H,FRANK GIBBARD, JR.
2382
A New Analytical Treatment of Heats of Dilution of Electrolyte Solutions1v2 by H. Frank Gibbard, Jr. Department of Chemistry, Southern Illinois University, Carbondale, Illinois 61901
(Received December 47, 1968)
A new analytical method is described for smoothing heat of dilution results for electrolyte solutions. The
method is consistent with the Debye-HiickeI theory and is much simpler than graphical methods described in the literature. The application of the analytical technique to the heats of dilution of aqueous sodium chloride is described.
Introduction I n the analysis of some recent vapor-liquid equilibrium measurements3 it was desirable to obtain an analytical expression for the heats of dilution of aqueous sodium chloride. The technique developed for smoothing heat of dilution data is more straightforward and involves considerably less labor than methods found in the particularly if electronic computers are available to perform the calculations. The change in state which occurs when an electrolyte solution is diluted a t constant temperature and pressure can be written
+ (nlf - nli)A(liquid) =
nzB-siA(solution)
nzB.nlfA(solution) (1) where n2 is the number of moles of the solute B, and n? and nlf are the initial and final numbers of moles of solvent A in the solution. The measured quantities are the initial and final weight molalities of the solute and the heat of dilution. The heat absorbed is equal to the change in the excess enthalpy HE of the solution. The excess enthalpy per mole of solute, HE/n2,is known8 as the “relative apparent molal heat content of the solute.” Young61° has devised the precise but tedious “chordarea” method for constructing a large-scale graph of the derivative of H E / n 2with respect to the square root of the molality from heats of dilution. The chord-area method was particularly useful in showing agreement of heat of dilution results with the Debye-Huckel limit= ing law for aqueous electrolytes of the 1-1, 1-2, and 2-1 valence typesng The limiting law for heats of dilution is now well established; the present paper describes a simple method for smoothing heat of dilution results which is consistent with the limiting law.
Procedure Any thermodynamic property of a solution can be derived from an equation for the excess Gibbs free energy GE as a function of pressure, temperature, and composition. Scatchard and PrentisslO have shown T h e Journal of Physical Chemistry
that the excess free energy of an electrolyte solution can be represented by a Debye-Huckel term plus a power series in the concentrations of the solutes and in the square root of the ionic strength. By the use of an approximate Debye-Hiickel term and the omission of terms in the square root of the ionic strength, eq 3 of Scatchard and Prentiss’O may be written”
GE
-2SXI”/”g1
E=
KU
+
%Btj ij
gl
+
where R is the gas constant, T is the absolute temperature, I is the ionic strength in moles per kilogram of solvent, 1 / ~is the “thickness of the ion atmosphere,” a is the distance of closest approach to the central ion, g1 is the weight of solvent in kilograms, and n, is the number of moles of solute species i. The Debye functions S and X are defined by the equations
8
N2es(dla/500)‘/’/(RTD1) X = [ K Q ( K U - 2) 2 In (1 K U ) ] / K % ~ =
+
+
(3)
(4)
(1) This work was begun a t Massachusetts Institute of Technology and was supported in part by the U. 8. Atomic Energy Commission under Contract AT-(30-1)-905. (2) Pfesented in part at the Midwest Regional Meeting of the American Chemical Society, Manhattan, Kansas, Oct 1968. (3) G. Scatchard, H.F. Gibbard, Jr., and R. A. Rousseau, “The Osmotic Coefficients of Aqueous Sodium Chloride,” Symposium on Water Desalination, 22nd Southwest Regional Meeting, American Chemical Society, Albuquerque, N. M.. Dec 1, 1966. (4) E. A. Gulbransen and A. L. Robinson, J. Amer, Chem. Soc., 56, 2637 (1934). (5) T. F. Young and 0. G. Vogel, ibid., 54, 3030 (1932). (6) T. F. Young and W. L. Groenier, ibid., 58, 187 (1936). (7) E. A. Guggenheim and J. E. Prue, Trans. Faraday Soc., 50, 710 (1954). (8) H. 8. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” 3rd ed, Reinhold Publishing Corp., New York, N. Y., 1968, p 332. (9) T. F. Young and P. Seligmann, J . Amer. Chem. Soc., 60, 2379 (1938). (10) G. Scatchard and S.S.Prentiss, ibid., 56, 2314 (1934). (11) G. Scatchard and L. F. Epstein, Chem. Rev., 30, 211 (1942).
HEATSOF DILUTION OF ELECTROLYTE SOLUTIONS
2383
where N is Avogadro's number, e is the protonic charge, and d l and D1 are the density and dielectric constant, respectively, of the solvent. The summations are taken over the solute species, and the B's, D's, etc., are functions of temperature. The quantity Ka is proportional to the square root of the ionic strength
Ka
=
a'11/2
The excess free energy per mole of solute is given by
(7) where m is the weight molality of the solute, and =
c 2.3
BdjVdV6
Bz =
c ijlc
v1v1vLD1jk;'
'
(8)
Differentiation of eq 7 with respect to temperature gives the excess enthalpy per mole of solute
where
The heat absorbed per mole of solute in the change of state ( 1 ) is given by
E ! nz
=
Z ( T 2 - g ) ( X f - Xi)
(12)
(5)
and a' is assumed t o be independent of temperature. This assumption greatly simplifies differentiation of the excess free energy with respect to temperature.'l For a single salt, one formula weight of which dissolves to give v i moles of ions of type i, eq 2 may be written
B 1
2
j = 1 jBi'm9+l]
+
Here the subscripts i and f denote the initial and final states. Equation 11 is used for smoothing heat of dilution data by subtracting the Debye-Huckel term from AH/nz; the resulting variable is a function of the w variables mrj-mif. The method of least squares is used t o determine the coefficients B,'. For the evaluation of the temperature dependence of activity and osmotic coefficients expressions for the relative partial molal enthalpies of the solute and solvent are useful. These are derived from eq 9
where
Y
=
+ a'11/2)
afP2/(1
(14)
Z = Y - x
(15)
hIanual calculations involving the X and 2 functions are simplified by the use of the table of Z as a function of Y published by Scatchard and Epstein." Example. Application to Aqueous Sodium Chloride. The procedure described in the previous section has been applied to the heats of dilution of aqueous sodium chloride. For this purpose 34 values of the heat of dilution at 25" with the initial and final molalities were taken from the results of Young and his coworkers.6Il2 A F O R T R ~ NI V computer program was written to carry out the least-squares analysis, and preliminary calculations were carried out on the IBM 360 computer of the Massachusetts Institute of Technology Computation Center. The final calculations were performed on the IBRll7040 computer of the Southern Illinois University Data Processing Center. The function S depends on the dielectric constant and density of the solvent. The values recommended for these properties for water by Harned and Owenla yield a value of 178.1 for the quantity (T2dS/dT) at 2 5 " ; this value was used in all of the sodium chloride calcula+;$;ns. Scatchard, et al.,3 obtained the value 180.9 for the same quantity by using the dielectric constants of Owen, et al.,14 and of Akerlof and Oshry15and the water densities of the International Critical Tables. For spdium chloride in water we chose a' equal to 1.5 at all temperatures. a Table I gives the parameters for cubic and quartic equations in the molality. Standard deviations for the cubic and quartic equations are 0.308 and 0.149 cal rnol-l, respectively. In Figure 1 deviations of the calculated heats of dilution from the experimental values are shown by lines connecting the initial and final molality. All of the deviations are shown except for two which coincide with deviations already plotted. The deviations very clearly show the superior precision of the shorter chords of Young and Machin. The quartic equation is necessary to represent the data (12) T. F.Young and J. S.Machin, J . Amer. Chem. Soc., 58, 2254 (1936). (13) Reference 8, p 173. (14) B. B. Owen, R. C. Miller, C. E. Milner, and K. L. Cogan, J . Phys. Chem., 65, 2065 (1961). (15) G.C. lkerlof and H. I. Oshry, J. Amer. Chem. Soc., 72, 2844 (1950). Volume 78, Number 7
July 1969
H. FRANK GIBBARD, JR.
2384
t
Figure 1. Deviations of AHE/nt from quartic equation: dotted lines, Young and Vogel; solid lines, Young and Machin.
within the apparent precision, but for the calculation of the change of osmotic or activity coefficients with temperature the difference between the cubic and quartic equations is negligible. For example, the difference in the quantity Rl - Rlofrom the two equations is 0.56 cal mol-’ for 6.0 m sodium chloride; the corresponding difference in the change of the osmotic coefficient over a 25” range is less than 0.0005.
Table I: Parameters for Sodium Chloride at 25” Cubic equation
Parameter
Bi‘ Bat B8 ’ B,’
-270 8756 28.40645 0.8731166 I
-
...
Quartic equation
- 288,2987 36.04629 2,423282 0.1053459
-
0
Figure 2.
puted by the analytical method with those found by integration of the chord-area plot. Since values computed by the graphical method using the same data are not available in the literature the author has redrawn Figure 4 of the thesis of Machin,’4 in which the quantity d(HE/n2)/dm’/’ is plotted against the square root of the molality. The only significant change made in Machin’s drawing was to give less weight to the most dilute point of Machin and no weight to the values of Gulbransen and R o b i n ~ o n . ~The same limiting slope of 472 cal mol-*’z was used in the computer calculations and in the graphical treatment of the data. The derivative curve was integrated numerically to give values of HE/n2 as a function of molality. The differences in HE/n2 from graphical and analytical treatments of the heats of dilution of sodium chloride are shown in Figure 2. The ordinate is the value of HE/n2 computed by graphical integration less the value computed from eq 9 with w equal to 4, obtained by interpolation in a computer-printed table of HE/n2 at 0.01 m intervals. A few entries from this table are given in Table I1 to show the size of H E / n 2in relation t o the differences shown in Figure 2. Values of the relative partial molal enthalpies in the third and fourth columns were computed using eq 11 and 12 and the parameters of Table I. The Journal of Physical Chemistry
I
20
I
I
3.0
I
I
4.0 5.0 rn (mole kg-’)
6.0
Comparison of graphical and analytical methods.
The most important comparison is given by the full curve, where the graphical and analytical methods have been applied to the same data. The results of the two methods agree within the expected precision of the graphical method. Only in dilute solution is the difference appreciable; here the data are sparse and the experimental precision is poorer. The broken curve corresponds to the values of H E / n 2given by Young and Vogel.5 Young and Vogel relied on their own measurements and three other sets of heats of dilution and heats of solution in constructing the chord-area plot. Here the deviations in dilute solutions are about the same
Table I1 : Thermal Properties of Aqueous Sodium Chloride at 25” 7%
It is interesting to compare values of ~ ~ / comn 2
I
10
mol kg-1
0.2 0.4 0.6 0.8 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
6.0
G1 - 8,0,
8, -
oal mol-1
oal mol-’
cal mol-‘
86.0 68.2 39.9
0.02 0.36 0.99 1.87 2.95 6.28 10 04 13.77 17.09 19.66 21.18 21.34 19.81 16.20 10.00
79.6 18.2 -51.8 -121.6 -188.5 -337.1 -487.0 -549.6 -616.8 -661.0 -683.8 -686.1 -668.4 -630.4 -570.7
HE/nz,
b.2
-24.5 -104.7 -178.3 -243.8 -300.6 -349.2 -389.8 -422.8 -448.4 -466.8 -478.1
I
R,o,
size as in the case of the full line but are opposite in sign. Recently, ParkeriT treated 11 sets of heats of dilution and heats of solution of NaCl by the chord-area method. Her values yield deviations which are shown by the filled circles. The dotted curve shows the deviations of the cubic equation from the quartic. (16) J. 8. Machin, thesis, The University of Chicago, 1932. The author is grateful to Dr. T. F. Young for making the thesis of Machin available. (17) V. B. Parker, “Thermal Properties of Aqueous Uni-univalent Electrolytes,” NSRDS-NBSB.
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HEATSOF DILUTION OF ELECTROLYTE SOLUTIONS The agreement of the results of the analytical and graphical methods has been shown to be excellent. The simplicity of the analytical method and the usefulness of analytical expressions should make the present method valuable to those who are interested in thermochemical properties of solutions.
Discussion The construction of the derivative curve in the chordarea method becomes particularly simple when the difference between the initial and final molality is very small. In this case the derivative curve should intersect each chord nearly a t its midpoint. Thus one can design experimental measurements using short chords to simplify the treatment of the results.12 A considerable advantage of the analytical method is that it treats short chords or long chords with equal ease. It, might be argued that the analytical method may miss real “wiggles” or “kinks” in the variable under consideration. I n fact deviation plots like Figure 1 would show up such peculiarities very well, perhaps better than any other method. If such deviation plots are always made, then the age of computers need not lead to “loss of integrity of the data.” The analytical method is well suited for critical studies like the one of Parker.” The assignment of different weights for different concentration ranges or for different investigators can be easily and most objectively incorporated into a least-squares program. The analytical method described for heats of dilution can be easily extended to the smoothing of heat capacity results by differentiation of eq 9 with respect to temperature. The advantages over graphical tech-
niques are not so large, because heat capacity measurements are made on solutions at constant composition. One experiment yields the excess heat capacity at one concentration, not the change in an excess property between two concentrations. Also the limiting slope for heat capacity is not well determined because the change of the dielectric constant with temperature is not known with sufficient accuracy. The simultaneous smoothing of heats of dilution at different temperatures can be accomplished by expanding each B, in eq 11 in a power series about a standard temperature T ,
The function
S = So
X then takes the form
+ SI/T + SzIn T + X3T
+ S4T2 + . . .
(17)
The coefficients Bji in eq 16 are determined by the method of least squares. The number of terms may be increased until the deviations from experimental results no longer vary systematically with temperature. The excess heat capacity and higher order temperature derivatives of the excess enthalpy may be easily determined from the coefficients B,, and &. Acknowledgment. The author is grateful to Professor George Scatchard for pointing out the need for an analytical treatment of heats of dilution and for his helpful comments.
Volume 73, Number 7
JuEg 1969
