Brian W. Clare School of Mathematical and Phys~calSciences Murdoch Unlverslty Murdoch, Western Australla 6153
Evaluation of Cation Hydrolysis Schemes with a Pocket Calculator
Two main kinds of problems arise in connection with ionic equilibria in solution. Given the analytic concentrations of the components and the stahility constants of the species present, the first problem is to determine the equilibrium concentrations of those species. The second, and by far the more difficult problem, is to determine the nature and stability constants bf the species involved from experimental data. The past three decades have seen a revolution in the treatmint of both of these problems. Prior to 1950, the first usually was treated by making chemically based approximations and neglecting species present in relatively low concentrations. This procedure resulted often in entirely different treatments of thisame system under different conditions, as may be seen, for example, in the usual undergraduate treatment of polybasic acid equilibria. The advent of the digital computer, and more recently, the programmable calculator, has alleviated this situation. A single program, solving a system of nonlinear eqnations, one equation for each component, can now account for all initial conditions in a given system. Even very complex systems can be handled on a quite inexpensive calculator ( I ). Prior to the advent of the computer, the problem of determining the identity and stahilitv constants of the soecies prese& was in an'even less satisfactory state. Graphical methods were devised and met with some success. They required considerable ingenuity, however, and had to be specially devised for each situation (2). w i t h the arrival of the computer, a much more systematic approach based upon the techniques of nonlinear regression hecame possible. Hypothetical species are postulated on theoretical grounds, by analogy with known systems, on the basis of crystallographic studies, or on the basis of preliminary graphical treatment. The stability constants of these species are refined. They will effectively vanish for species not present in significant amounts and become constant for important species provided that all major species have been included. Many such species can be handled simultaneously on a large computer. Complexity Constants by Nonlinear Regression
The basic methods of nonlinear regression have been discussed by Wentworth (4) and by Draper and Smith (5).In the context of complex equilibria, if one has a system with a metal ion M and n different ligands Li, forming a number of complexes of the form MmL1,,L2,, . . . , with stability constants of the form
In principle one need only know the analytic concentration of the metal, all of the ligands, and the equilibrium concentration of only one l i ~ a n dsay , LI, a t a range of analytic concentrations in order to derive all of the stability cor&ants. Knowing the equilibrium concentration [L,], one can defme a ligand number n as the number of moles of bound L1 per mole of metal ion M: -
np =
L1°
- IL11 MO
(1)
where L10 and Mn represent analytic concentrations. This number is accessible experimentally. If we knew all of the stability constants of the species present containing both M 784 1 Journal of Chemical Education
and Li, ii could also be calculated, by solving the n nonlinear mass balance eqnations in t h e n variables [MI and [L;]
i = 2 ton
and then evaluating
where m, 11,12. . .1, are the number of M, L1, Lp. . . Ln entities in the snecies. and the summation extends over all soecies present. With a uostulated set of soecies and their stabilitv constants &, we can therefore form (it, - ri,) = f, for each set of analvtic concentrations measured. and thus we can minimize X/1, where the iummiltio~~ (.xtt:nd+w r r :ill ~rt~i~:r\.ntitms. This minimization mav be ncromvlished hv the Gauss Newton algorithm. For the purpose oithis pap&, the three parameter case will be discussed. Suooose we have three stability constants a,(3, and y, and a n&ber of obseyations, C ; . y k wish to minimize Z(Ti, - &I2 = &(f(a, 0 , y ; Ci)Izwhere Ci represents the set of measured analytic concentrations. If (a:, Pi, y i ) is an approximation to the set of parameters which minimize Zf2,then a better approximation, according to the Gauss Newton algorithm, is given by a;+,= a ;- 6a P;+r = Bz - 6 8 ?;+I = ~i - 6 7
where ba, 6P, and dy are given by the solution of the system of linear equations
where the summations are over all measured sets of initial concentrations Ci, and f a , for example, is bflba evaluated a t ail Pi, 7 ; . Calculators and Programming
Two models of pocket calculators exist which have sufficient Dower to handle sienificant orohlems of the tvDe considered Hbove. These are tlhe ~ e w l e t t - ~ a c k a ~r d~ 6 7 " s nthe d Texas Instruments TI59. The programs described here were developed for the HP97, a portable printing calculator. They are, however, applicable without modification to the HP67. The HP67 has 26 data registers. This is sufficient for a Gauss Newton regression with up to four parameters. Using four parameters, however, uses so much memory that there is little available for other purposes, and thus restricts the problems to which the program can be applied. A three parameter regression program will he described and is listed in Tahle 1. I t should he noted that the derivatives which appear in equation system (3) do not necessitate analytic differentiation. In the program of Tahle l A , steps 1-80 are the Gauss Newton aleorithm. The remainder will be exolained in due course. In p&icular the subroutine under ladel C (steps 81-185) represent the function f. whose sum of sanares is to be minimized. The three paramet& to he varied & stored in R7, Rg and Rg. Each parameter is, in turn, increased by 1%. The function f is evaluated, and the parameter is returned to its original
This program is quite generally useful and has been applied in this laboratory to the fitting of conductivity data to the Fuoss Onsager Skinner (1965) equation, to the fit tin^ of acid base citration curves, and to the analysis of solubi1it;studies involving acid base equilibria. It is a simple exercise to modify the program for one or two parameters, and some four parameter cases can also he handled. The Newton Raphson rontine can he extended to equation systems in up to nine or more variables (1, 7). This routine is useful in handling problems of type 1.It'is less useful for problems of type 2 (with more than one ligand) as there is difficulty in supplying an appropriate starting approximation.
Table 2.
- ,
7 iic %i
H y d r o l y t i c Equilibria
The above, rather general discussion may now be applied to the restricted case of hydrolytic equilibria. Here, there is only one ligand (hydroxide). Its equilibrium concentration can readily be determined potentiometrically, by the glass electrode. The experimental ligand number Ti, is given by T + [H+]- K,I[H+] Fi, = MO
where T is the amount of hvdroxide titrated into an initiallv stoichiometri(! solution d the metal salt. The ai~algticconcentration d t h e ration is 11".h series ot hvdrolvsis r,n,ducts of the form M,(OH), (omitting charge;) is formkd, with various values of rn and p. mM"+ + pHzO = Mm(OH),(m"-~)+ + pHt Their stability constants are given by
The total concentration of M in the solution is given by the mass balance equation
1 m&,,[M"+]m/[H+]P
(4)
p.m
where the summation is over all species. This equation is to be solved by the Newton Raphson routine of card A, and it is to he programmed under label E. Since we know the p H and have hypothesized the P,,,, it is an equation in one variable, IMn+l. ~ h value k of ii" is then obtained from the equation -
nc =
1
(&PP~.~[M"+I'"/[H+ lp
(5)
The function f = ii, - Ti, can then be formed, and the program of Tahle 1 can evaluate any hydrolysis scheme containing three parameters (i.e., three hydrolysis products). One need only program the metal balance equation (eqn. 4) under label E, to be solved by the Newton Raphson routine, and the hydroxide balance equation (eqn. (5))under label 2. The H y d r o l y s i s o f U ( V I )
As a specific example, consider the hydrolysis of hexavalent uranium. This process has been studied by many workers (8-11 ). Even in the strongest acid, the cation present is U0z2+. We will therefore take this as M. Baes and Meyer (8) consider that the main hydrolysed species present in an hydrolyzed uranyl solution are (U0&OH)z2+, (UOz)a(OH)s+, and to a lesser extent, U020H+. Dunsmore, et al (9) consider that (U0&0HWt, rather than UOzOH+ is present. In the Baes and Meyer scheme, eqn. (4) becomes [U0s2+]+ [UOZOH+] + 2[(UOz)dOH)~2+] + 3[(U02)d0H)stl = [U022+]+ &l[U022+]/[H+]+ 2B22[U0~2+]2/[H+12 + 3Ps3[U022+]3/TH+]5( 6 ) With Pll in R7, 8 2 2 in Re, and P53 in R9, [H+] in Ro, ? in ie R1 and U0 in Rz, this equation is solved for [U0z2+] by the Newton Raphson routine of Tahle lA, steps 81-120. The expression is programmed under label E, program steps 153-185, exactly as it is written. Equation (5)now becomes U"
786 1 Journal of Chemical Education
pH
3
Model 1
Mcdel2
3.039 3.485 3.788 4.072 4.428 3.200 3.705 4.047 4.331 4.706 3.118 3.678 3.973 4.176 4.572 3.433 3.767 4.106 4.387 4.650 5.068 3.591 4.165 4.488 4.808 5.061
0.049 0.231 0.507 0.860 1.287 0.056 0.301 0.677 1.061 1.446 0.022 0.181 0.436 0.704 1.249 0.039 0.129 0.377 0.756 1.145 1.536 0.035 0.319 0.764 1.228 1.462
-0.0007 0.0059 -0.0007 -0.0035 0.0004 -0.0032 0.0019 0.0021 0.0093 -0.0044 -0.0011 0.0021 0.0024 0.0005 0.0008 -0.0049 -0.0062 -0.0043 0.0014 0.0033 --0.0262 0.0039 -0.0054 -0.0034 0.0068 -0.0019 0.0068
0.0012 0.0098 0.0009 -0.0063 -0.0028 -0.0024 0.0045 0.0016 0.0066 -0.0052 -0.0018 0.0028 0.0034 -0.0001 -0.7913 X -0.0076 -0.0092 -0.0056 0.0014 0.0050 -0.0228 -0.0006 -0.0094 -0.0029 0.0114 0.0032 0.0073
Standard deviations:
=
Ma = [M"+] +
Potentlometric Titration of Uranyl Nitrate (Ref. 8)
1
lo-'
IBn[UO?+]l[Ht] + 2B~~[U0z2+I2/[H+l2
+ 5/3~~[U02+13/[H+lsl steps 120-152. The function Fi, - Ti, is calculated bv. .propram . under label 2, just as written. The data of Baes and Mever ( 8 ) are shown in Table 2. It has been partly processed, in that we are supplied with UOand fi,. The data, [H+],ii,, and Uo are loaded into RQ,RI, and Rz, and are then recorded on magnetic cards; one point per card. An estimated set of values of P11,822, and P53is stored in R7, R8, and Rs, respectively. An approximation to [U022+]for the first point is stored in RE ready for the Newton Raphson routine. The approximations for the s3/' are obtained by trial, entering selected experimental points and pressing C to obtain the residual .f.. and adiustine the B's to obtain a small value of f.. remembering for instance that the most hydrolyzed species are formed at the highest pH, and the most condensed species a t the highest cation concentration. (Thus a high Uo and high DH point should be used when adiustine 452.) he program is then run as fol n e r f:ii i ~ tl h r nhore methudb miiy seem, they h a w thrir limitntioni. T h e usual p r u l h m i s e x ~ e r i m e n t aac~:ur;~cv. l The more complex the model, t h e greater t h e demands on t h e experimenter, with regard t o both range a n d accuracy of data. The Experiment
The uranyl problem (Table 2) was chosen both because it extends to the utmost the power of the HP67, and because it is reasonably tractable experimentally, for those who may wish to design laboratory exercises from this paper. Uranyl nitrate is readily available in pure Ibrm, and it crystallizes well. If thesolutions are made from UOrH20, hvwever, there is always an excess of acid, and this must be deter-
mined and allowed for in the calculation uf ii. The reference ( 8 )discusses this, and also the liquid junction potential. The latter may, however, he ignored frx many purposes. Carbonate free sodium hydroxide should be used, and for preference, a three figure pH meter, especiallv if fine distinctions between minor species are to be attempted: Many other systems could be chosen. With some, however, highii values cannot be achieved without precipitation. Uranyl is good in this respect, as the initial precipitate on mivingof the U02Z+ solution with the alkali readily redissolves. In all cases, a high concentration of supporting electrolyte is needed to huffer the activity coefficients. If ionic strength depends markedly on the position uf the equilibria, the calculations are greatly complicated. Baer and Mesmer (12) have oruvided an excellent summarv of the
Literature Cited I l l Clsre,B. W.. Chrm. Rrii.. submitfad 1978. 121 Hussufti, F. J. C. and Ros~ati, H. S.. "The Determination of Stability C,mrlan~1." McGraw-Hill.NeuYrk,I W I . isi Rydheq,J..Acta C l w m Scand.. I5.17211196li. (41 Wentworth. W. E., J. CHEM. EDIIC., 42.98 119Bhi. IS1 Dleper. N. R. and Smith. H.."Applied Repremion Analysis? Wiley Interscienca, New York. 1966. Chapter 10. (6) Hewlett I'ackard Cllmpsny. "HP2SAppIiraliunrPmgrams.'.p.78. 171 Hewlett Packard Company. HP 67/97 llrerr Lilrrary. Program 02648D. 181 R a e , C..'E and Meyer, N. .I.. Inor*.. Chml.. I . 780 (19621. 191 Dunsmure. H. S.. Hiefanm, S.. and Sillen. I.. C., A m Chem, Scand.. 17. 2844
.
,."
,,oe,, ,,,,,.
(101 Ahrland, S., Aeio C h e m Scond.. 3,374 119491. (111 Hu8h.R. M..Juhnson, J. S..and Klaus. K.A..Inor~. ('hem.. 1.378 (19621. (12) R a c . C. F.and Memer. H. F:."The Hydndyris ~~fCsfions." Wiley Inter~cience. New
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Volume 56. Number 12, December 1979 1 787
