Determination of Hydration Number of an Electrolyte (7) R. F. Platford and T. Dafoe, J . Mar. Res., 23, 63 (1965). (8) K. H. Khoo, R. W. Ramette, C. H. Culberson, and R. G. Bates, Anal. Chem., 49, 29 (1977). (9) J. R. Campbell and G. H. Nancollas, J. Pbys. Chem., 73, 1735 (1969). (10) K. S. Pitzer and G. Mayorga, J . Solution Chem., 3, 539 (1974). (1 1) L. 8. Yeatts and W. L. Marshall, J. Chem. Erg. Data, 17, 163 (1972). (12) C. C. Briggs and T. H. Lilley, unpublished work. (13) W. L. Marshall and R. Slusher, J . Pbys. Chem., 70, 4015 (1966). (14) B. Elgquist and M. Wedborg, Mar. Chem., 3, 215 (1975). (15) F. C. Gilbert and W. C. Gilpin, J. Soc. Chem. Ind., 65, 111 (1946). (16) A. Manuelli, Ann. Chim. Appl., 5, 13 (1916). (17) E. Posnjak, Am. J . Sci., 238, 559 (1940). London, Ser. A , 232, (18) P. G. M. Brown and J. E. Prue, Proc. R. SOC. 320 (1955). (19) B. T. Doherty and D. R. Kester, J. Mar. Res., 32, 285 (1974). (20) T. H. Lilley and C. C. Briggs, Proc. R. SOC.London, Ser. A , 349, 355 (1976). (211 R. A. Robinson and R. H. Wood. J. Solution Chem.. 1. 481 (1972). (22j M. Whitfield, ”Chemical Oceanography”, Vol. 1, 2nd ed, Academic Press, London, 1975, Chapter 2. (23) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions”, Butterworths, London, 1968. (24) K. S. Pitzer, J . Chem. SOC.,Faraday Trans., 68, 101 (1972). (25) K. S. Pitzer and G . Mayorga, J . Phys. Chem., 77, 2300 (1973). (26) K. S.Pitzer, R. N. Roy, and L. F. Silvester, J. Am. Chem. Sac., 99, 4930 (1977). (27) C. J. Downes, J. Chem. Thermodyn., 6, 317 (1974). (28) W. W. Watson, R. H. Wood, and F. J. Millero, ACS Symp. Ser., No. 18, chapter 6 (1975). (29) Y. C. Wu, R. M. Rush, and G. Scatchard, J. Phys. Chem., 72, 4048 (1968).
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(30) C. J. Downes and K. S. Pitzer, J . Solution Cbem., 5, 389 (1976). (31) J. Padova and D. Saad, J. Solution Chem., 8, 57 (1977). (32) K. H. Khoo, C-Y. Chan, and T. K. Lim, J. Solution Chem., 6, 651 (1977). (33) K. H. Khoo, C-Y. Chan, and T. K. Lim, J . Solution Chem., 6, 855 (1977). (34) J. B. Macaskill, D. R. White, R. A. Robinson, and R. G. Bates, J. Solution Chem., in press. (35) S.Z. Lewin and J. E. Vance, J . Am. Chem. Soc., 74, 1433 (1952). (36) S.Tsunogai, M. Nishirnura, and S.Nakaya, Talanta, 15, 385 (1968). (37) N. A. North, Geochim. Cosmochim. Acta, 38, 1075 (1974). (38) H. S.Harned and J. C. Hecker, J. Am. Chem. Soc., 56, 650 (1934). (39) R. A. Robinson, J. M. Wilson, and R. H. Stokes, J. Am. Chem. Soc., 63, 1011 (1941). (40) A. Zirino and S. Yamarnoto, Limnol. Oceanogr., 17, 661 (1972). (41) D. R. Kester and R. M. Pytkowicz, Limnol. Oceanogr., 14,686 (1969). (42) J. W. Davis and A. G. Collins, Envlron. Sci. Techno/.,5, 1039 (1971). (43) Y. C. Wu, R. M. Rush, and G. Scatchard, J. Phys. Chem., 73, 2047 (1969). (44) R . F. Platford, Can. J. Chem., 45, 821 (1967). (45) C. C. Ternpleton, J. Chem. Eng. Data, 5, 514 (1960). (46) J. B. Macaskill, R. A. Robinson, and R. G. Bates, J. Solution Chem., 6, 385 (1977). (47) R. M. Rush and R. A. Robinson, J. Tenn. Acad. Sci., 43, 22 (1968). (46) R. A. Robinson and V. E. Bower, J . Res. Natl. Bur. Stand., Sect. A , 70, 313 (1966). (49) R. A. Robinson and A. K. Covlngton, J. Res. Natl. Bur. Stand., Sect. A , 72, 239 (1968). (50) C. J. Downes, J. Solution Chem., 4, 191 (1975). (51) R. A. Robinson, R. F. Platford, and C. W. Childs, J. Solution Chem., 1, 167 (1972).
Determination of Hydration Number of an Electrolyte by Vapor Pressure Measurements Chai-fu Pan Department of Chemistry, Alabama State University, Montgomery, Alabama 36 10 1 (Received January 6, 1978; Revised Manuscript Received June 28, 1978) Publication costs assisted by Alabama State University
By using the Stokes-Robinson model of hydration with mathematical derivations, it can be shown that the limiting value of the ratio of the measured vapor pressure of an electrolyte solution to the calculated vapor pressure of an “ideal Debye-Huckel solution” equals the ratio of the mole fraction of water if the hydration effect is considered to the mole fraction of water if hydration is ignored. A hydration number for the electrolyte is thus obtained.
Introduction The thermodynamic properties of electrolyte solutions are determined by several physical factors which are believed inseparable. From the activity coefficient point of view, Stokes and Robinson’s theory1 was important. They proposed a model for electrolyte solutions up to a few molal in concentration in which the Debye-Huckel treatment of interionic interactions was combined with the idea that the solute species in solution are hydrated ions. However, the hydration number h in their parameter equations has lost its original meaning. In fact, the hydration numbers of individual ions or electrolytes resulting from different measurements and interpretations of various properties of solutions are considerably different. Bockris and Reddy,2 Robinson and stoke^,^ as well as Hinton and ami^,^ have given detailed surveys of the results obtained by investigators using various concepts and methods. By using Stokes and Robinson’s concept of hydration,l the hydration number of an electrolyte can be derived from the experimental vapor pressure data. The detailed procedure is presented in this communication. 0022-3654/78/2082-2699$0 1.OO/O
Derivation and Application For a nonassociative dilute aqueous electrolyte solution, the measured vapor pressure of the solution usually deviates from that of the “ideal Debye-Huckel solution”. According to Stokes and Robinson’s idea,l this is primarily due to the hydration of the ions. The so-called “ideal Debye-Huckel solution” is just a hypothetical solution which may be defined in this way: “It is an electrolyte solution which deviates from an ideal solution only because of the interionic interactions. It follows the Debye-Huckel law exactly.” The Debye-Huckel law may be expressed by the following equation5
in which fk,DH is the mean rational activity coefficient calculated from the Debye-Huckel equation, m is the molality of the solution, and K1 and K2 are constants. For aqueous solutions a t 25 “C,for 1:l electrolytes, K1 = -1.17604, K2 = 0.3286186, and for 2:l electrolytes, K1 = 0 1978 American Chemical Society
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The Journal of Physical Chemistry, Vol. 82, No. 25, 1978
Chai-fu Pan
TABLE I: Values for the Calculation of the Hydration Number of Potassium m aw,expt GW,DH NW h 0.1 0.996 668 0.996 699 54 0.996 409 85 4.850 0.2 0.993 443 0.993 548 07 0.992 845 39 4.043 0.3 0.990 25 0.990 462 49 0.989 306 34 3.640 0.4 0.987 09 0.987 421 72 0.985 792 43 3.206 0.5 0.983 94 0.984 416 06 0.982 303 39 2.954
-4.07393, Kz = 0.5691829, where 9 is the ion-size parameter in A units. The ion-size parameter 9, or closest distance of approach, may be chosen as the sum of the crystallographic radii of the cation and anion of the electrolyte in an “ideal Debye-Huckel solution”, because it is assumed that there are no ion-solven t in teradions. On account of the hydration effect, the “true” molality of an electrolyte solution, m’, is different from the conventional stoichiometric molality m.l They are related to the hydration number by the expression m’ = m / ( l 0.018hm). The “true” mole fraction of water, N,’, also is different from the conventional stoichiometric mole fraction of ~ a t e rN,, , ~ Le., N , = 55.508/(55.508 + um); N‘, = (55.508 - hm)/(55.508 - hm + um), where u is the stoichiometric ion number of the electrolyte. The vapor pressure of an “ideal Debye--Huckel solution” will deviate from the ideal value calculated from Raoult’s law. The deviation may be described by f,, the activity coefficient of water in an “ideal Debye-Huckel solution”. Accordingly, we put PDH= i P N w (2) where P D H is the vapor pressure of the “ideal DebyeHuckel solution”, and Po is the vapor pressure of pure water. Similarly, for a real solution, we have P e x p t = f,’~Nw’ (3) where Pexpt is the actual vapor pressure measured experimentally, and ’,f is the activity coefficient of water in a real solution. Equations 2 and 3 lead to the result Pexpt - - - -f,’N,‘ (4) PDH fwNw In a very dilute solution, the Debye-Huckel law reduces to the limiting law. By use of the Gibbs-Duhem relation3 and series expansion, it can be shown that In f , -KNg3l2 (5) where N B is the mole fraction of the electrolyte and K is a constant for a given charge type of the solute, Le., K = (55.508)‘JzKl/3 ~ ~ 1Thus ~. f , = ~ - K N B ~ / ’ 1 - KNB3/2 = 1 - K ( l - N,)3J2 (6) Therefore, the ratio f,’/f, in eq 4 gives ’,f_ - 1 - K ( l - Nw’)3/2 1 - K(1 - N,)3’2 f, = [l - K ( l - N,’)3/2][1+ K(l - N,)3/2 + ...I (7) As m 0 , N , 1 and N‘, 1; by putting (1- N,) = t, and (1 - N,’) = E ’ , eq 7 can be written as follows: (8) f w ’ / f w = 1 - Kef312 + Kt312 - K2t31’26‘312 + ... Also, as m 0, the ratio N,’/N, in eq 4 is Nw‘ - - - -1-- E’ - (1 - €’)(I t + €2 + ...) N, 1--E = 1 t - t’ + t 2 - tt’ - €2.5’ + (9) In studying eq 8 and 9, it is obvious that as m 0, both f,’/f, 1 and N,’/N, 1; however, the former ap-
- -
-
-
-
+
+
-
-
...
Chloride at 25 “ C fw
fw ’
h
1.000 290 7 1.000 707 7 1.001 168 6 1.001 652 8 1.002 150 7
1.00 029 53 1.00 072 92 1.00 122 09 1.00 175 03 1.00 230 84
5.548 4.850 4.504 4.108 3.882
proaches unity faster than the latter does as we consider more terms of the series. Therefore, eq 4 has the following limiting behavior:
The Debye-Huckel vapor pressure P D H is simply related to $DH by the definition of the molal osmotic coefficient: i.e.
pDH = pOe-vm9~n/5h.508 (11) in which $DH is the molal osmotic coefficient of the “ideal Debye-Huckel solution”. It takes the form5 K1 4 D H = 1 + ---[(I + K2rn1iz)- 2 In (1 + K73m Kzm1J2) - (1 + K2m1/z)-1] - 0.009um (12) The value of Pexpt (or can be obtained from the measurement of vapor pressure of the solution at some lower concentrations. The Debye-Huckel vapor P D H (or u,,DH) at the corresponding concentrations can be calculated from eq 11 and 12. By examining eq 10, we may set f,’/f, = 1 in eq 4 from which values of h at different concentrations can be evaluated. The best value of h is obtained by extrapolating the plot of h vs. m to m = 0. By substituting the approximate limiting value of h into the Debye-Huckel expression, eq 11, we obtain the computed values off, and f,’ at each concentration; i.e. V~$DH
l n f w = -- In N , 55.508 ’~DH’
- In N,’ 55.508 By putting these values off, and ’,f into eq 4, we obtain a new series of values of h as a better approximation for extrapolation. This cycle may be repeated again. In applying the above equations to evaluate the hydration number of an electrolyte, aqueous solutions of potassium chloride are chosen as examples. The experimental values of the activity of water are taken from Robinson and stoke^.^ The value of 9 is obtaiped from Pauling.6 For potassium chloride, 9 = 3.14 A. The key values for calculating h by use of eq 10 and the computed values of h are presented in Table I. The plot of h vs. m is shown in Figure 1 (the lower curve). The limiting value of h obtained from the extrapolation of the curve to m = 0 is approximately 6.5. By using h = 6.5, values off, and f,’ in eq 13 and 14 are computed and also compiled in Table I. By putting these values off,’ and f, into eq 4, a new set of values of h is obtained, which is listed in the last column of Table I. The plot of new values of h vs. m is more smooth for extrapolation. It is also shown in Figure 1 (the upper curve). The new limiting value of h confirms the accuracy of the initial extrapolation. The hydration number for potassium chloride is therefore about 6.5.
In ’,f
= --
Discussion The hydration number of an electrolyte should be independent of concentration in dilute solutions for which
The Journal of Physical Chemistry, Vol. 82,No. 25, 1978 2701
A Slmple Model for Solvated Electrons
responsible for the deviation of a real solution from the “ideal Debye-Huckel solution”. In fact, some other factors also contribute to the deviation. For example, Desnoyers and Conway7have shown the effect of mutual salting-out of ions on the activity coefficient; Ramanathan and Friedmans in their theory of ion-ion interactions call this the cavity effect. Statistical mechanic deficiencies in the Debye-Huckel approach as well as structural and cooperative effectsg may also be important. Acknowledgment. The author thanks Professor W. J. Argersinger, Jr., and one of the reviewers for making suggestions to improve the manuscript.
References and Notes
0 4
05
m Figure 1. The hydration number of potassium chloride.
there are very many water molecules present for each ion. However, Figure 1 shows that the values of h depend on m in the concentration range up to 0.5 m where there are still more than 50 water molecules per ion. In these dilute regions h should be nearly constant if hydration alone is
R. H. Stokes and R. A. Robinson, J . Am. Chem. SOC.,70, 1970 (1948). J. O’M. Bockris and A. K. N. Reddy, “Modern Electrochemistry”, Plenum Press, New York, N.Y., 1970, pp 117-132. R. A. Robinson and R. H. Stokes, “Electrolyte Solutions”, 2nd ed., revised, Butterworths, London, 1985, pp 29, 31, 34, 238-251, 476. J. F. Hinton and E. S. Amis, Chem. Rev., 71, 627 (1971). C. Pan, J . Chem. Eng. Data, 22, 234 (1977). L. Pauling, “The Nature of the Chemical Bond”, 3rd ed., Cornell University Press, Ithaca, N.Y., 1960, p 518. J. E.Desnoyers and B. E. Conway, J. Phys. Chem., 88, 2305 (1964). P. S. Ramanathan and H. L. Friedman, J . Chem. Phys., 54, 1086 (1971). F. Vaslow in “Water and Aqueous Solutions: Structure, Thermodynamics, and Transport Processes”, R. A. Horne, Ed., Wiley-Interscience, New York, N.Y., 1972, Chapter 12.
A Simple Model for Solvated Electrons Marc De Backer, Jean-Pierre Lelleur, and GBrard Lepoutre” Laboratoire de Chimie Physlque,‘ 59046 Lille, France (Received July 11, 1978) Publication costs assisted by CNRS
The absorption spectrum of electrons solvated in polarsolvents is interpreted in terms of a rectangular potential well with spherical symmetry. The depth and radius of the well are calculated for several solvents and some of their mixtures. The results are realistic in terms of a simple molecular model with a single solvation shell and without significant long-range interactions.
In a previous paper De Backer et a1.2 give a new interpretation of the absorption spectrum of solvated electrons. They use a rectangular potential well with spherical symmetry and two parameters: depth and radius. The whole absorption spectrum is used for calculating “experimental moments”. “Theoretical moments” are calculated from the wave function of the fundamental state of the well, as algebraic functions of the two parameters (the wave function is known exactly). A system of equations is obtained, which is solved by introducing the values of the experimental moments. The depth and width of the well are thus obtained. This calculation was first made for very dilute metalammonia solutions in order to test the validity of the rnodeL2 The present paper applies the method to electron spectra in water-ammonia and water-ethylenediamine mixture^,^ in ammonia at several temperatures4 and several alcohols.6 Results are interpreted in terms of the molecular properties of the solvents.
Calculations and Results In each case two experimental moments have been used to compute the parameters of the well (depth and width). 0022-3654/78/2082-2701$01 .OO/O
TABLE I: Parameters of t h e W e l l for E l e c t r o n s in Liquid A m m o n i a
To,“C E,,,
eV
V,,
eV
a, A
E(ls),ae V r e f
0.881 3.99 0.727 4 -75 1.92 -65 0.858 1.87 4.01 0.696 4 -55 0.844 1.82 3.97 0.643 4 t20 0.669 1.49 4.00 0.410 3 a E( 1s) is given as a n absolute value. This energy is characteristic of a bound state. E,, is t h e p o s i t i o n of t h e m a x i m u m of t h e spectrum.
The third moment is then calculated by means of these two values, and compared to the third experimental moment. The agreement is good, within 20%. It is then possible to use the same parameters to calculate the energy levels of bound states in the In all cases, only one bound state is found to be allowed in the well. Optical transitions occur therefore from the bound state to the continuum. The results for electrons in ammonia at various temperatures are reported in Table I. Raising the temperature has a negligible effect on the radius of the well. The depth and position of the bound state have a temperature 0 1978 American Chemical Society
