2548
CHAI-FUPAN
New Forms of McKay-Perring Equations by Chai-fu Pan Department of Chemistry, Alabama State College, Montgomery, Alabama 36101
(Received January 17,1968)
Some new forms of McKay-Perring equations may be derived from McKay-Perring equations, equations of Randall and White, and Robinson and Sinclair. The derived equations can be applied to calculate the mean activity coefficients of electrolytes of the same charge type in mixed solutions if sufficient information about the osmotic coefficients of the mixture is available. This can be obtained from isopiestic vapor pressure measurements. A t fixed total molality, the osmotic coefficients of the mixture may be expressed as a polynomial function of the stoichiometric molality fraction of an electrolyte, from which the integral parts in the derived equations can easily be evaluated. Alternate forms of these equations can also be used to obtain the relative activity coefficients if osmotic data for solutions in dilute region are not available.
Introduction McKayl first pointed out that activities of nonvolatile solutes in a mixed solution can be determined from measurements of vapor pressure alone. In ternary aqueous systems, the activities of the three components are connected by the cross-diff erentiation equation. From a series of transformations and the application of the Gibbs-Duhem equation to the reference electrolyte, McKay and Perring2 were able to derive equations which are suitable for integration to evaluate the mean activity coefficients of the component solutes by the isopiestic method. These equations are valid when isopiestic measurements are made on a number of mixtures in which electrolytes B and C are present in different concentration fractions but at constant water activity. Robinson3-6 has given numerical illustrations of the McKay-Perring equations. However, he limited himself to the case for which the reference electrolyte is solute B or C, a component in the test mixture. Electrolyte B or C should not be chosen as the reference if its thermodynamic properties (e.g., osmotic or activity coefficients) are not available. Bonner and Holland7 have given equations in a more general form, independent of the kind of electrolyte used as a reference. Eevertheless, following the original derivation of McKay and Perring,2 they retain the function m, which is any linear combination of the molalities of electrolytes B and C in their equations. This makes the integral parts of their equations somewhat more complicated in numerical calculations. New forms of LlcKay-Perring equations can be derived from 3lcKay-Perring equations, equations of Randall and White, and Robinson and Sinclair. The derived equations may be applied to evaluate activity coefficients of electrolytes of the same charge type in mixed solutions if sufficient information about the osmotic coefficientsof the mixture is available. Derivation of Relations For single electrolyte solutions, the mean activity The Journal of Physical Chemistry
coefficient of the solute, y, is related to the practical osmotic coefficient of the solution, 4, by the Randall and White equations In
y =
(4 - 1)
+ Sm(4 - 1) d l n m m i 0
(1)
where m is the molality of the solution. By applying the Gibbs-Duhem equation to a pair of isopiestic solutions, Robinson and SinclairQwere able to derive an equation to compute the mean activity coefficient of an electrolyte at any concentration providing the mean activity coefficient of the reference electrolyte is known. The Robinson and Sinclair equation is In y
=
In
YR
+ In R +
where R is the isopiestic ratio, y is the mean activity coefficient of the test electrolyte at molality m, YR is the mean activity coefficient of the reference electrolyte at molality mR, and mR and m are the equilibrium molalities of the reference electrolyte and test electrolyte, respectively. The McKay-Perring equations may be written in a more convenient form5J
(1) H. A. C . McKay, Nature, 169,464 (1952). (2) H. A. C. McKay and J. K. Perring, Trans. Faraday Soc., 49,
163 (1953). (3) 3.A. Robinson, Trans. Faraday Soc., 49, 1411 (1953). (4) R. A. Robinson, J . Phys. Chem., 65,662 (1961). (5) R. A. Robinson and V. E. Bower, J . Res. Nat. Bur. Stand., 69A, 19 (1965). (6) R. A. Robinson and V. E. Bower, ibid., 69A, 439 (1965). (7) 0.D.Bonner and V. F. Holland, J . Amer. Chem. SOC.,77, 5828 (1955). (8) M. Randall and A. M. White, ibid., 48, 2514 (1926). (9) R. A. Robinson and D. A. Sinolair, ibid., 56, 1830 (1934).
NEWFORMS OF RTcKAY-PERRING EQUATIONS In
YB
In yc
= In YR
+ In R +
In
+ In R +
=
YR
2549
Recall the relationship for the isopiestic solutions in question
(3)
Thus
and where YR is the mean activity coefficient of the reference electrolyte at molality mR which is isopiestic with the mixture containing electrolytes B and C; the molalities of electrolytes B and C in the mixture are mB and mc, respectively; +R is the osmotic coefficient of the reference electrolyte at molality mR; R is the isopiestic ratio, which is defined as
R =
+
From eq 8 and 9 and with the relationship
m-e obtain
VRmR VBmB
(9)
VCmC
(4)
v i is the number of ions per “molecule” of electrolyte i; and yc are the stoichiometric ionic fraction of electro-
YB
at constant
YB.
Thus, expression 7 becomes
lytes B and C in the mixed solution respectively; ycSm
i.e.
(*)
dlnm
m = o ~ Y Bm YB
=
VBmB VBmB
+
(12)
and
VCmC -yBJrn
(5) yc =
vCmC YBWZB
+ wine
By comparison of eq 2 with eq 3, it is easy to see that the contribution to In Y B and In yc, due to the presence of another solute, is given by the corresponding first term in the bracket in eq 3 over the interval of integration; i.e.
(6)
and
For simplicity, we limit ourselves to the case where B, C, and the reference electrolyte are all of the 1: 1 charge type. That is V B = vc = UR = 2; then YB = mg/in, yc = mc/m, and R = nzR/m, where in = mB mc. At constant YB, the terms in (6) can then be written as
+
and
(7)
m=o
(*) ?YB
dlnm m
By placing a contribution term due to the presence of a “foreign” salt in the Robinson and Sinclair equation, the RIcKay-Perring equation can be formulated. The contribution terms have been transformed to be “harmonic” to the terms in the Randall and White equation. By adding the corresponding terms of expression 6 to the right-hand side of the Robinson and Sinclair equation, and expressing the left-hand side In y as In YB or In ycr we obtain the AIcKay-Perring equation. Similarly, by adding the corresponding terms of expression 12 (equivalent to the terms in expression 6) to the right-hand side of the Randall and White equation (equivalent to the Robinson and Sinclair equation) , and expressing the left-hand side In y as In YB or In ye,new forms of XcKay-Perring equations are obtained In
YB
=
(4 - 1)
+
m
y e smm =
(*)
~~ Y Bm
YBS~ (*)
dlnm
dlnm
m = o ~ Y Bm
From eq 13 it is easy to show that for two-salt solutions, eq 1 becomes Volume 72,Number 7 July 1968
CHAI-FUPAN
2550 YB
In YB
+ yc In
YC =
(4 - 1)
+ Sm (4 - 1) d In m m=O
We also obtain a simple relationship between InYB YC
=sm(””> m=O a y B
(14)
YB and yc
dlnm
(15)
m
The integrations in eq 13 are performed at fixed values of Y B ; the integrals may be evaluated by graphical methods. Equation 13 can be applied to single electrolytes as well as to mixtures containing electrolytes B and C of the 1: 1 charge type. Following the above derivation, one can show that eq 13 is also valid for mixtures of B and C of the same charge type. The reference salt may be the same or different. As YB + 0 or Y B + 1, eq 13 can be used to evaluate the limiting value of y for a solute in a mixture of total molality m.
Discussion Since the isopiestic method is not applicable in concentrations less than 0.1 m, extrapolations at lower concentrations yield crude estimates of the required area. Due to the difficulty of extrapolating, alternate forms of eq 13 may be used to evaluate the relative values of the mean activity coefficient of the solutes, i.e. In
2 = (4 - 4’) + YB
sm
(4 - 1) d In m
m=m’
ycsm m=m’
In
YC
YC
=
(4
- 4’) +
+
m
(””> d l n m ~ Y B m
(4 - 1) d In m -
m=m’
where YB’ and yc’ are activity coefficients of electrolytes B and C in the mixture at total molality m’, respectively, and 4’ is the osmotic coefficient of the mixture at total molality m’, all at the same value of YB as in the solution of molality m. One advantage of the derived equations over McKayPerring equations lies in the fact that theoretical guides for extrapolation to infinite dilution are more readily expressible in terms of 4 than in terms of R. The osmotic coefficient of the mixed solution at constant total molality may be expressed as a polynomial function of yB. 10-12 The polynomial functions at different fixed values of m can be obtained by the interpolation of isopiestic data. If Harned’s rulell holds for both electrolytes B and C, the osmotic coefficient of the mixture is a quadratic function of YB. If a higher order p term is necessary to be added to modThe Journal of Physical Chemistry
ify Harned’s rule, the osmotic coefficient in the mixture is then a cubic function of YB. The osmotic coefficient in aqueous mixtures of NaOH-NaC1l3 and NaC1-KC14 should thus be expressed as cubic functions of YB. Still higher terms are required to show the deviations from Harned’s rule when larger chemical interaction is encountered. For dilute 1: 1electrolyte solutions, the mean activity coefficient of the solute may be expressed by a modified Debye-Huckel law of the form
where A , B, C, D,. . . are constants for a given system at constant temperature; it is an average ion-size parameter assumed to be constant for a given solution. The constants A and B here are slightly different from the values tabulated by Robinson and Stokes14 at different temperatures by a factor d”’; d is the density of s01vent.l~ Constants C, D, . . . may be determined experimentally. The osmotic coefficient of the solution can then be derived by the relation16
For very dilute solutions, function of m in the form
4 = 1
may be expressed as a
+ Klml/’ + Kzm + K3mS/’+ . . .
(19)
where Kt’s are constants. Equation 19 suggests that for very dilute 1:l-1:l mixtures, 4 retains the same form. However, the constants KzJKs, . . . are now functions of y ~ (1 ; Kim"') are Debye-Huckel limiting law terms, they should be independent of composition. Hence, at a fixed value of yB, the function ( b + / b y B ) m may be expressed as a polynomial in powers of ml”, but starting from a linear term in m ; in the limit of very low concentration, the derivative is a linear function of m. Scatchard’s e q ~ a t i o n ~for ~ ’ 4~ *in 1: 1-1 : 1 mixtures can be written as
+
4
=
YB~BO
+ yc4co + ‘/~YBYCPO
(20)
(10) B. B. Owen and T . F. Cooke, Jr., J . Amer. Chem. Soc., 59, 2273 (1937). (11) H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” 3rd ed, Reinhold Publishing Carp., New York, N . Y., 1958, Chapter 14. (12) G. N. Lewis and M. Randall (revised by K. S. Pitzer and L. Brewer) “Thermodynamics,” 2nd ed, McGraw-Hill Book Co. Inc., New York, N. Y., 1961, pp 570, 571. (13) H. S.Harned and M. A. Cook, J . Amer. Chem. Soc., 5 9 , 1890 (1937). (14) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed, Butterworth and Co. Ltd., London, 1959, p 468. (15) Reference 12, p 338. (16) Reference 14, p 34. (17) G. Scatchard, J . Amer. Chem. Soc., 83, 2636 (1961). (18) R. M. Rush and R. A. Robinson, J . Tenn. Acad. Sci., in press.
NEWFORMS OF MCKAY-PERRING EQUATIONS
2551
where superscript 0 refers to a solution of a single salt at molality m; Po is a function of m but not of YB, so that
but In
YBO
= ( 4 ~ '- 1)
+ smm=i4~o - 1) d In m
(24)
Thus Thus from eq 13 In
YB
=
+ yc4c0 -
YB+BO
+ Jmm=o(?/Bd~o + yc4co - 1) d In m + Y C S ~ = ~ (-~ B 4co) O d In m + '/~YBYCPO+ 1
Scatchard has also defined a function Bo such that
m
' / 2 s m m = ? ~ ~ cdP ~ In
m
+
Hence
' l z ~ c(YC ~ -~ YB)POd In m 9%-0
In
YB
=
YB,
eq 22 can be simplified, i.e.
+ yc4c0 1 + Jm=o(4~O- 1) d h m +
YB~BO
m
+ ' / z Y c ~PO~d In m
'/ZYBYCPO
m=O
1
+ YCZ(BO- Po)
(27) This is Scatchard's equation for In YB, assuming that there is no term in ?JBYC(YB - yc) in 4. Rush and Robinson18 used Scatchard's equation to treat the NaCl-KCl-H20 system. They evaluated functions of YB and yc which are consistent with the above derivations using the new forms of McKay-Perring equations. YCPO
At constant
Volume 72, Number 7 July 1968
