136
AH',,, AS',,,
J . Phys. Chem. 1984, 88, 136-139
+
= AH+,$,+ 2 AH~,(t-BuO2)- AH$,(04) Acvap[T =
+ R In (V,) + T C + AC,,,
AS'l,,
TO
- 2TB1
In (TT0/TB2)
where the superscript ' indicates the 1-atm standard state, TB is the boiling point of t-BuO,, To is the boiling point of tBu04-t-Bu, AHT:,(t-Bu02) and AHZp(O4) are the heats of vaporization at the boiling points, V, is the molar volume of the solvent, and T C is Trouton's constant. It is observed e m p i r i ~ a l l y ~that ' , ~ ~AC,,, 50.2 J mol-' K-' and so it was estimated that AHT;,(O,) = 41.8 kJ mol-' with To = 467 K (based on an average of values for dodecane and ditert-amyl peroxide) and AH:;,(t-BuO2) = 27.2 kJ mol-' with TB = 323 K (based on the values of 2,2-dimethylbutane). R In (V,) 20.9 J mol-' K-l for many solvents, and so
-
-
AHDg,300= AH+1,300
ASog,30o
=
AS+1,300
+ 18.8 = 58.6 kJ mol-'
+ 83.7 = 238.5 J mol-'
K-I
The value for AHDg,300 is almost a factor of 2 larger than that deduced by Nangia and Benson22from the same experimental data. This discrepancy arises from the use of different hydro(43) R. Shaw, J . Chem. Eng. Data, 14, 461 (1969)
carbon species as models for t-Bu02 and t-Bu204and different from the boiling point of the liquids methods for correcting AHvap to 300 K. This indicates that the real uncertainty in the bond dissociation enthalpy at 300 K in the gas phase is at least 1 2 4 kJ mol-'. The value for ASog,30o = 238.5 J mol-' K-l is substantially greater than the value of 150.6 J mol-] K-I obtained by additivity methods where group values were estimated by assuming a "normal" structure and frequencies. On the basis of this value of the entropy of H204would be only 217.6 J mol-' K-', which implies a tighter molecule than even that suggested by Giguere (Table IV). If, on the other hand, the additivity value of aSDg,300 is assumed correct and converted to AS+1,,40,then a value of 220.0 J mol-' is obtained. This may be combined with the experimental value of AG+1,140 = 16.7 kJ mol-' to yield AH+l,,40 = 47.7 kJ mol-'. However, these parameters are not consistent with the reported temperature dependence of the equilibrium constant. It thus appears that, while a looser H204 species than expected from additivity methods is required by the kinetic data, a tighter one is needed to fit the equilibrium data. However, it should be borne in mind that the uncertainties in this kind of analysis are large and so the reliability of the gas-phase thermodynamic parameters is difficult to assess. Registry No. Hydroperoxo, 3170-83-0.
Diffusion of Carbon Dioxide in Heptane Hani Saad and Esin Gulari* Department of Chemical and Metallurigical Engineering, Wayne State University, Detroit, Michigan 48202 (Received: April 18, 1983)
Photon correlation spectroscopy (PCS) was used to measure the binary diffusion coefficient and viscosity in the C02-n-heptane system in the two-phase region. The measurements covered a temperature range of 10-50 OC and a pressure range of 1-8 MPa. At constant temperature, DABwent through a minimum while the viscosity decreased monotonically. D A B p / Treached a value of 2.2 X dyn/K which remained independent of temperature and pressure for Xco, > 75%.
Introduction Diffusion of dissolved COz in liquid hydrocarbons has been of interest because of recent uses of CO, as a miscible solvent in tertiary-oil recovery and supercritical extraction processes. Even under moderate pressures, ranging from 1 to 10 MPa, C 0 2 dissolves in the gasoline-range hydrocarbons in appreciable quantites. For example, at 10 'C, 12% and 56% increases in the volume of the liquid phase occur when C 0 2 is injected into n-heptane at 2.8 and 3.4 MPa, respectively. A shortage of reliable diffusion data exists because conventional methods, based on measuring the transfer of gas into or out of a liquid or measuring concentration gradients in a diffusing system as a function of time, are difficult to perform at elevated pressures and temperatures. Furthermore, diffusion of C 0 2 in liquid hydrocarbons is about 1 order of magnitude faster than that of C 0 2 and other gases in water.' Photon correlation spectroscopy (PCS) of light scattered from concentration fluctuations at the microscopic level provides an ideal method to measure binary diffusion coefficients without perturbing the thermodynamic equlibrium.2 It is also possible to measure liquid viscosities by determining the diffusion coefficient of known, uniform-sized, probe particles suspended in the liquid at very dilute concentrations. In this limit, the viscosity is inversely related to the diffusion coefficient of the probe particles (1) D. M. Himmelblau, Chem. Reu., 64, 527 (1964).
(2) K. J. Czworniak, H. C. Anderson, and R. Pecora, Chem. Phys., 11,451
( 1975).
0022-3654/84/2088-Ol36$01.50/0
by the Stokes-Einstein r e l a t i ~ n . ~We have used PCS to measure the diffusion coefficient of C 0 2 in n-heptane, in the liquid phase along the two-phase (gas-liquid) boundary as a function of temperature and pressure. We have also measured the corresponding viscosities of the liquid phase using inverse micelles of bis(2ethylhexyl) sulfosuccinate (AOT) as probe particles. The theories for diffusion in liquids are not rigorous and apply in the limit of negligible volumetric expansion for species similar in molecular size and interaction potentials. It is impossible to predict diffusion coefficients from molecular parameters without introducing empirical parameters. However, in both the statistical mechanical approach and the hydrodynamic approach to describe diffusion in liquids, viscosity plays an important role. In this study, simultaneous measurements of the binary diffusion coefficients and viscosities are used to determine the effect of temperature, pressure, and concentration on the diffusivity of C 0 2 .
Experimental Section The light-scattering spectrometer used for the photon-correlation experiments is essentially similar to those described in the litera t ~ r e . The ~ range of scattering angles of the spectrometer with the high-pressure scattering cell was 6-1 1 ' for homodyne detection. We used a Spectra-Physics Model 165 argon-ion laser (3) B. J. Berne and R. Pecora, "Dynamic Light Scattering", Wiley, New York, 1976, pp 83-6. (4) Es. Gulari and B. Chu, Biopolymers, 18, 2943 (1979).
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 1 , 1984 137
Diffusion of Carbon Dioxide in Heptane operating at about 400 mW and Xo = 514.5 nm. The correlation functions were computed by a Malvern Scientific Model K7025 real-time multibit correlator. The sample cell, machined out of a solid brass block, was fitted with flanged, 1.25 cm thick quartz windows and tested up to 20 MPa. The optical path length of the cell was 7 cm. The inside surfaces of the cell were blackened and a red glass tube was inserted into the cell to absorb the reflected green light and provide a stray-light-free background. The cell was fitted into a thermostated jacket the temperature of which was controlled within fO.l "C. The temperature was measured by a thermistor embedded in the cell block. The pressure was monitored by a Dynisco Model PT520G-5M pressure transducer, the sensitivity and the reproducibility of which were h0.005 and h0.03 MPa, respectively. For each loading of the cell, both CO, and n-heptane were introduced gravimetrically from high-pressure sample cylinders equipped with quick-connect type fittings. The evacuated cell was first charged with n-heptane; then CO, was injected into the cell containing n-heptane until the liquid reached the desired level and the cell was closed off. COz in the loading line was condensed back into the sample cylinder by emersing the cylinder into liquid nitrogen. The amounts were determined from the differences in the weights of the sample cylinders before and after loading. Spectrophotometric-grade n-heptane and bone-dry CO, were used without further purification. Fluka purum grade AOT was purified by first dissolving in methanol, filtering through active charcoal, and then extracted with heptane followed by drying under vacuum.
Methods of Data Analysis Measurements of transport properties of liquids using quasielastic light scattering are based on the following principles. In a fluid, light is scattered by optical inhomogeneities or fluctuations in the local dielectric constant, &(r,t), at the microscopic level. In a binary mixture, &(r,t) is related to fluctuations in the local thermodynamic quantities such as pressure, temperature, and concentration. Gc(r,t) = ( d e / d P )T,cGP(rJ) + (at / a %,c3 T(r,t) + (a€/ac)P,dc(r,t) (1) These time-dependent fluctuations modulate the incident electric field and produce the altered time dependence of the scattered electric field that contains information about the modes of fluctuation and dissipation and hence the transport properties of the scattering medium. The scattered electric field can be described through a derivation of an expression for the time and spatial dependence of the fluctuations in the dielectric constant. Mountain and Deutch5 have used thermodynamic fluctuation theory in conjunction with the macroscopic equations of continuity, motion, heat conduction, and mass diffusion to obtain the decay rate of fluctuations in the dielectric constant and the shape of the resulting frequency spectra. The temperature and concentration fluctuations are exponentially decaying functions localized in space.
GT(K,t) = GT(K,O) exp[-xK%]
(2)
GC(K,t) = 6C(K,o) eXp[-D~&t] (3) where x is the thermal diffusivity and DAB is the binary diffusion coefficient, and K = (4na/X0) sin (8/2); n is the refractive index, Xo is the wavelength of the incident light under vacuum, and 0 is the scattering angle. The time dependence of these fluctuations is quantitatively measured by measuring the field autocorrelation function. Under the condition x >> DAB,the field autocorrelation function is a sum of two exponentials: $I)
(KJ) = ( a e / a r ) , % (taT(K)1)2exp[-xfW
/ ac)P,Tz = A eXp[-xPt]
+
( l6 c(K) I ) exp [-DABK2t1
+ B eXp[-D~#t]
(4)
( 5 ) R. D. Mountain and J. M. Deutch, J . Chem. Phys., 50, 1103 (1969).
0 60 053 0 5 6 ; 0 6 t T 07408$70 '084, 086
t
50
0:
P
2
i
t t t P P
4 6
I t
t
;
8
YPIhP
6
Pressure ( MPa)
4
Figure 1. Plots of temperature and pressure along constant-composition lines. Open symbols represent thermodynamic states where DABmeasurements were made with overall Xco, = 0.30, 0.53, 0.60, 0.69, 0.74, 0.84, and 0.89, respectively. Closed symbols correspond to states containing 5% w/w AOT dissolved in heptane and at which simultaneous p and DABmeasurements were made with overall X,,, = 0.56,0.72,0.78,
and 0.86. When colloidal-sized particles are dispersed in a binary liquid mixture at very low concentrations, such a system will comprise a dilute ternary solution. In the limit where the multicomponent diffusion coefficients are replaced by the effective diffusivities, the space and time evolution of the fluctuations in the local dielectric constant can be calculated in an approach analogous to the one used for the binary system. Then the field autocorrelation function is the sum of three exponentials. If x >> D, >> D,, with the appropriate choice of correlation times, D1 and D, can be resolved experimentally. In this study, the dilute ternary systems were mixtures of COz in n-heptane containing AOT inverse micelles as probe particles where D1 could be identified as DAB and D, as D,, the diffusion coefficient of the particles. Then, from measurements of D, and knowledge of the hydrodynamic size of the particles, r,, the viscosity of the medium, p, could be calculated from the Stokes-Einstein relation with kB and T being Boltzmann's constant and the absolute temperature respectively.
Results and Discussion All the measurements reported in this investigation were made in the two-phase region. In Figure 1, the thermodynamic states scanned for the diffusion coefficient and viscosity measurements are plotted along various constant overall composition curves. We studied seven binary mixtures ranging from 30 to 89 mol % CO,. After sufficient time was allowed for equlibrium at each state, the correlation functions were measured for various delay times and scattering angles. In homodyne detection, the intensity autocorrelation function, G(,)(T)is measured. C(,)(T)is related to g(')(T) given in eq 4 by
G ( 2 ) (= ~ )€411
+ fllg(1)(~)12]
(6)
where B , is the background and p is an adjustable factor. Therefore, the square roots of the measured correlation functions were fitted to eq 4 by using a nonlinear least-squares algorithm. x values were varied parametrically and the values of x and DAB corresponding to the best fit were chosen. The contribution of the concentration fluctuations to the scattered light was dominant because the refractive indices of CO, and n-heptane are different by -30%. However, we could see deviations from a single-exponential behavior at shorter times and at lower concentrations of C 0 2 . In Figure 2 a typical measured correlation function is shown along with the deviation plot of the double-exponential fit used in determining x and DAB. On the same plot the singleexponential function fitted to the tail end of the correlation function is also shown. Values of x and DAB along seven constant-composition curves are listed in Table I. The errors assigned to DAB include the reproducibility of the thermodynamic state and the effect of the
138 The Journal of Physical Chemistry, Vol. 88, No. 1. 1984
Saad and Gulari r
1x10
-5
32 48 64 Delay chonnel number Figure 2. Plot of measured correlation function vs. delay channel number (closed circles) at Xco2 = 0.74, T = 19.5 O C , P = 4.29 MPa, for delay - Yfitted)/ time = 1 ps, 0 = 9.07O. Percent deviations [=lOO(Ymasured Ymeasurcd] of the double-exponential fit with x = 0.6 X lo-' cm2/s and DAB= 4.13 X cm2/s are also shown. The curve corresponds to the I6
0
single-exponential fit after deletion of the first 25 data paints yielding DAB = 4.10 X cm2/s. TABLE I: Values of D A B and x
0.30 i. 0.01
0.53 t 0.01
0.60 i. 0.01
0.69
0.74
t
0.01
i. 0.01
10.0 19.9 29.9 40.0 50.0 9.9 19.5 29.6 40.0 50.0 9.9 19.5 29.6 40.0 50.0 9.9 19.5 29.6 40.0 50.1 9.9 19.5 29.0 40.0 50.1
0.84 t 0.01
0.89
t
0.01
9.9 19.5 29.6 40.0 10.0 19.5 29.5 40.0
1.72 1.90
2.13 2.34 2.56 2.67 3.04 3.42 3.77 4.07 2.98 3.48 4.01 4.55 5.05 3.29 3.87 4.46 5.06 5.59 3.56 4.29 5.06 5.87 6.60 3.75 4.66 5.69 6.85 4.07 5.03 6.24 7.31
0.7 i. 0.1 0.7 i. 0.1 0.7 i. 0.1 0.65 i. 0.1 0.6 t 0.1 0.7 t 0.1 0.7 i: 0.1 0.6 t 0.1 0.6 i: 0.1 0.6 f. 0.1 0.7 * 0.1 0.6 t 0.1 0.6 i: 0.1 0.6 i. 0.1 0.6 i. 0.1
0.6 t 0.1 0.6 i. 0.1 0.6 i: 0.1 0.55 i. 0.1 0.55 i. 0.1 0.6 i. 0.1 0.6 t 0.1 0.5
i
0.1
0.5 i: 0.1 0.5 i. 0.1 0.5 i; 0.1 0.5 i. 0.1 0.5 i 0.1 0.5 i 0.1 0.5 t 0.1 0.4 i. 0.1 0.4 t 0.1 0.3 i: 0.1
4.74 f. 0.20 5.63 i. 0.15 6.61 t 0.13 6.65 t 0.30 7.72 f 0.13 4.28 t 0.05 5.00 i. 0.05 5.73 * 0.05 6.53 i. 0.05 7.30 t 0.05 4.06 i 0.05 4.62 * 0.05 5.39 i 0.05 6.17 i. 0.05 6.96 i. 0.05 3.70 t 0.05 4.39 i 0.05 5.19 t 0.05 5.90 i. 0.05 6.60 t 0.10 3.47 i. 0.05 4.13 i. 0.05 4.77 t 0.05 5.40 i. 0.05 6.14 i 0.10 3.62 i 0.05 4.45 i. 0.05 5.12 t 0.05 5.60 i 0.05 4.89 i: 0.10 5.46 t 0.10 6.16 * 0.10 5.92 t 0.20
parametric variation of x. x values varied from 0.7 X to 0.3 X cm2/s over the range of the data. The trends were such that x decreased with increasing temperature and COz content. x of heptane at 20 O C is 0.89 X loT3cm2/s. x of saturated liquid COz at 20 O C was measured6 and reported as 0.2 X cmz/s. Given these values for the pure components, x values listed in Table I for various mixtures seem to be resonable. When D A B is plotted vs. pressure along isotherms, it goes through a minimum as seen in Figure 3. Using a wetted-wall column apparatus, Davies et ala7have measured the diffusion coefficient of COz in (6) E. Reile and J. Straub, "Proceedings of the 8th Symposium on Thermophysical Properties", American Society of Mechanical Engineers, New Yoik,. 1982, p 463. (7) G. A. Davies, A. B. Ponter, and K. Craine, Can. J. Chem. Eng., 45, 372 (1967).
l hA l 7.
t
l
l
2
3
I
I
I
I
4 P(MPa) 5
I
I
7
6
I
I
8
Figure 3. Plot of diffusion coefficient vs. pressure along isotherms. Open symbols represent DAB measurements of the binary systems; solid symbols
represent DAB measured simultaneously with viscosity for systems containing AOT micelles.
b
77 %
3x10-
5
0
20
40 Temp ( " C )
1
60
Figure 4. Plot of minimum value of
D A B along five isotherms vs. temperature. The overall composition of the mixture at which the minimum DABoccurred is indicated as mole percent COz.
heptane to be 6.03 X cmz/s at 25 "C. Their value is in good agreement with out measurements of D A B = (5.63 f 0.13) X and (6.61 f 0.13) X cmz/s a t 20 and 30 O C , respectively, for a loading of overall Xco2 = 0.3. D A B measurements were made in the composition range for which the total pressure remained linear with the overall Xco,. We find that the minimum value of D A B corresponds to an overall composition of 75 f 1 mol % COz and increases linearly with temperature as shown in Figure 4. For purposes of determining the behavior of the viscosity in the range where D A B goes through a minimum, we have studied six mixtures of COz in n-heptane containing AOT dissolved at a concentration of 5% by weight. The thermodynamic states along four of these constant-composition curves are shown in Figure 1. In hydrocarbons, AOT forms monodisperse inverse micelles, which are 1.55 A 0.05 nm in radius, and this size is independent of pressure and is very slightly dependent on temperature as reported by Zulauf and Eicke.* We have confirmed their results and the sizes of AOT micelles that we have measured in n-heptane ranged from 1.55 nm at 20 O C to 1.45 nm at 60 OC. With AOT micelles as probe particles DAB/DPvaried from -5 to 10. The correlation functions were measured for delay times of 1, 5, and 10 g s per channel corresponding to correlation times of 0.064,0.32, and 0.64 (8) M. Zulauf and H. F. Eicke, J . Phys. Chem., 83, 480 (1979).
Diffusion of Carbon Dioxide in Heptane
The Journal of Physical Chemistry, Vol. 88, No. 1, 1984 139
-
TABLE 11: Diffusion Coefficient of AOT Micelles and the Corresponding Viscosities overall
xc0,
0.56 i 0.01
Y
106D , T, “C P, M P ~
9.9 19.9 30.0 40.0 49.9
b,QcP
cm2L
3.02 3.48 3.92 4.33 4.68 3.54 4.23 4.92
5.90i- 0.02 6.64 i 0.04 7.20i- 0.05 8.01 i 0.03 8.70i 0.05 0.72 i 0.01 10.0 7.40 * 0.10 20.0 8.40i 0.10 30.0 9 . 3 0 i 0.10 40.0 5.58 10.00 t 0.10 49.9 6.17 10.70 i 0.10 0.78 i- 0.01 10.0 3.77 8.71 i: 0.10 20.0 4.63 9.91 i. 0.10 30.0 5.57 11.70 f 0.10 40.0 6.52 12.60 i- 0.10 50.0 7.45 13.80i 0.20 0.87 f 0.01 10.0 4.01 9.30 i: 0.20 20.0 4.91 10.60 i 0.20 29.9 5.88 11.90 i 0.20 40.1 6.92 13.20t 0.20 50.0 7.78 13.60 i 0.20 a Viscosities are calculated for rH = 1.50 nm.
0.234 * 0.001 0.215 f 0.001 0.205 i: 0.01 0.191 i: 0.01 0.181 i 0.01 0.187 i 0.02 0.170 i: 0.02 0.159 i 0.02 0.153 i: 0.02 0.147 i 0.02 0.159 i: 0.02 0.144 i 0.02 0.126 i 0.02 0.121 i 0.02 0.114 i 0.02 0.149 i 0.03 0.135 i 0.03 0.124 i 0.02 0.1 16 i: 0.02 0.1 16 f 0.02
< ’
03
t I
3
4
5
I
I
I
6
7
8
0
On the basis of the statistical mechanical approach by Bearmang DA,/A/FTis independent of composition. F [=(a In a l a In is a thermodynamic factor with a and C being the activity and concentration, respectively. According to Hartley-Crank theoryl0 based on the hydrodynamic approach DA&/FT is linear in volume fractions and is independent of temperature. We have no means of estimating F, which required knowledge of the liquid-phase compositions. However, in a plot of DAB/A/Tvs.pressure (Figure 6 ) we observe that, over a pressure range that corresponds to overall Xco, > 75%, DAB/A/ T remains independent of pressure and temperature. For lower C 0 2 contents the behavior of DAB/A/T is complex and data on liquid- and gas-phase compositions are needed to understand the dependence of DAB on temperature, pressure, and concentration.
Pressure ( MPa)
Figure 5. Plot of viscosity vs. pressure along isotherms. The points along isotherms represent various compositions. ms with a 64-channel correlator. These data were superimposed and analyzed in terms of double-exponential fits to determine the short- and long-time behavior of the measured correlation functions and hence obtain both DAB and D,. DAB values measured for the seeded system (solid symbols on each isotherm) are in excellent agreement with DAB measurements of the binary system (open symbols on each isotherm) as shown in Figure 3. The liquid viscosities calculated from the measured diffusion coefficients of the probe particles are listed in Table 11. When the viscosity is plotted vs. pressure along isotherms it decreases montonically as seen in Figure 5 .
7
dyn/K.
-
2
6
Figure 6. Plot of D,B~/Tvs.pressure. The lines indicate D,Bp/T values along constant-tempertaturelines which merge to a value of 2.2 X
II ( C P )
O.I2
5
Pressure ( MPa)
0.20 -
016
4
,
Summary We find that PCS is a very suitable method for studying diffusion of dissolved gases in liquids at nonambient conditions which is very difficult by others methods. We have introduced a convenient method for measuring liquid viscosities using probe particles. In this investigationAOT micelles of 1.50-nm radii were used as probe particles. The reliability of the viscosity values obtained depends heavily on the assumptions that the micelles are of known uniform size and that there are no interactions between the scattering micelles. In the composition and temperature ranges of this study which are far away from any critical loci, diffusion of C 0 2 in n-heptane exhibits complex behavior. At constant temperature, we observe that DAB goes through a minimum while the viscosity is decreasing continuously. For overall Xco, > 75%, DABfi/T remains independent of pressure and temperature at a value of 2.2 x 10-lo dyn/K. Further studies on the equlibrium phase compositions are in progress. These measurements should provide means to estimate the thermodynamic factor F and explain the composition dependence of measured DAB. Acknowledgment. This work was supported by National Science Foundation Grant No. CPE 80-21952. Registry No. Carbon dioxide, 124-38-9; heptane, 142-82-5. (9) R. J. Bearman, J . Phys. Chem., 65, 1961 (1961). (10) G. S. Hartley and J. Crank, Trans. Faraday SOC.,45, 801 (1949).
