3165
NOTES
which is 1.58 X ~ m mo1e-1,18 . ~ 1.1 is the dipole moment of the water molecule, T is degrees Kelvin, and g is the Kirkwood correlation factor that accounts for orientation between neighboring molecules) was used to calculate p2g from the various values of dielectric constant. Densities of water were obtained from the tabulation of V o u k a l ~ v i t c halong ~ ~ the liquidvapor equilibrium curve, and from those values of K e n n e d ~ ~ Oand - ~ ~Sharp26at higher temperatures and pressures. Values of p2g calculated in this manner from several sets of data are shown in Figure 1. At low densities (and of necessity at high temperatures for water, which has a critical temperature of 374" and critical density of 0.32 g. c m . 7 , short-range interactions between molecules (e.g., hydrogen bonding) should become small, and, as density approaches zero, the Kirkwood correlation factor theoretically should
Estimation of the Dielectric Constant of Water to 800"
by Arvin S. Quist and William L. Marshall Reactor Chemistry D i v f i n , Oak Ridge N a t w d hboratwu, Oak Ridge, Tennessee (Received March 8,1966)
The electrical conductances of dilute aqueous solutions of KzS042and other salts have been measured in this laboratory at temperatures to 800" and pressures to 4000 bars. In order to use the Onsager limiting law3 to calculate the theoretical variation of conductance with concentration, the dielectric constant of the solvent must be known. Although this constant for water has not been measured a t temperatures above 393", it has been estimated to 800" and densities of 1.0 g. ~ m . by - ~ F r a n ~ k . These ~ estimates were based (1) Research sponsored by the U. S. Atomic Energy Commission on a graphical fit of the Kirkwood equation for the under contract with the Union Carbide Corp. dielectric constant of polar liquid^^*^ to several sets of (2) A. S. Quist, E. U. Franck, H. R. Jolley, and W. L. Marshall, J . Phys. Chem., 67, 2453 (1963). experimental data. Since 1956 additional results on (3) L. Onsager, Physik. Z . , 28,277 (1927). water have been published. Therefore, we have (4) E. U. Franck, 2.physik. Chem. (Frankfurt), 8 , 107 (1956). reviewed the previous and more recent data and have (5) J. G. Kirkwood, J. Chem. Phys., 7 , 911 (1939). applied computer techniques in an attempt to obtain (6) G. Oster and J. G. Kirkwood, ibid., 11, 175 (1943). better estimates of this constant a t high temperatures (7) J. Wyman, Jr., Phys. Rev., 35, 623 (1930). and pressures. (8) J. Wyman, Jr., and E. N. Ingalls, J . Am. Chem. SOC., 60, 1182 (1938). The experimental data used by Franck were those of Wyman788 (0 to loo", 1 atm.), Akerlof and Oshryg (9) G. C. Akerlof and H. I. Oshry, ibid., 72, 2844 (1950). (10) J. K. Fogo, S. W. Benson, and C. S. Copeland, J. Chem. Phys., (110 to 370" in the presence of vapor), and Fogo, 22, 209 (1954). Benson, and Copelandlo (377 to 393" a t densities from (11) 5. Kyropoulos, 2.Physik, 40, 507 (1926). 0.2 to 0.5 g. ~ m . - ~ ) .Other earlier measurements (12) W. L. Lees, Dissertation, June 1949, Department of Physics, Harvard University. include those of Kyropoulos" (20" to pressures of (13) F. E. Harris, E. W. Haycock, and B. J. Alder, J . Chem. Phys., 3000 kg. cm.-2 or 2942 bars), Lees12 (0 to 50" at pres21, 1943 (1953). sures to 12,000 kg. cm.-2 or 11,770 bars), Harris, (14) B. K. P. Scaife, PTOC. Phys. SOC.(London), B68, 790 (1955). Haycock, and Alder13(25.6" to pressures of 127 atm.), (15) B . B. Owen, R. C. Miller, C. E. Milner, and H. L. Cogan, and ScaifeI4 (20' at pressures to 6000 kg. cm.-2 or J. Phys. Chem., 65, 2065 (1961). (16) G. A. Vidulich and R. L. Kay, ibid., 66, 383 (1962). 5884 bars). More recent measurements are those (17) A. W. Lawson and A. J. Hughes in "High Pressure Physics and of Owen, et aZ.15 (0 to 70" a t pressures to 1000 bars), Chemistry," Vol. I, R. S. Bradley, Ed., Academic Press Inc., New Vidulich and Kayl6 (0 to 40" at atmospheric pressure), York, N . Y., 1963, Chapter 4, iv. and T. E. Gier and H. S. Young" (at 200, 250, 301, (18) R. M. Waxler and C. E. Weir, J . Res. Natl. Bur. Std., A67, 163 (1963). and 350" at pressures to 2000 bars). (19) M. P. Voukalovitch, "Thermodynamic Properties of Water and By using a modification of Franck's a p p r ~ e c h , ~ Steam," 6th Ed., Veb Verlag Technick, Berlin, 1958. the Kirkwood equation (20) G. C. Kennedy, Am. J. Sci., 248, 540 (1950). (2e
+ 1]1(e - 1) - 4 9tS
y a 3M
+
g)
(1)
(where e is the dielectric constant, N is Avogadro's number, k is the Boltemann constant, M is the molecular weight in grams mole-l, CY is the polarizability,
(21) G. C. Kennedy, ibid., 255, 724 (1957). (22) G. C. Kennedy, W. L.Knight, and W. T. Holser, ibid., 256,590 (1958). (23) W. T. Holser and G . C. Kennedy, ibid., 256, 744 (1958). (24) W. T. Holser and G. C. Kennedy, ibid., 257, 71 (1959). (25) W. E. Sharp, "The Thermodynamic Functions for Water in The Range -10 to 1000°C. and 1 to 250,000 Bars," University of California Radiation Laboratory Report UCRL-7118, 1962.
Volume 69,Number 0 September 1966
NOTES
3166
I
I
1
I
I
I
I 1
BENSON, COPELAND +GlER AND YOUNG o FOGO,
t5
m
10
01
than the experimental value at 200". The reverse behavior would be expected. Several types of equations estimating p 2 g as a function of temperature and density were fitted to the data in Figure 1 (excepting those of Gier and Young) by the use of a generalized least-squares program.27 The equation that best fitted the data had the form
t 5
0
0
0.2
0.4 0.6 0;8 DENSITY ( g-cm-3)
1.0
1.2 '
where AI, Az, and As are adjustable parameters, d is the density in g. cm.-S, and f(T) is a function of temperature (OK.) represented by either of the three equations
f(T) = T43
Figure 1. Calculated values of p z g (from the Kirkwood equation) for water.
f(T) = TPA4 f(T) =
approach a value of unity. Accordingly, the dipole moment of the water molecule would approach 1.87, the value reported for an isolated water molecule.26 If the Kirkwood equation is correct, the term pzg should then approach 3.50 (1.872) at low densities and high temperatures. This behavior is observed in Figure 1, where it is seen that p2g extrapolates to approximately 3.50 at zero density. This observation based on experimental measurements substantiates the use of the Kirkwood equation. Values of p2g obtained from the data of Owen, e2 UZ.,'~ and Lees12 initially decrease as the density increases from approximately 1 g. C M . - ~to higher values along the 0 and 10" isotherms. The viscosity of water in this low-temperature region also initially decreases with increasing density. Both pzg and viscosity isotherms at 20" and higher show a steady increase with increasing density. The points at 360 and 370" in Figure 1 calculated from the constants of Akerlof and Oshry deviate from the smooth curve. A corresponding deviation in dielectric constant was mentioned by Akerlof and Oshry. Consequently, these two points were omitted in the determination of equations for p 2 g as a function of tempera,ture and density. Those values of p2g obtained from the data of Gier and Young show some unexpected behavior. At 250, 301, and 350°, with increasing density they extrapolate to p2g values higher than the calculated values at 200" (compared at the same density at densities higher than approximately 0.9 g. cm.-:i). Also, when their experimental dielectric constants are plotted against density (not shown), at a density of 1.0 g. cm.-3 the extrapolated value for the dielectric constant at 250" is somewhat greater The Journal of Phy&
Chemistry
e--44(T)
where Ad is another adjustable parameter. In obtaining the parameters for eq. 2 through 5, several combinations of the various sets of experimental data were tried. Those parameters considered most reliable were obtained by combining and using the data of Owen, et ~ 1 . ' (88 ~ points, 0-70", by 10" intervals, 1lo00 bars, by 100-bar intervals), L e e P (22 points), and Akerlof and Oshryg to 350" (25 points, every 10" from 110 to 350°, as given in the table in their paper), and three values joining Owen's data to the AkerlofOshry data (80, 90, 100"). All data were given equal weight. Each of the functions represented by the combination of eq. 2 with eq. 3, 4, or 5 fitted the experimental data equally well as judged from the standard error of each parameter and variance of fit obtained from the lea&-squares program. The parameters obtained for the combinations of eq. 2 with eq. Table I: Values for Parameters in Eq. 2 and 3, 2 and 4, and 2and5 calcd. e, calcd. at 800° at 200° and and 1.0 e. 1.0 e. cm.-' om.-'
e,
Eq.
AI
AI
. ( 2 ) a n d ( 3 ) 175 136 (2)and(4) 1 1 . 0 167 (2)and(5) 12.2 1 3 . 7
Aa
-111 -94 -9.8
A4
... 0.35 0.00113
15.1 16.9 12.9
42.3 44.1 42.6
(20) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2nd Ed., Butterworths Scientific Publications, London, 1969,p. 1. (27) M. H. Lietzke, "A Generalized Least Squares Program for the I.B.M. 7090 Computer," USAEC Report ORNL-3269,1962.
NOTES
3167
Table 11: Dielectric Constant of Water Calculated from Eq. 1-3 Using the Parameters from Table I Temp., OC.
0 25 100 200 300 400 500 600 700 800
Density, g. em.-8
r
0.0001
1.001 1.001
1.000 1.000 1* 000
0.1
1.83 1.71 1.63 1.56 1.51
0.2
3.13 2.81 2.58 2.40 2.26
0.3
4.9 4.3 3.9 3.5 3.2
0.4
7.1 6.1 5.4 4.9 4.5
0.5
9.7 8.3 7.3 6.5 5.9
0.6
12.7 10.8 9.4 8.3 7.5
0.7
(20.0)" 16.0 13.5 11.7 10.4 9.3
0.8
(34.3)" 23.8 19.5 16.5 14.2 12.5 11.2
0.9
(88.4)" (78.1)" (54.6)" 36.4 28.4 23.2 19.5 16.8 14.8 13.2
1.0
88.4 78.5 58.0 42.3 33.0 26.8 22.6 19.4 17.0 15.1
" For liquid in equilibrium with vapor at the temperatures shown in the heading.
3, 4, or 5 are given in Table I. For comparison, calculated values are included for the dielectric constant of water at 200 and 800" at a density of 1.0 g. ~ m . using these parameters. Since eq. 2 and 3 gave an intermediate value at 800", this combination was used to calculate the dielectric constants for water at integral temperatures and densities given in Table 11. The values are generally somewhat lower (10-2074, at 1.0 g. 3% at 0.6 g. no difference at 0.2 g. c m . 3 than those of F r a n ~ k . With ~ eq. 2 and 3 the average deviation of the calculated dielectric constants from the experimental values used to obtain the parameters was 1.1%. the value of p2g At densities from 0 to 1.0 g. for water might be expected to lie between the limits of 3.50 (where g is unity) and 15.55 (Figure 1, at 0" and d = 1.0 g. ~ m . - ~ ) .By substituting these limiting values for pzg into the Kirkwood equation, boundaries of 6.53 and 23.1 for the dielectric constant were obtained at 800" and a density of 1.0 g. The calculated constant (15.1) given in Table I1 lies within these limits. When only the data along the liquid-vapor equilibrium curve were used to obtain the parameters, it was found that a combination of eq. 2 and 4 gave the best fit, using parameters of 43.0462 (A1), 43.3064 ( A z ) ,- 14.7365 (A3),and 0.318561 (A4). The average deviation of the calculated dielectric constants from the experimental values (from 0-350") was 0.12% with the largest difference (0.39%) at 350". Acknowledgments. The authors wish to thank Professor J. E. Ricci, New York University, for helpful discussions on this work. The advice of Dr. M. H. Lietzke, ORNL, on the use of his computer program is gratefully acknowledged.
Dissociation Studies in High Dielectric
~
Solvents. V.
Magnesium Sulfate as an
Unassociated Salt in N-Methylformamide
by Gyan P. Johari and P. H. Tewari chemistry Department, University of Gorakhpur, Cforakhpur,India (Received May 19,1966)
We have demonstrated that MgSO4 behaves as a slightly associated salt in formamide, and the FuossOnsager theory predicts its conductance data satisfactorily.' This salt has been found as completely unassociated in another amide solvent, N-methylformamide (D = 182.4).2 The solvent was purified by keeping it over anhydrous &C&, refluxing, and fractionally distilling it at least four times under reduced pressure (specific conductance 0.8-1.5 X 10-6 mho cm.-l, viscosity 0.0165 poise, and density 0.99885 g./ ml.; viscosity and density values are in good agreement with the literature valuea). The apparatus and the modus operandi for the conductance measurements have been described in previous communications. The conductance of the solution was obtained by subtracting the conductance of the solvent of the same batch determined separately in a cell. All resistance measurements were obtained from solutions prepared from solvents which were purified less than 24 hr. before. (1) G. P.Johari and P. H. Tewari, J. Phys. Chem., 69, 696 (1965). (2) G. R. Leader and J. F. Gormley, J . Am. Chem. SOC., 73, 5731
(1951). (3) C. M.French and K. H. Glover, Trans. Faraday Sac., 51, 1417 (1955).
Volume 69,Number 9 September 1966
