J . Phys. Chem. 1984,88, 5409-5412
5409
coefficients in eq 13 and 14 indicate that the rate-determining step is between I1 and I11 and is relatively close to 11, Le., kz3 >> k32 in reaction 8. It is noteworthy that the dependences of y2kf and kbon K , in the present system are moderate compared to those in the protolysis of carboxylic acids in aqueous s o l u t i ~ and n~~~~~ the base equilibrium of amines in a SDS micellar solution.6 These results imply that kal and k23 are almost of the same order and one cannot be neglected compared to the other in eq 9 and 10. Consequently, together with the steady-state requirement of 11, we can deduce the following relationship for the rate constants in reaction 8: kZl
6
PKa
Figure 6. K, dependences of the rate constants of the protolysis of carboxylic acids in the presence of DAC (0.20 M) (large circles) and in the absence of the micelles (small circles, data from ref 22). The carboxylic acids cited for the latter case are not corresponding to those for the former case.
constant Kb. Assuming that k12,kZ1>> k23, k32 in eq 9 and 10, the overall rate constants are given by kf = (k12/k21)k23 kb =
k32
(11)
(12)
These equations provided a satisfactory interpretation for the Kb dependences of the rate constants. As a consequence, it was concluded that the intra-ion-pair proton transfer, corresponding to the process I1 * I11 in reaction 8, is the rate-determining step in the base equilibrium of amines in a micellar system of SDS. In the present system, the K, dependences of y2kfand kb, shown in Figure 6, are expressed by y2kf = 109.OK -0.18 a (13) kb = 109.OK 0.82 a (14) According to the Bronsted catalysis the Bronsted coefficient, the index of Ka or Kb, expresses the degree of proton transfer in the rate-determining step. Since the degree of proton transfer is zero in I and I1 while unity in I11 in reaction 8, the Bronsted (23) Breslow, R.“OrganicReaction Mechanisms”; W. A. Benjamin: New
York, 1969.
k23
>> k129
k32
(15)
As shown above, the micellar effect on the protolysis of aromatic carboxylic acids is smaller than those on the base equilibrium of amines, Le., statically, the difference of the pK, values (ApK,) in the presence and in the absence of the DAC micelle is 0.9-1.3 which is smaller than the corresponding value of pKb in the amineSDS system, 1.4-1.8: and, kinetically, the pK, dependences of the rate constants are closer to those in an aqueous solution in the absence of a m i ~ e l l e ’ ~compared ,~* to the corresponding ones in the amine-SDS system.6 The reason for this small micellar effect is not clear in the present stage. As shown by NMR s t ~ d i e sthe , ~aromatic ~ ~ ~ ~carboxylic ~ ~ ~ acid is solubilized at the surface of the cationic micelles with their aryl groups fitting betwcen the ammonium head groups of the surfactant and the carboxyl groups protruding into the water-rich Stern layer. According to this picture, the interaction of the carboxyl group and the cationic ammonium head groups seems to be weaker than that of the amine group and the sulfate groups in the base equilibrium of amines in the SDS micelle.6 However, our preliminary experiments showed that the ApK, values of the n-alkylcarboxylic acid-DAC system were almost similar to those of the aromatic carboxylic acid-DAC systems: this fact contradicts the above idea. The reason may rather be ascribed to the difference of the properties of the micelle, e.g., charge density which is dependent on the degree of packing of surfactant molecules or the degree of binding of counterions. Further detailed studies with micelles of various detergents will contribute to clarifying the mechanism of the micellar effects on the protolysis of carboxylic acids and the base equilibrium of amines. Registry No. o-N02BA, 552-16-9; o-ClBA, 118-91-2;m-NO,BA, 121-92-6;m-ClBA, 535-80-8;BA, 65-85-0; DAC, 929-73-7. (24) Bunton, C. A.; Minch, M. J.; Hidalgo, J.; Sepulveda, L. J. Am. Chem. SOC.1973, 95, 3262. (25) Erickson, J. C.; Gillberg, G. Acia Chem. Scand. 1966, 20, 2019.
Partial Molar Volumes of Atmospheric Gases in Water N. Bignell CSIRO Division of Applied Physics, Lindfield, New South Wales, Australia 2070 (Received: March 5, 1984; In Final Form: May 14, 1984) The change in the density of water when nitrogen, oxygen, and argon are dissolved has been measured from 3 to 21 OC and the weighted sum of the results compared with the result for air. Good agreement was found, contrary to earlier reports. The partial molar volumes for the three gases have been calculated. These results are discussed in the light of current theories of liquid water. Introduction This paper presents the results of measurements of the change in the density of water when oxygen, nitrogen, and argon are dissolved in it. These gases are of course the main constituents of air, and a previous paper’ has reported the change in water (1) Bignell, N. Metrologia 1983, 19, 57.
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density on saturation with air over the same temperature range, 3-21 ‘(2. b.uder2 f m n d that the separate effects of the gases on the density were not additive and suggested a, more detailed investigation. The results of measurements reported here show that if the individual changes are added with their appropriate (2) Lauder, I. Aust. J . Chem. 1959, 12, 40.
0 1984 American Chemical Society
5410
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984
mole fraction weighting, then the measured change in density for air is obtained. The measurement of density change enables the partial molar volumes of the gases in water to be calculated and usefully extends the data available. Current theories of liquid ~ a t e r regard ~ . ~ it as a disordered network of hydrogen bonds below a percolation temperatureS so that a continuous chain of hydrogen bonds exists throughout. Stillinger and others6 have suggested that water molecules form themselves into relatively unstrained, bulky, hydrogen-bonded polyhedra as the temperature drops to 0 OC and below. This is a mechanism for decreasing the density as the temperature drops whereas ordinary thermal expansion decreases the density as the temperature rises, hence the origin of the well-known density maximum. A nonpolar solute molecule must displace the network which is thought to form a solvation cage around it. Though it is an imperfect polyhedron, Stillinger claims that it can serve as a template and a stimulus for the formation of attached polyhedra. This structuring of the water molecules in the vicinity of nonpolar solute molecules is known as hydrophobic hydration. The temperature of maximum density (TMD) is expected to rise in solutions of nonpolar molecules because of their stimulating effect on the growth of these bulky polyhedra. The TMD has been estimated from the results of the measurements presented here. Hydrophobic interaction refers to the situation which applies when two or more solute molecules are present, and it is generally thought to be an attractive interaction since two nonpolar molecules close together can share a solvation cage with less total ordering of the water molecules. Pratt and Chandler' and Pangali Rao and Berne8 have calculated, by different methods, that the potential of mean force between two solute particles exhibits two minima, one which corresponds to the particles in contact, as conventional views of the hydrophobic interaction would have expected, and another which corresponds to the solute molecules separated by a water molecule. The energy difference between the two levels is about half the value of Boltzmann's constant times room temperature so that the relative numbers of pairs in the two states will change significantly from 0 to 30 O C . Since the lower state has less volume, we expect to see some increase in the average volume occupied per solute molecule as the temperature rises. This is working in the opposite direction to the above-mentioned effect, the formation of polyhedra which join to the solvation cage and increase the volume per solute molecule as the temperature falls. From the density measurements the volume occupied per solute molecule has been calculated as a function of temperature.
Experimental Details Only a short account will be given here as the magnetic float apparatus and the procedure have been described in detail elsewhere.g The apparatus consists of a vacuum-insulated, temperature-regulated water bath in which is a glass vessel containing the water sample and a float. The 126-mL glass float containing a small magnet is sensed optically and its submerged position controlled by variation of the current in a coil. When this changes by Ai, the change in density, Ap, is given by Ap = cAi where c is a constant depending on gravitational acceleration, the volume of the float, the strength of the magnet, and the geometry of the coil magnetic system. This geometry has been chosen so that the magnet is (3/2)1/2rfrom the plane of the coil, where r is its radius. For this condition coil temperature changes have a negligible effect on the value of c . ~The laboratory air conditioning is excellent but has an upper limit of about 22 "C. If the sample temperature exceeds this, then convection effects disturb (3) Stillinger, F. H. Science 1980, 209, 451. (4) Angell, C. A. Recherche 1982, 13, 584. (5) Stanley, H. E.; Teixeira, J. J . Chem. Phys. 1980, 73, 3404. (6) Geiger, A.; Rahman, A.; Stillinger, F. H. J. Chem. Phys. 1979, 70,263. (7) Pratt, L. R.; Chandler, D. J . Chem. Phys. 1977, 67, 3683. (8) Pangali, C.; Rao, M.; Berne, B. J. J . Chem. Phys. 1979, 71, 2975. (9) Bignell, N. J . Phys. E 1982, 15, 378.
Bignell 17 35
16
15
e
E'
14
34
?
E 13
Y D
?
52 -
33
2
g Y
1
g o
32
Y2
31
4 zj
4
8 -1 * t -2 n
E 7
s?a
' 2
-3
2
30
-4
-5
9a
-6 -7
0
2
4
6
8
10 12
14
16 18
20 22
TEMPERATURE ( " C )
Figure 1. Graphs of the changes in density for standard atmospheric pressure, Ap, and the partial molar volumes, plotted against temperature for nitrogen, oxygen, and argon. The graph of Ap for air and the curve calculated from the separate measurements are also shown.
vg,
the float and measurements are very difficult. For the same reason measurements below 4 OC become more difficult, though much more slowly, so that work a t 3 O C is still satisfactory. The water is rendered air-free by boiling and is stored under its own vapor pressure. It is brought to approximately the temperature of measurement before transfer to avoid volume hysteresis effects in the float. An excess is transferred and the initial portion allowed to flow to waste to ensure that the sample is uncontaminated by air. When temperature equilibrium is reached, a reading of the current is taken. Gas is then bubbled through the sample for 15 h, a time sufficient for saturation, and a further reading of the current taken to give Ai and hence Ap. The oxygen used was 99.999%, the nitrogen 99.95%, and the argon 99.99%. Since the change in density is a bulk property and the effect is proportional to the partial pressure of the gas, then extremely pure gases are not needed.
Results The density changes associated with the saturation of the water by argon, nitrogen, oxygen, and air have been measured over the temperature range 3-21 "C.In the course of the measurement, atmospheric pressure changed from day to day, affecting the partial pressure, P,, of the solute gas, which is given by the equation where PAtm is the atmospheric pressure and pwgh is the pressure due to a head, h, of water. The term 2 y / r is the excess pressure inside a bubble of radius r in a liquid of surface tension y and P,( T ) is the vapor pressure of water at the temperature T OC. The density change was corrected to that for a partial pressure of 101.325 kPa by assuming that it is proportional to Pg. The lowering of the vapor pressure of water according to Raoult's law by the dissolution of the gas is small enough to be neglected. The observed changes in the density of water are shown in Figure 1, where the dots represent the means of repeated readings at that temperature and a straight line or quadratic of best fit is also shown. The equations to these are given in Table I. The standard deviation of repeated readings at the one temperature is about 0.15 X kg m-3. The dashed lines surrounding the argon line represent 95% confidence limits; for the other curves slightly better confidence limits will apply since more points were taken.
Partial Molar Volumes of Atmospheric Gases in Water TABLE i: Experimental Curves"
curve Density Change on Saturation with Gas at 101.325 kPa, Ap,b and Weighted-Sum Curve Ap = -6.827 + 0.172t - 0.00243t2 N2 (66 PtS) Ap = 0.721 + 0.131t - 0.00457t2 0 2 (76 ~ t s ) Ap = 16.842 - 0.18962 Ar (31 pts) Ap = -4.911 + 0.1672 - 0.00277t2 air (77 pts) sum Ap = -5.005 + 0.160t - 0.00286t2 Partial Molar Volumes, P = 34.49 - 0.00822 P = 31.729 - 0.0919t + 0.00267 t2 Y = 32.676 - 0.06709t
N2 0 2
Ar
"The equations for Ap(t) and P&f)for nitrogen, oxygen, and argon; for air is also given and the weighted sum of Ap(t) curves, 0.78N2 + 0.2102 + O.01Ar. bAp/(10-3 kg m-3), t/'C cV/(10-3 m3 mol-'), t/'C. Ap(t)
For a system of amounts ng and n, of gas and water the volume Vis given by the standard result
v = ngVg + nwVw where Vgand V, are the partial molar volumes of gas and water. If M g and M , are the molar weights, then the density of the solution is P
= (ngMg + nwMw)/(ng& + nwVw)
where X, = n,/n, is approximately equal to the mole fraction for these dilute solutions. Because X, is small, we can write V, = V,, the molar volume of water, and
Using
pw
(P
= Mw/Vwwe have
- P w ) / P w = AP/Pw =
X , M , / M W
- XgVgPw/Mw
Hence r g
= M J P W - APMw/(Pw2Xg)
X , can be estimated in the case of pure gases by using the Henry's law coefficient, H , and the partial pressure, Pg,as Xg = P,/H and hence
The partial molar volume Vg will in general be a function of concentration. However, as shown by Girard and Coarasa,lo the change in density for air is proportional to the quantity dissolved, and if this is true for the constituents of air separately, then the term ApM,/(pw2Xg)will remain unchanged as Xg goes to zero and hence the value of Vgfound here should be the same as that a t infinite dilution. The partial molar volumes for oxygen, nitrogen, and argon have been calculated by using eq 1 and the values of H quoted by Kell." The values at the various temperatures have been fitted to curves, as for the density changes, and are shown in Figure 1, and the
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5411 equations are given in Table I. The values given by Moore et al." and by Kell" show that there is considerable scatter in the literature values. The accuracies of the results given here, based on the standard deviation of repeated readings of the density change, are as follows (in units of m3): argon, 0.08; oxygen, 0.10; nitrogen, 0.18.
Discussion Lauder2 measured the change in the density of water on saturation with nitrogen, oxygen, and air at 0 "C and found values of -9.3, +1.8, and -4.7 ppm atm-'. H e then added the values for nitrogen and oxygen in the proportion 0.78 and 0.21 and obtained -6.9 ppm atm-'. The results presented here do not support this nonadditivity. If the curves representing the density changes as functions of temperature are added in the proportions above, and including 0.01 times the argon result, then the resultant is in agreement with the experimentally obtained curve for air within the limits of experimental error. These two curves are plotted in Figure 1 and their equations are given in Table I. Though Lauder's result for air agrees with the result here, his values for oxygen and especially for nitrogen do not. Maharajh and Walkley,16 in measurements on gas mixtures, found the saturation solubility of each gas reduced below the Henry's law value. According to their results for mixtures of nitrogen and oxygen at 25 "C the appropriate factors by which to multiply the density curves are not 0.78 and 0.21 but 0.71 and 0.15. If the curves are added in this proportion, with again 0.01 times the argon result, then a curve is obtained with departures from the measured result for air in the other direction and of about the same magnitude, Le., within experimental error. Density measurement is not in fact a sufficiently sensitive technique to investigate these departures from Henry's law, particularly as the effect of nitrogen on the density is of opposite sign to that of argon and oxygen so there tends to be a compensation. Further data on this effect at other temperatures than 25 "C and for gases of the composition of air would be welcome. The measurement of the change in the density reported here allows an estimate of the change in the temperature of maximum density (TMD). The percolation theory of Stanley and TeixeiraS and the specific prediction of Stillinger3 indicate that this temperature should increase for argon, a definitely nonpolar solute. However, the Ap-T curve for argon drops with rising T , which would indicate the opposite. This, however, is not for a solution of constant composition and merely reflects the change in the solubility with temperature. When this is allowed for with Henry's law, using the values quoted by Kell," we can obtain the density change per mole. By combining this result with the density curve for pure water and finding the TMD, one obtains a value for the change in the TMD of 252 K mol-' for these dilute solutions. For oxygen the value obtained is 221 K mol-' using the Henry's law constants quoted by Kell," and for nitrogen 38 K mol-', using the solubility data given by Battino,I2 and 12 K mol-' using the Henry's law values of Kell." There are obviously great differences in the effects on the density of the three solutes studied but it must be remembered that they have different molecular weights and different solubilities. The partial molar volume takes account of these differences. When it is expressed as the volume per molecule (partial molecular volume) in units of the average volume occupied per water molecule, a more revealing quantity is obtained, denoted here by f. Thus,
f = Mg/Mw (10) Girard, G.; Coarasa, M. J. "Proceedings of the Second International Conference on Precision Measurements and Fundamental Constants, June 1981". (11) (12) (13) (14) (15)
Kell, G. S. J. Phys. Chem. ReJ Data 1977, 6, 1109. Battino, R.Solubility Data Ser. 1982, 1 1 , 1. Figure 7a of ref 3. Figure 7c of ref 3. Stillinger, F. H.; Weber, T. A. J. Phys. Chem. 1983, 87, 2833. (16) Maharajh, D. M.; Walkley, J. J. Chem. SOC.,Faraday Tram. 1 1973, 69, 842. (17) Moore, J. C.; et al. J . Chem. Eng. Data 1982, 27, 22.
-
APH/(PwPg)
For the calculation off the values of A p ( r ) from the fitted curves have been used rather than the experimental points, as was used for the partial molar volume. The values off obtained are shown in Figure 2. It must be remembered that these are the volumes occupied per molecule, not per atom, though for argon this is the same thing. The f vs. T curve for argon in Figure 2 tends asymptotically to a value of about 1.72 at high temperatures but rises more and more steeply as T falls to 0 OC. This could be explained by
J . Phys. Chem. 1984,88, 5412-5416
5412
can be similarly analyzed to give an f value of 1.67, a little lower than observed. A further polyhedron14of intermediate complexity, 15 vertices, consists of two parallel pentagons linked vertex to vertex by bonds to an oxygen. The volume of the inscribed sphere of this figure yields anfvalue of 1.73. This excellent agreement must not of course be taken to mean that this polyhedron and no other is the one to be found in the structure. The volume of an inscribed sphere is not necessarily the correct one to take, though if the argon atom were to vanish, the polyhedron would not collapse to zero volume so that the full geometric volume of the solid figure would be an overestimate. It also should be noted that the polyhedron would be expected to be distorted6 even when averaged over time and Many argon atoms as we effectively have done here. The almost constant volume occupied by nitrogen and the minimum at about 17 OC for oxygen certainly require explanation. The increasing occupation of the upper minimum in the potential of mean force curve for a pair of solute particles mentioned in the Intrduction may explain the oxygen minimum but this effect is hard to quantify. Perhaps the argon curve would show a similar rise if continued to higher temperatures. That the volume occupied by nitrogen remains relatively uninfluenced by temperature and the very low value for the change in the TMD suggest a lack of interaction with the network of the type described for argon or alternatively a so complete involvement that it becomes part of it and hence the volume occupied relative to the network remains unchanged. As Stillinger and Weber have remarked in a recent paper,15 molecular dynamics studies of quenches of solutions of nonpolar molecules should show some of the structures mentioned here. It is hoped that the form of the f vs. T curves found here will be confirmed by such studies. Registry No. N2, 7727-37-9;02,7782-44-7;Ar, 7440-37-1;water, 7732-18-5.
VOLUME OCCUPIED PER MOLECULE
lSg5
F- L N2
1.90 7 r
c
I
1 . 7 O t ' " " " " " " " " " " " " 0 5 10
TEMPERATURE
15
20
25
( C)
Figure 2. Graphs off, the volume occupied per solute molecule, plotted against the temperature for nitrogen, oxygen, and argon. The unit off is the average volume occupied by a water molecule, m,/p,.
equating the value of 1.72 with the volume of the inscribed sphere of the solvation cage formed around the argon by the hydrogenbonded network. As the temperature falls, the presence of this polyhedral cage stimulates the formation of polyhedra adjoining it, thus increasingf, the effective volume occupied per argon atom. It has been noted3g6that the nearly tetrahedral bond angles applicable to water are also present in certain polyhedra. Stillinger3 gives some of these as possible examples of solvation cages, one of which is a dodecahedron. The volume of the inscribed sphere of a dodecahedron with an edge of 2.64 A (Stillinger3) gives an fvalue of 3.56, much higher than the value 1.72. Another much simpler polyhedron of only eight vertices suggested by Stillinger"
Calculating Ternary Diffusivities with Dominant Cross Terms W. Jay Lechnickt and Joseph A. Shaeiwitz* Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61 801 (Received: March 5, 1984; In Final Form: June 1, 1984)
A pattern search method has been developed to calculate ternary diffusivities for situations in which one of the cross-term coefficients dominates, because previous methods provded unable to handle such a situation. The method employs direct calculation of concentrations and residuals instead of approximating them through a Taylor series expansion. Weighting factors are included in order to reduce the statistical importance of the larger concentration changes caused by the magnitude of the cross-term diffusivity relative to that due to the main-term diffusivity. Results show that this method is indeed superior for systems with large cross-term diffusivities and that this method gives identical results for systems in which the cross-term diffusivities are smaller than the main terms.
Introduction The diaphragm cell is well suited for studying ternary diffusion in those systems where concentrations are readily determined by means of physical and/or chemical analysis. Any difficulty associated with the diaphragm cell lies in the data analysis, so much of the previous work has focused on developing a sound statistical method for analyzing the concentration data to obtain the ternary diffusion coefficients. For example, Burchard and Toor' replaced the exponential factors in the diaphragm cell equations by a linear expansion approximation and determined the coefficients through an analysis of variance. Kelly and Stokes2 performed a series of experiments of varying duration in which the initial concentration Present Address: AMOCO Research Center, Naperville, IL 60566. Department of Chemical Engineering, West Virginia University, Morgantown, WV 26506.
* Present Address:
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difference between compartments of one component was equal to zero and were able to obtain excellent results. Kim3 developed a theory, primarily for use in quaternary systems but certainly adaptable to ternary systems, in which coefficients were determined by integrating the concentration differences over time. All of these methods have limitations on their use either because of experimental inconvenience of complications (ref 2 and 3) or theoretical limitations (ref 1). Cussler and Dunlop4 solved most of these problems by using a least-squares method based on the optimization technique of Newton-Raphson in which they assumed an initial value and then computed a correction term. Through the (1) Burchard, J. K.; Toor, H. L., J . Phys. Chem. 1962, 66,2015. (2) Kelly, F. J.; Stokes, R. H. "Electrolytes", Pesce, B.; Ed.; Pergamon Press: Oxford, 1962; pp 96-100. (3) Kim, H. J . Phys. Chem. 1966, 70, 562. (4) Cussler, E.L.; Dunlop, P. J. J . Phys. Chem. 1966, 70, 1889.
0 1984 American Chemical Society
