is also positive for a fdte,positive x.
Literature Cited ~~
It is s a c i e n t to show that the numerator is greater than zero.
" "
" "
" "
~
1. M m h a l L A G. Biophyshl ChmisIq; Wiley New Yo*, 1981; pp 7M4. 2 , Laidler, K J. Physical Chemistry uith Blobghl Applieatians; Benjamin/Cum"?in@, 1978; pp 4M-411. (The special cases of independent binding sites and of eoowrative hi"di"p:.m mn8;dered.I 3. See sillen, L G.; M&U, A. E. Wahllity Constants":Supplement No. 1, Spedal Pvhlieation No. 25; The Chemieal Society: Landon, 1911. and later ualumes. 4. Denhigh, K ThoPmncipka of Chemieol Equi1ibri.m:Cambridge University, 1971. 5. S"?ith.W.R.;Mieaen,R. W J Chprn.Edu. 1989.66.489. 6. Weltin, E. J Chem. Edue. 1891,68,486. 7. Thi8ideak nofpaltieularlyneru; we have used it in t h e c l s s m m c o r m a n y y Y3'
9. We1tin.E.J. Chrm. Edur. 1992,69,392. 10. Cobranchi, D. P,Eyring,E. M. J Chem. Edue. l89l,MI,40.
=C C + i , K & p + J - ' i.oj=o
Noting that the diagonal term i =j is zero, we can write the double sum as
The second term, aRer interchanging the summations and relabelling the dummy summation indices, is
Thus, we arrive a t the desired result.
forKi>O,Kj>O, a n d x 2 0 . Dividing the last expression by fix)', we find that&) is positive, approaching zero asymptotically for large x as
Equilibrium Calculations are Easier Than You Think -But You Do Have To Think! E. Weltin University of Vermont, Burlington, VT 05405
A short time ago Cobranchi and Eping (I)published an article on calculating equilibrium concentrations for a challenging equilibrium system by the continuation method, a general technique designed to generate all possible solutions of the mathematical equations. Other general methods to solve the nonlinear systems exist, and Smith and Missen (2)have given a number of techniques to calculate chemical equilibria based on Gibbs &ee energy considerations. We have previously shown that the equilibrium concentrations are uniquely determined by the initial concentrations for single-equation systems (3.4) competing and coupled equations (5) Extensions to stepwise binding of ligands or dissociation of polyprotic acids are straightforward. The common theme of our approach is to explicitly take into account the following chemical requirements.
Thus, the function g(x), which is zero a t x = 0, does not have local maxima or minima a t positive x, but grows monotonically toward its asymptotic value of n. This propertv is directlv ex~loitedin the calculation of maxima of the concentraiions of intermediate complexes by eq 3. It also assures that y' (the derivative of eq 2)and h' (the derivative of eq 61 are greater than or equal to 1. Thus. the total L in ea 2 is monotonicallv increasine with increasing x. As a consiquence, not only 'lls y uniquely determined by x, the reverse is also true: x is uniquely determined by y, although not by a very simple functional relationship. An analogous result holds for the function h. The maximum of a n intermediate complex [MLjl is reached a t the points where the derivative with respect to x i s zero.
This limits each reaction step to a chemically acceptable interval in which the functions of concentrations important to the calculation of equilibria have a simple, systematic mathematical behavior. As a consequence, the initial concentrations that determine the mass and charge conservation equations lead to a unique equilibrium state. The numerically simple bisection methods may be used to solve equilibrium problems.
which is satisfied if
The Corn~lexationof Thallium Ion with Nitrite Ion The complexation of T1+by NO< in aqueous solution discussed in ref I is of special interest because it represents a combination of the cases we have considered separately.
-
A concentration (or partial pressure) can never be negative. Equilibrium constants may range over many orders of
magnitude but can only take on positive fmite values.
Equilibria H++ NO,'
that is fix) - xf (x) = 0
which is the same as eq 3.
zHNO,
HNO, + Ht Z? NO' + HzO
+ H,O
2HN02
N,03
TI++ rNO,l-
zTINO,
K, = 103.16w1
K2=
(1)
M'
K, = l
~l y r l ~
K, =
w1
Volume 70 Number 7 July 1993
(2) .
(3) (4)
571
~
~
Applying the Chemical Constraints Clearly, z is limited to the range 0 to q.For any given z the following equation is easily evaluated.
Conservation Equations
-eT 1
[[NO]'] + [H' I + [TI'] -[OK1 - [NOZ17= q [HN021 + [NO27 + 2[Nz081+ [NO'] + [TIN021=c,
(6)
[TI']+ [llNO21= q
(8)
y=- z
K4
(7)
The first three equilibria (eqs 1-3) are stepwise processes, whereas the reaction shown by eq 4 is competing with that shown by eq 1, and the reaction shown by eq 5 has reagents in common with the reactions shown by eqs 1 3 . Equations 7 and 8 represent total nitrite and total thallium species, while a nonzero charge q in eq 6 balances the counter ions from the original thallium salt that do not take part explicitly in any of the equilibria. In ref 1 the , CT equal to lo3 system of equations is solved for q, c ~and M each.
The value ofy itself can not exceed c ~ . This constraint narrows the acceptable range of z to the interval
In spite of the apparent complexity, eq 10 is, for given y and z , a simple quadratic equation in x.
with b=Kc
Solving the System Directly
We show here that by using the chemical constraints it is possible to solve the system of equations directly for the single chemically significant solution. One or more concentrations are selected as independent variables. In the bisection method they are varied systematically in the chemically acceptable range until all conservation equations are satisfied. Choosing the Independent Variables for Mathematical Convenience The variables are not necessarily the quantities under direct experimental control but are chosen for mathematical convenience. The three conservation equations may be satisfied by three independent variables. For the equilibria shown by eqs 1-3, obvious choices are
and Y = [NO27
Then [Wl=x [NO27 = Y
(11)
a = 2&b2
+ K2b
e=y+Koz-c,
The coefficients b a n d a cannot be negative. Consequently, only the positive solution of the quadratic equation is chemically acceptable.
Finally, to solve eq 9, we introduce the following function. f=K2bx2+x+z-(KJZ)-~
Substituting eqs 11 and 12, we can see that this is a function ofz that is easily evaluated for any z in the acceptable range. Bisecting the Interval A preliminary calculation shows that f(z) is less than q a t zland greater than q a t zh. It increases smoothly with increasing z in the chemically allowed interval. Consequently, there is only one chemically acceptable solution of f l z ) = q. There is a very efficient and easy way to find as close a bracket to the solution as one desires: Bisect the interval. In other words, using
[HNOzI = Klxy [NOt] = K&& [NZO~I =K ~ ( K ~ x ~ ) ~
For eq 4 we select
we replace zl with z iff < q, and we replace zh with z iff > q. With the final z = [TI+]
z = [TI']
Y = [[NOJl
Then [TI+]= z [TlN021= Koz
For eq 5 we select [OH7 = K&
Thus, the conservation equations are
z+K&=+
These three simultaneous equations must be solved for the three unknowns: x , y, and z. 572
Journal of Chemical Education
and z = [Hi]
the concentrations of the remaining species are calculated using the known equilibrium constants by the expressions eiven ., above. The actual computational loopcomprises seven lines in a BASIC or Pascal promam. or it can eenilv he imdemented on a programmable calculator. The computation times are less than 1s each for both the preliminary calculation and the solution of eq 9 on an AT&T PC 6300 (an XT-class machine without math co~rocessor).This compares to 1 min per root for a total of 2'5 mi" on an AT-CI~S;machine with 80287 coprocesmr quoted tw Cobranchi and Eyring . - for the continuation methid. For the conservation conditions
A simple, but effective test for correctness of the solution is to recalculate the equilibrium constants and the conservation relations from the calculated equilibrium wncentrations. However, to obtain good agreement, especially if the concentrations varv over many orders of magnitude, one finds that it is .generally n e c e ~ s &tocalculate~hecon~ centrations to a precmion well beyond what is considered chemical
and q.=lo3M
the results are1 [H+l=5.600 x
lo4
M
[NO2-]= 5.561 x lo4M
[mo2I = 4.399 x lo4
M
[NO+]= 1.957x lo-'' M [N20, I = 3.067 x lo4 M
[Tlq= 9.961 x lo4 M = 3.921 x lo4 M [TINO~I [OK1 = 1.786 x 10-l1 M
here is a small discrepancy with ref 1, which gives Vli] = 9.996 x lo4 M and [TINOa = 3.935 x t @ M. We have verified that our 2General-purposeeqJilibrium algorithms do, of WLrse, exist. A reviewer has ponted out tnat the EOS software (Tscnnlcal Database Systems. New York) can solve the problem in one operation. I cannot comment directly on this softwareas I have never used it. I would like to point out, however, that the wncentrations of species in one system may vary over many orders of magnitude (lo-'' to 10" M in the present example).This is bound to lead to convergence difficulties.
Conclusions The calculation of equilibrium concentrations is important in chemical education and research. Using the fact that equilibrium constants are positive and paying attention to the chemical restriction that concentrations cannot be negative, we can eliminate from consideration all spurious mathematical solutions that have no chemical significance. What remains is a single solution of the nonlinear system that can be found easily by a numerical method. For the classes of equilibrium problems that we have discussed previously, it is quite easy to design general computer programs that fully automate the calculation of the equilibria. It is well worth the effort, on the other hand, to analyze complicated systems like the present one individually2 because this leads to a n efficient computation of equilibrium concentrations. Literature Cited 1.Cobranchi, D. P.; Eyring, E.M.J
C k m Educ. 1891,68,40. 2. Smith, W. R;Miasen, R. W. ChemMlReoction Eqzilibrium Analyak: ThooryandAIgorithms; Wiley: New York,1982. 3. Weltin, E. J. C k m .E d m . 1890,67,548. 4. Weltin, E. J. C k m Educ. 1831,68,486. 5. Weltin, E.J. C k m Educ. ISSB, 69,392.
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