase with radial distance from the cavity center and are especially significant at radial distances near the cavity wall where the potential energy becomes repulsive. For molecules whose sizes approach cavity dimensions, the small Langmuir constant values indicate that the cell potential is mainly repulsive; variations in cell potential with angular position can cause significant discrepancies between exact calculations of the Langmuir constant, and calculations using the smoothed cell approach wherein the fluctuations are smoothed out. For very small molecules, the Langmuir constants are again relatively small due to the fact that attractive cell potentials that contribute to the integrand in expressions for the Langmuir constant are appreciable only a t radial distances near the cavity wall. At these radial distances, fluctuationsin cell potential with angular position become significant thus resulting in deviations between the true Langmuir constant and the smoothed cell Langmuir constant. Linear Molecules. For the case of linear molecules, the Langmuir constant (eq 2) leads to
Structure 11, Small Cavity
Structure 11, Large Cavity
0.5
300.0
X
1.181
0.0
0.0
1.52 5.27 X lo2 4.51 X lo6 7.02 x 103
1.35 5.95 X 102 5.54 X lo6 5.06 x 103
1.126 0.886 0.814 1.385
0.0
0.0
X
2.78
Parameter values are typical of those used in semiempirical calculations of hydrate equilibria.I2 C = Langmuir constant from exact calculations. C = Langmuir constant from the smoothed-cell method. a
size. As is seen from Table 111, these relatively large Q* values are approached at small values of the Langmuir constant. Qualitatively, this can be interpreted as due to large fluctuations
The cell potential W = W(r,O,qi,B’,qi?is a function of both the angular position of the center of mass of the molecule and the orientation of the molecule as defined by the Eulerian angles 8’ and 4’. In calculations of the cell potential we have used a form of the Kihara potential to determine binary guest-host interactions. Thus, the linear guest molecule is treated as being r o d - ~ h a p e d ~ . ~ and the binary guest-host interaction is represented as where p is the shortest distance between the rod-shaped core of the guest molecule (of core length 1) and a water molecule (of core length 0); pm is the shortest distance between molecular cores at minimum potential e.* Thus, the parameters 1, pm, and t are fixed energy and size parameters for a given guest-host binary interaction. For a linear guest molecule located at the radial, angular, and orientational coordinates, r,O,qi,O‘,qi‘, the quantity p for each individual binary interaction can be obtained from the geometrical considerations of dropping a perpendicular from the host molecule to the rod-shaped core of the guest as shown in Figure 3. The length of this perpendicular is, by definition, p; if the perpendicular falls outside the length of the rod core, then the distance between the end of the rod closest to the host molecule and the host molecule is in this case the quantity p. The cell potential W(r,8,qi,et,qi3 is the sum of the binary interactions between the linear molecule a t coordinates r,e,@,O’,@’withthe host molecules of the hydrate cavity. Calculations for the smoothed cell potential W*(r,O,qi,&‘,qi?were done using the following approach. We take an elemental area R2 sin 0‘‘ d8” dqi”on the surface of the smoothed cell, where R (9) Kihara, T. Reu. Mod. Phys. 1953, 25, 831
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3283
Cell Theory of Langmuir Constants 3.0
I
1
I
I
lo5
1
lo4
-
2.0
Q*
lo3
c
lo2
lo1
I
0.0
2.2s
I
I
2.7s
I
I
3.2s
3.7s
10-1 loo
t , , , , l
3.25
3.0
3.0
2.0
2.0
1.0
1.0
3.75
4.25
Q*
0.0
0.0
2.2s
2.7s
3.2s
3.75
I . d d 2 - J 3.5
4.0
4.5
(A)
P,
’m ( A ) Figure 4. The effect of intermolecular size and energy parameters on Q* = C/C“ for linear molecules: (a) structure I, small cavity; (b) structure I, large cavit ; (c) structure 11, small cavity; (d) structure 11, large cavity. (-) t/k = 130 K. (---) t/k = 300 K. (0) I = 1.0 A, (A)I = 1.75 A, (v)I = 2.0
x.
is the cell radius, and O”,Q,” are the angular coordinates of the elemental area. If z is the coordination number of the host lattice, the number of smeared water molecules in the surface element is
(
--&)It2
sin 6” doff d 4 ”
The potential energy of interaction between a linear guest molecule at (r,8,4,0f,Q,’)with this surface element is Z
I’* = e{(pm/p)12 - 2 ( ~ ~ / p ) ~ sin ] - 8” d8” dQ,” 4A
(9)
where p = p(r,~,Q,,6f,Q,f,0‘f,~f’) is the shortest distance from the surface element to the core of the linear guest in this case. Thus, integrating over the surface of the sphere we get the smoothed cell potential W*(r,8,4,O‘,Q,3= E 4 1r 2 0r ~ r ( ( p m / p ) 1 2- ( p , / ~ ) ~sin ]
6” d8” d 4 ” (10)
and the expression for the smoothed cell Langmuir constant C* is
sin 8 d r d6 dQ, sin 6’ d8’ db’ (11) The fivefold integrals of eq 7 and 11 were again evaluated with
a modified Gaussian ten-point integration routine.’ Figure 4a-d and Table IV show the results for the Langmuir constants and for Q* = C / C . To minimize computer time, integration interval splitting was not used to obtain increased accuracy; however, the fivefold integrals were computed with double-precision arithmetic. Results for Q* again indicate the minor deviations from unity for linear guest molecules in the small cavity of structure I and in both cavities of structure 11. Q*values differ significantly from unity for the large cavity of structure I due to the marked effect of cavity asphericity. Additionally, a minimum in Q* was not indicated for molecules in the large cavity of structure I. If the molecular parameter pm is decreased to values less than 3.25 A, the smallest value shown in Figure 4b, the results are not realistic since such low values of pm indicate molecules that are too small to be contained in the cavity. For example, Figure 5 shows that a linear guest with molecular parameters, pm = 3.0 A, 1 = 1.0 A, elk = 130.0 K, when oriented at d = n / 2 , Q, = A, 8’ = n / 2 , 4‘ = A, experiences an attractive interaction even at the cavity wall, whereas its interaction is highly repulsive at similar radial distances when it is a t angular and orientational coordinates, 8 = 0.0, 4 = A, 8‘ = 0.0, 4’ = A radians. This suggests that the molecule is not stable in the hydrate cavity and can penetrate through the lattice at certain orientations. Trends observed in Q* as a function of molecular parameters for linear molecules are roughly similar to the trends observed for spherical molecules. Thus Q* values become significantly greater than unity when the size of the linear molecule (as in-
3284 The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 800.0
1
0
I
I 1
I
I
1
8 = 0.0
Cavity Surface
g = n
3
-
8 ' = 0.0
!a'= v
1
I
---c
(radians) 8 ' = n/2
g'=
l7
8 = n/2 ,0= n
I
I
0.0
4
m u
.i
4J a!
-400. rl
4
u
U
John and Holder TABLE I V Langmuir Constants of Linear Guest Molecules as a Function of Molecular Parameterso pmin,A 1, A elk, K C, atm-' C*, atm-' Q* = C/C* Structure I, Small Cavity 2.25 1.o 130.0 4.38 X lo-' 3.69 X lo-' 1.190 9.61 x 10-3 9.60 x 10-3 2.75 1.001 3.25 3.75 X 4.22 X 0.888 6.87 x 10-3 6.34 x 10-3 3.75 1.084 2.25 1.o 300.0 4.49 X 3.50 X 1.283 9.23 X IO-' 1.03 X 10' 0.898 2.75 2.82 X lo2 3.25 0.684 1.73 X lo2 3.75 0.106 8.59 X 10' 8.10 X 10' 2.25 1.15 130.0 4.52 X 3.64 X 1.243 1.062 2.75 1.01 x 10-2 9.56 x 10-3 1.009 3.25 1.51 X 1.50 X 3.601 1.97 X lo-' 5.48 X lo-' 3.75
Structure I, Large Cavity
-800.
-1200. 0.0
1.0
2.0
3.0
Radial Distance lrl
4.0
(2)
Figure 5. The effect of molecular orientation on the cell potential between a linear guest molecule and a hydrate cavity (structure I, large cavity). Effective Kihara interaction parameters: 1 = 1.0 A, pm = 3.0 A, elk = 130 K.
dicated by pm and 1) approaches cavity dimensions and the Langmuir constant approaches zero. This is due to the effect of potential fluctuations with angular position and guest molecule orientation. These potential fluctuations are marked for large linear molecules in the large cavity of structure I which is spherically asymmetric, and thus leads to significant deviations between the exact Langmuir constant and the smoothed-cell Langmuir constant. Another observed trend is the effect of the energy parameter t; for both spherical and linear guest molecules, an increased sensitivity of Q*as a function of the size parameter (a or p,) is noticed as t increases.
Conclusions and Significance The results discussed above indicate that the actual value of the Langmuir constant (C) can deviate from the value obtained by using the smoothed-cell approach. The effect of these deviations on hydrate equilibrium calculations is discussed below. In determining hydrate equilibrium, certain reference properties of hydrate structures such as ApaB, the chemical potential difference between water in the metastable @ phase and the pure a phase (water/ice) at Oo C, 0 atm, and e,. uw,the molecular parameters of water in the hydrate lattice, have to be evaluated accurately. These reference properties are obtained by fitting experimental hydrate data for perfectly spherical guest molecules such as Ar to the mathematical equations describing hydrate eq~ilibria.'*~J The precise determination of the reference properties is important to subsequent hydrate equilibrium calculations for other gas species. It has been assumed in previous research that the LJD smoothed-cell approach to calculating Langmuir constants for small spherical molecules and fitting the hydrate data for such molecules using eq 1 leads to reasonably accurate values for the reference properties. The present work shows that, even for spherical molecules, the Langmuir constant differs from that obtained when the LJD cell theory approach is used; thus improved accuracy in reference property determination can be achieved by using the Q*correction factors to correct the smoothed-cell values. The main problem with hydrate calculations, however, lies with the prediction of hydrate equilibrium for large spherically asym-
'
(10) Holder, G. D.; Corbin, G.; Papadopoulos, K.I d . Eng. Chem. Fundam. 1980, 19, 282. (11) Mazo, R. M.Mol. Phys. 1964, 8, 515.
3.25 3.5 3.75 4.25
1.O
130.0
3.25 3.5 3.75 4.0
1.75
3.25 3.5 3.75 4.25
1.o
300.0
2.25 2.75 3.25 3.75
1.o
130.0
4.60 X lo-) 1.03 X 4.11 X 7.12 x 10-3
3.69 X 9.65 X low3 4.25 X 5.76 x 10-3
1.25 1.063 0.967 1.236
2.25 2.75 3.25 3.75
1.75
130.0
io-*
1.o
3.64 X 9.59 x 1.46 X 3.73 x 3.54 X 1.06 X 2.64 X 6.76 X 1.66 X
1.279 1.092 1.119 9.79
2.25 2.75 3.25 3.75 4.0
4.66 X 1.05 x 10-2 1.63 X 3.65 x 10-7 5.09 X 1.08 X loo 2.23 X lo2 1.19 X lo2 2.68 X
10' lo2 10' 10"
1.439 1.02 0.843 1.764 16.19
3.5 4.0 4.5 4.75
1.o
130.0
1.19 X lo-' 1.15 1.21 1.44 X
1.20 X lo-' 1.22 1.18 1.13 X
0.992 0.944 1.029 1.263
3.5 4.0 4.5 4.75
2.0
130.0
0.972 1.024 2.284
1.o
300.0
1.68 X 4.94 X 1.83 x 0.0 2.40 X 3.62 x 5.71 X 5.28 X
lo-' lo-' 10-5
3.5 4.0 4.5 4.75
1.64 X lo-' 5.06 X IO-' 4.18 x 10-5 0.0 2.20 X lo2 3.03 x 105 5.82 X lo6 9.09 X lo2
lo2 105 lo6 lo2
0.9 16 0.835 1.019 1.724
130.0
4.37 X 9.72 X 1.56 X 6.25 x 4.65 X 7.2 X 3.62 X 7.41 X 4.92 X 9.65 X 1.28 x 1.66 x
lo-' 10-4
lo4
10' lo2 104 io0
3.45 X 7.57 X 1.02 X 4.17 x 3.15 X 3.48 X 5.13 X 3.36 X 3.87 X 7.54 X 6.41 x 2.22 x
lo-' 10-5 lo-' loo2 10" 10' lo2 104 10-3
1.266 1.284 1 .53 1 14.979 1.479 2.069 7.058 220.2 1.272 1.280 1.991 748.3
Structure 11, Small Cavity
300.0
10-3
Structure 11, Large Cavity
metric guest species. If the smoothed-cell LJD theory is used to calculate the Langmuir constants for such species it is found that the equilibrium fugacities obtained from van der Waals model' are far lower than those obtained experimentally. As eq 1 indicates, the inverse relationship between C andfimples that the Langmuir constants calculated are too high; as a consequence, the smoothed-cell Langmuir constants C* must be corrected by Q* values less than unity to obtain accurate values, C. The calculations presented here for linear molecules, however, show that Q* values greater than unity are obtained as guest size approaches cavity dimensions. Mazo" has proved from theoretical considerations that the effect of cell potential fluctuations with molecular orientation is to increasethe value of the configurational integral and thus the Langmuir constants. Thus Q* values greater than unity are theoretically valid as long as cell potential fluc-
Cell Theory of Langmuir Constants
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3285 correlations were developed such that calculated equilibrium pressures agreed with experimental values. These correlations are shown in Figure 6 which indicates that the empirical Q*s decrease exponentially with molecular size and molecular nonsphericity as indicated by the acentric factor, w . Substantial reduction in Q* values is observed in particular for molecules which enter the large cavity of structure I1 which is the largest cavity, and also the most spherically symmetric of hydrate cavities. In or lower, significant distortion order to obtain Q* values of of the cavity structure may be taking place during enclathration of large nonspherical guest species, such as n-butane. Crystallographic studies of the hydrate structure for hydrates of such species have not been carried out and such evidence will be very useful in validating cavity distortions.
Nomenclature a
A
C
f
h k I
4 U W E
(R-a)kTo Figure 6. Empirical Q*correlations showing how Q*would have to vary with molecular size in order to accurately predict equilibrium pressures. (k is the Boltzmann constant, w, the Pitzer acentric factor, and Tois 273.15 K.)
tuations occur during rotation due to the discrete locations of the water molecules. If, however, the amplitude of the potential fluctuations is so high that the guest species becomes trapped in a potential well, its motion becomes one of torsional oscillations within the well. The consequent localization of the guest molecule implies a decrease in the free volume available to the guest species resulting in a reduced Langmuir constant. Thus Q* values less than unity can be obtained only when high rotational barriers cause localization of the guest species. Unsymmetric interactions between large guest species and the host cavity can cause cavity distortions resulting in large potential barriers to rotation. We thus propose that Q*values less than unity may be due to cavity distortions, causing significant localizations of asymmetric guest species. In an earlier study12we assumed that Q*values were less than unity due to cavity distortions and obtained empirical correlations for Q* based on molecular size and energy parameters. The (12) John, V. T., Papadopoulos, K. D., Holder, G. D. AIChE J . 31,252 (1985).
r
R T W W X 2
a
P
r e
8 8’ 8” P Y
P Pm
a
4 4‘ 4” w
effective core radius in the Kihara binary interaction potential applied to spherical molecules (A) amplitude of potential variation with angular position (erg/mol) Langmuir constant (atm-I) fugacity (atm) molecular partition function of an encaged species Boltzmann’s constant (erg/(mol/K)) length of rod-shaped core in the Kihara binary interaction potential applied to linear molecules molecular partition of a gaseous species with the volume term excluded radial distance of the center of mass of an encaged species from the cavity center (A) cell radius (A) temperature (K) exact cell potential (erg/mol) smoothed-cell potential (erg/mol) intermolecular distance (A) coordination number pure water/ice phase metastable phase of empty hydrate lattices Kihara pair potential (erg/mol) depth of intermolecular potential well (erg/mol) angular coordinate of the center of mass of an encaged species (radians) orientational coordinate of a linear guest molecule (radians) angular position of elemental area on the cavity surface (radians) chemical potential (erg/mol) number of cavities per water molecule shortest distance from a point on the cavity surface to the core of a linear guest molecule (A) the value of p at which the pair potential r = e (A) distance parameter in the Kihara pair potential for spherical molecules; the distance at which r = 0 (A) angular coordinate of the center of mass of an encaged species (radians) orientational coordinate of a linear guest molecule (radians) angular position of elemental area on the cavity surface (radians) acentric factor
