PREDICTION OF DISSOCIATION PRESSURES OF MIXED GAS HYDRATES FROM DATA FOR HYDRATES OF PURE GASES WITH WATER I S A M U N A G A T A A N D R l K l K O B A Y A S H I Department of Chemical Engineering, William Marsh Rice University, Houston, Tex.
The dissociation pressures of ternary gas hydrates predicted using only data for binary hydrates are compared with experimental values. The methane-propane-hydrate crystallizes in Structure I1 for the pressure range studied here, while the methane and propane hydrates crystallize in Structures I and II, respectively. Predicted results are in close agreement with experimental data. The methane-ethane hydrates both crystallize in Structure !. The predicted hydrate pressures are higher than the observed data for the methane-rich concentration regions.
N A
previous paper (9) the present authors showed that the
I Kihara potential is better than the Lennard-Jones potential
in describing the dissociation pressures of methane hydrate, nitrogen hydrate, and the ternary hydrate of methane, nitrogen, and water, based on the assumption that encaged gas molecules rotate freely in the vicinity of the center of the cage. In the present paper the predicted results of the dissociation pressures of the other ternary hydrates including methane, using only parameters determined from data for hydrates of pure gases and water, are presented. Theory
Calculation of Chemical Potential Difference below Ice Point
Ap (cal./mole) for gas hydrates of Structure I a t 273' calculated by : Ap = 167 -/- 0.073P
[Ap/RT]T =
+ Z CJ,fJ)
(1 )
where Ap is the difference in the chemical potential between the empty gas hydrate lattice and ordinary ice,fJ is the fugacity of solute J in the hydrate, y J , is the fraction of solute included in the cavity of radius a,, and v, denotes the number of type i molecules per molecule of water. I n the case of gas hydrates of Structure I, V I = 1 / 2 3 and v 2 = 3 / 2 3 ; in the case of gas hydrates of Structure 11, V I = 2/27 and v z = 1/17 (70). CJL is the Langmuir constant of solute J in the cavity of radius at. w ( r ) is the spherically symmetrical potential in the cavity, with r measured from the center. McKoy and Sinanoilu (8) derived w ( r ) for spherical and line molecules of the Kihara potential. The present authors (9) obtained the general expression of w ( r ) for rodlike molecules, which reduced to ~ ( r for ) spherical or line ones as the limiting case. T h e chemical potential difference, Ap', above the ice point can be expressed by : AP' = R
T { ~In] ( 1
+ cJ1fJ) + vzln (1 + cJzfJ) + In
xW)
(4)
This equation involves the assumption that the aqueous solution in equilibrium with the hydrate forms an ideal solution. xzois the mole fraction of water in the water-rich liquid phase. 466
l&EC FUNDAMENTALS
(5)
where P i s the dissociation pressure of a gas hydrate a t 273' K. (atmospheres) and A p for the propane hydrate of Structure I1 a t 270' K. is 196 cal. per mole (77). Integration of A p / R T between T and 273' K. along the equilibrium curve gives
Van der Waals and Platteeuw ( 7 7) derived the following general equations relating the thermodynamic properties of gas hydrates:
Ap = RT Zvc In ( 1
K.is
-
sy
(AH/RT)dT
:s
+ (AV/RT)
dP
dT
(6)
where AV is the difference between the molar volume of hydrate and that of ice and the value of AV is 3.0 ml. per mole for Structure I and 3.3 ml. per mole for Structure I1 (70). A H is unknown but we assumed A H 0 for hydrates of Structure I1 as van der Waals and Platteeuw suggested A H 'v 0 for hydrates of Structure I ( 7 7 ) . The value of d P / d T is well expressed from the data of Deaton and Frost (#).
dP - = A exp ( A T dT
- B)
(7)
where A and B are given by: A
Ethane Propane
0.04460 0.04627
B 10.6260 12.1008
Ap's of methane hydrate are available (9). Numerical Calculations
The ternary hydrate consisting of methane, propane, and water forms a solid solution of Structure I1 over a considerable range of pressure, temperature, and composition, although the pure methane and pure propane hydrates crystallize in Structures I and 11, respectively (7 7). Since the propane molecules occupy only the larger cavities, it is possible to obtain the value of Cp, ( C p , = 0) a t various temperatures below the ice point from Equation 1.
T h e relationship between Cpa and absolute temperature can be well expressed by:
CPz = exp
(18.276
- 0.04613 T )
Then the values of C p , a t temperatures near but above the ice point were calculated by using Equation 8. T h e values of C,l and C,w, for methane in Structure I1 follow from Equation 3 by taking the values of the parameters of the Kihara potential of a spherical core reported in the previous paper ( 9 ) ( ~ / k= 149.3’ K., p m = 3.312 A., core radius c = 0.32 A , ) and the appropriate structural constants for Structure I1 (70) (a, = 3.91 A., a2 = 4.73 A . , 21 = 20, and 2 2 = 28). Integrating the equation expressing the pressure effect on the chemical potential difference, AM’, a t constant temperature yields :
of propane in water was considered t.o be negligible (7) a t the pressures covered by these calculations. The solubility data of ethane and methane in water were taken from Culberson and McKetta (2, 3 ) . Theoretical fits to the experimental values of ethane hydrate using three kinds of the Kihara potential (Table I) were obtained. I n all three cases it was presumed that the ethane molecules could enter both cages. Table I also includes the results based on the assumption that the ethane molecules could occupy only larger cavities. Equation 9 suggests the following argument: The dissociation pressures of the ethane hydrate are lower than those of the methane hydrate a t any fixed temperature, so that A p ’ of the ethane hydrate should be I
40
+ AV’(P - Po)
Ap’ = AMO’
E X P T L AT 277.6’K DEATON B FROST (41
(9)
If propane is used as the reference substance (5),the chemical potential difference of the ternary system of Structure I1 can be calculated. A p ’ = ApCaHs‘f 0.1186
I
I
I
(P - P c ~ H ~ )
CALC
-ONLY L A R G E --- BOTH CAGES
CAGE
(10) 30
where the units of chemical potential a n d pressure are calories per mole and atmospheres. T h e general equation for the ternary system may be expressed as follows: AP’ = RT{ v 1 In (1
+ vZ
+ cB,fB)+ In (1 + + cB2fB) + In
rte
Lz 3
ctlfA
CAJA
HItG
w
v)
xm}
wl
: 20 a
(11)
where fA is the fugacity of component A . Subscripts A , B and 1, 2 represent the gas components and type of cavity, respectively. T h e chemical potential difference of the ternary system can be calculatedl from Equation 10. T h e pressure a t a fixed gas composition and temperature were predicted by using Equations 10 and 11. T h e mole fraction of methane bound in the solid phase of hydrate Y,, was calculated by:
IO I
0
Figures 1 and 2 show a satisfactory agreement with the calculated and experimental results. T h e fugacities of propane were obtained from Hougen, Watson, and Ragatz (7) and Hoffman, Welker, Rao, and Weber ( 6 ) . Solubility
Table 1. Rod
T, ’K. 260 264 268 272
277.5 282.3 Parameters used I , A. C, A. pn, A.
elk, O K.
c1,
I
80
100
M O L E % IN M E T H A N E
AP,
CI, l/atm.
llatm.
0.3531 0.2964 0.2501 0,2121
2.990 2.495 2.093 1.765
Eq. 7, ca1.l mole 159.3 161.7 164.1 166.7
1.407 1.163
P ’> Eq. 4 193.2 226.9
l/atm.
llatm.
2.696 2.254 1.894 1.600
mole 159.2 161.6 164.0 166.6
1.278 1.058
P I, Eq. 4 193.7 228.1
1.30 0.32 3.200 181.2
I
60
Figure 1. Pressure-temperature diagram for methane-propane-water system
tal./
0,5832 0.4851 0.4057 0,3411
0,2710 0,2233
I
40
Chemical Potential and Langmuir Constant of Ethane Hydrate Line Sphere AI*, Eq. 7,
CZ,
I
20
0.1704 0.1418
CZ,
1.540 0.0 3.480 180.5
A&
AP,
GI, llatm. 0.5632 0.4682 0.3914 0.3289
0.2611 0,2150
2.725 2.276 1.911 1.613
Eq. 1, Gal./ mole 159.4 161.8 164.2 166.7
1.288 1.065
AI*I , Eq. 4 193.7 228.0
CZ, l/atm.
0.0 0.77 3.175 162.3
VOL. 5
Eq. I,
CZ,
Gal./
l/Atm.
3.784 3.176 2.667 2.239
Mole 159.2 161.9 164.3 166.6
1.760 1.427
Eq. 4 189.9 216.2
4,
Gal./ Mo’e, ExptE. 159.3 161.8 164.2 166.7
AP ’,
c1 = 0 CZ = exp(12.701
- 0.U4373T)
NO. 4 N O V E M B E R 1 9 6 6
467
I
I
I
I
I
50
I
40
40
30
20 30
i
t
a
-
i l-
W K
a
IO
3
v ) 8
W
Ln W
a 3
a
20
v) v)
a
W
a
6
HE+ G
0.
10 0
-
7 L
I
273
L I t G
I
l
275
l
EXPTL. DEATON 8 FROST (4) CALC.
I
TEMPERATURE, 0
0
I
I
I
I
20
40
60
80
1
277
I
279
l
l
281
O K
Figure 4. Pressure-composition diagram for methaneethane-water system
IO0
MOLE % IN METHANE
Figure 2. Pressure-composition diagram for rnethanepropane-water system
,
100
I
I
higher concentration range of methane in ethane, as shown in Figures 3 and 4. An assumption that methane, ethane, and water molecules might form the hydrate of Structure I1 does not lead to better agreement between the predicted values and the experimental results.
I
EXPTL. DEATON 8 FROST ( 4 )
0
BO
I
40
1 Nomenclature
ai C
a
a
"
6
41
I
274
I
I
I
I
1
276
278
280
282
284
TEMPERATURE,
O K
Figure 3. Pressure-temperature diagram for methaneethane-water system
lower than the corresponding value of the methane hydrate. A comparison of all calculated Apl's for the ethane hydrate (Table I ) and 225.1 cal. per gram mole for the methane hydrate a t 282.3' K . shows that A p t based on the assumption that the ethane molecules occupy only larger cavities meets this condition. However, the calculated values are not in good agreement with the experimental results (4) in the 468
I&EC FUNDAMENTALS
cell radius
Langmuir constant of solute J in cavity i Langmuir constant fJ = fugacity of solute J in hydrate G = gas = hydrate of Structure I = hydrate of Structure I1 = molar enthalpy difference = ice Z = Boltzmann constant k L1 = water-rich liquid = pressure P = radial position 7 = gas constant R T = temperature A V = molar volume difference = mole fraction of water in water-rich liquid phase = mole fraction of type i occupied by a gas molecule J = mole fraction of solute M bound in solid phase of hydrate = structure constant = mole fraction of solute propane CJi
c
W
=
= core radius = =
GREEKLETTERS pm = energy and distance parameters in Kihara potential Y; = number of cavities of type i per mole of water
E,
= chemical potential w ( r ) = spherically symmetrical Kihara potential
p
Literature Cited
(1) Azarnoosh, A., McKetta, J. J., Petrol. R@ner 37, No. 11, 275 (1958). (2) Culberson, 0. L., w[cKetta, J. J., Petrol. Trans. A T M E 1891 319 (1950). (3) Zbid., 192, 223 (1951). ( 4 ) Deaton, W. M., Frost, E. M., U. S. Dept. Interior, Bur. Mines, Monograph 8 (1946). (5) Din, F., “Thermodynamic Functions of Gases,’’ Vol. 3, Butterworth, London, 1961. (6) Hoffman, D. S., Welker, J. R., Rao, V. N. P., IVeber, J. H.,
A.2.Ch.E. J . 10, 901 (1964). (7) Hougen, 0. A., Watson, K. M., Ragatz, R. A., “Chemical Process Principles,” Part 11, 2nd ed., Wiley, New York, 1959. (8) McICoy, V., Sinanog’lu, O., J . Chem. Phys. 38,2946 (1963). (9) Nagata, I., Kobayashi, R., IND.ENG.CHEM.FUNDAMENTALS, 5, 344 (1966). (10) von Stackelberg, M., Muller, H. R., 2. Elektrochem. 5 8 , 25 (1954). (11) Waals, J. H. van der, Platteeuw, J. C., Advan. Chem. Phys. 2, l(1959).
RECEIVED for review March 21, 1966 ACCEPTEDJune 9, 1966 Work performed under the auspices of the National Aeronautics and Space Administration Research Grant NSG6-59.
A Q U E O U S OXIDATION OF E L E M E N T A L S U L F U R F A T H l H A B A S H I A N D E R W I N L. BAUER Department of Metallurgy, Montana College of Mineral Science and Technology, Butte, Mont.
+
+
The reaction SI 1l / 2 0 2 H2O + HzS04 i s greatly dependent on temperature and oxygen partial pressure. Below ,the melting point of sulfur it is extremely slow, and above this temperature the rate i s appreciable and increases rapidly with temperature. The reaction i s chemically controlled, and the activation energy is 11-75 kcal. per mole. The rate is proportional to p~:’~, thus suggesting that it takes place in two steps: S -t 1 l / 2 0 2 +- SO3 and SO3 H20 +- H2SO4, the first being the rate-determining step. Sulfur dioxide is not an intermediate reaction product. The rate i s affected by the presence of some foreign ions.
+
HE aqueous oxidation of elemental sulfur is of importance Tto the chemical industry as well as to hydrometallurgy. Thus, Schoeffel (6) and Lukas and Zizka ( 5 ) suggested that sulfuric acid can be manufactured directly and quantitatively in an autoclave by heating an aqueous suspension of elemental sulfur to a temperature of 300’ C. under an oxygen partial pressure of 30 atm. I n hydrometallurgy, the aqueous oxidation of sulfide ores is gaining importance as a method for the recovery of nonferrous metals (7, 2). If the aqueous oxidation of sulfide ores is conducted at a temperature below 120’ C. and in acidic medium, demental sulfur is always a reaction product. Once this temperature is exceeded, no sulfur is formed. I n spite of its importance, there is not a single published report on the kinetics a.nd mechanism of this reaction. The present work was undertaken to throw some light on this reaction.
Experimental
The reaction was conducted in a 1-gallon autoclave (Autoclave Engineers Corp.), made of stainless steel and provided with a loose liner, therniocouple well, stirrer, and cooling coil made of titanium. Temperature was automatically controlled to =t2’ C. The reaction mixture consisted of 200 grams of pure sulfur powder and 1900 ml. of distilled water. Speed of stirring was 600 r.p.m. and oxygen partial pressure was 30 p.s.i. unless otherwise st,ated. The mixture was heated to the desired temperature, then oxygen was admitted to the required pressure. From this moment the reaction time was measured. When the reaction time was reached, heating was stopped, the oxygen supply was shut off, and cooling water was admitted rapidly. I t tarkes about 2 minutes to cool the autoclave contents to room temperature. The apparatus was then opened, the reaction mixture filtered, and an aliquot of the filtrate titrated with 0.1N N a O H solution using phenolphthalein indicator. 11:was then possible to calculate the number of moles of H2S04 formed under the conditions of the experiment.
Results
Rate of Reaction. The amount of sulfur reacting under a certain set of conditions was found to be linear with respect to time, as shown in Figure 1. The slopes of these straight lines are the reaction rates. Effect of Temperature. When the reaction was studied a t a temperature below the melting point of sulfur, the rate was extremely slow. Thus, at 60” C. and 6 0 - p s i . oxygen pressure, no acid was detected after 2 hours, and when the temperature was increased to 90’ C., only 0.002 mole of HzS04 was formed. When, however, the temperature was increased above the melting point of sulfur, the rate increased appreciably, as illustrated in Figure 1. From these data, the activation energy in the temperature range 130’ to 170’ C. was calculated (Figure 2) and found to be 11.75 kcal. per mole. Effect of Stirring. T h e effect of stirring was studied a t 130’ C. (Table I). I t is apparent that the ratk is independent of stirring in the range 400 to 900 r.p.m. Effect of Oxygen Pressure. The effect of oxygen pressure was studied at 130’ C. (Table 11). These data fit a straight line only when the rate is plotted against fi,,?’* as shown in Figure 3. Effect of Hydrogen Ion Concentration. This effect was studied a t 130’ C. by adding variable amounts of H2S04 to
Table I. Effect of Speed of Stirring (Temperature 130” C.,po, = 30 p.s.i.) Speed of Stirring, Rate, Mole R.P.M. H ~ S 0 4 / 2Hr. 0
0.0011
400
0.0061 0.0068 0.0061
600
900
VOL. 5
NO. 4
NOVEMBER 1 9 6 6
469
