Calculation of Titration Curves by the Next Guess Factor Method S. Chaston University of Canberra, Belconnen, A.C.T. Australia 2616
The next guess factor (NGF) iteration method of calculating equilibrium concentrations in complex chemical systems was described recently in this Journal ( I ) . This article extends the method to the calculation of all points on titration curves for mono-, di-, and triprotic acids and their bases. The NGF method is an alternative method that does not require the solution of nth-degree polynomials, as described in a recent article (21, and makes it possible to obtain equilibrium concentrations by calculation with a pocket calculator. Method for Obtaining NGFs
The approximate equilibrium concentration of one (or two) of the species in the titration is chosen to start off the first iteration. For example, in Table 1a n estimate of the approximate value of [HB] is used as the basis for calculating all of the other species concentrations and the first NGF. The second and improved value of [HBI is then obtained by multiplying the first NGF by the first [HBI. The second [HBI is used in the second iteration, which produces a second NGF, and so on to the third iteration. The number of iterations carried out depends on how good the first approximate [HBI value was and how many significant figures are desired. I t is usually possible to achieve three significant figures within four iterations. The mathematical expressions for obtaining NGFs are ratios that are usually based on mass or charge balances. However, NGFs may also be obtained from equations that are based on equilibrium constants ( I ) .Each NGF must he equal in value to exactly 1when the true concentration values for all of the species concentrations are obtained in the calculation contain in its denominator the guessed concentration from which it was calculated, such as [HB] (eq 11,Table 1) Spreadsheets are ideal for these calculations because i t is possible to get a large number of new points on a titration curve within a minute of changing the K, or LB values. When a spreadsheet is used, there is no need to choose a good approximate first value, such as for [HBI in the above example, because over 100 iterations can be carried out each minute.
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Monoprotic Acids The next guess factors (NGFs) for the titration curves for monoprotic acids are derived from eq 1,which is a combination of eqs 2 and 3.
+[OK1 + [GI1 = [Na+l+[Ht]
+ [HB]
(1)
Mass Balance
Charge Balance Equation 1applies to all acids and bases regardless of the values of the ionic charges on HB and B. The value of LB is obtained from eq 4 when a weak acid is titrated by a Table 1. Calculation of the Titration Cuwe pH vs. PercentTitration Titrating Weak Acid HB with Strong Base [HB],
NGF=
= NGFx [HBIOI~ after the firstguess.
ZB-[Nail
+ [OH7 + [Cl7
from 0 to 100%
(7)
(11)
[ H I + IHBI
NGF=
[OH7 from 100to 200% [ H I + [HB] - Z B + [Nat]
% titration = Vb when
pH =-log [ H I
(12)
+
Va = 100 and Ca = Oo
(1 3)
(14)
The NGFs of eqs 11 and 12 work best for 16 Mlues from 1 to lo-'! in eq 11 [CIl represents any strong acid that may also be mixed with the weak acid in the sample. For eqs 11 and 12, at0'Aand loo%, the square root of the NGFwarb more efficiently.
Table 3. Calculation of the Titration Cuwe pH vs. Percent Titration Titrating Weak Acid HzB with Strong Base
Table 2. Calculation of the Titration Curve pH vs. Percent Titration Titrating Weak Base NaB with Strong Acid [HzB],w
NGF =
[Hfl [Cll - [Nat]+ [OH1+ [ E l
strong base, and from eq 5 when a weak base is titrated by a strong acid.
where C, and Cb are the concentrations of the acid and base; and V. and Vb are their volumes added to the sample flask. The NGF value in eq 6 is obtained from the the ratio of the left-hand side over the righehand side of eq 1.
5 + [ o m + IC11 [Na']
[ H z B ] after ~ ~ the first guess.
(23)
[E21new= NGF(q)x [ E Z l oafter ~ the first guess.
(24)
% titration = Vb when Va = 100 and Ca = Cb
(29)
pH = 4og [Ht]
(14)
above 100%
%titration = V, when Vb = 100 and Ca = Cb (22) pH = 4og [Ht] (14) The NGFs of eqs 20 and 21 work best over the 16 range of 10*to lo-". For eq 20, in the absence of strong base in the sarnpla,[Na*] = XB. In sqs 20 and 21, at 0% and 100%. the square mot of the NGFworks more eflidently
NGF =
= NGYM) x
+ [HI] + [HBI
For particular points on the titration curve, eq 6 can sometimes be modified to work more efficiently, that is, to produce the desired result in fewer iterations. These modifications are discussed below. Calculation Format
The format for the calculation of points on the curve for the titration of a weak acid by a strong base is displayed in Table 1,and for the titration of a weak base in Table 2. The equations in the tables can be entered into a column on a spreadsheet and then copied into as many adjacent columns as desired. Each column produces one point on the titration m e . The first value in the column for IHB]in Table 1 should be an educated guess of its approximate value if it is necessary to minimize the number of iterations, but the second and subsequent guesses are obtained from eq 7 in Table 1. With each guess for [HB] a new iteration is begun, and all oftheother values in the equations in the table (eqs 7-14) can be recalculated. Each new iteration produces more exact values than the previous one, and the number of iterations that are carried out depends on the number of significant figures required. Use of the Equations
In Table 1, eqs 11 and 12 are modified forms of eq 6. Equation 11 is obtained when the value of [ N a l is sub-
tracted from both the numerator and denominator of eq 6. This prodnces values in the numerator that are closer but slizhtlv neater than the lHBl value a t each mint: it also makes.ti;e iteration procedure more efficient. Above 100Ci, ea 1 1 is replaced bv e4 12 i n which IOH I has the dominant role in thk numerator and a slightly greater value than [HBI. At the 0% and 100%points the square root of the ratio in the NGF is found to work more efficiently. At these points the NGFs approximate to
respectively, which causes extreme fluctuations in the [HBI., values in eq 7 if the square roots of these ratios are not used. In Table 2 the same considerations apply as in Table 1. The calculation of a point on the monoprotic acidlbase titration curve by the NGF method can be done on a pocket calculator when only two or three sigmficant figures are required because only three or four iterations are usually needed to achieve this accuracy when a reasonable approximation of [HBI is chosen initially. A class of 30 students may calculate three points each, and their results can be recorded on a large graph paper on a large pin board with a drawing pin for each point. Ionic strength and activity coefficients may also be incorporated into the calculation formats in these calculations. Diprotic Acids To get the calcnlation started for each point on a titration curve for a di~roticacid. the eauilibrium concentrations of two species must be guessed. i n Table 3, [HzBI and [B2-I are these concentrations, but a combination of any two of [H2Bl,[HB-I, [BLl, or [HI] could be used. Two NGF equations are therefore needed: N G F W in eq 27 based on the mass balance for the system *NGF(q)in eq 28 based on a combination of the charge balance and the mass balance
These equations are valid regardless of the charges on H7B. - . HB. and B. After the first iteration, the new value of [HBI for the second iteration is obtained by multiplying its old value by Volume 70 Number 11 November 1993
879
Table 4. Calculation of the Titration Curve pH vs. Percent Titration Titrating Weak Acid HsB with Strong Base
the numerators in their respective NGFs. However, very good first estimates of their concentrations under these circumstances can be obtained from the initial approximations:
The number of iterations required at 100% can also be minimized using the relationship
which is particularly accurate when K d i s very much smaller than Kal. The calculation for the titration curve of Na2Bby strong acid has the same format as in Table 3. The only difference is in the NGF(q), which is given in eq 30, where [CI-I represents the amount of strong acid added.
Triprotic Acids As in the t~trationof l HzR I, two guesses are needed to get the calculation started on the titration curve of a t r i ~ n d i c acid with a strong base. In Table 4, approximate values for [H3Bl and [B3-I are selected as starting values for the calculation. The calculation format is essentially the same as for diprotic acids. Titration curves for tetraprotic acids, H4B,can also be calculated in this way. % titration = Vb when Va = 100 and Ca = Cb
DH = 4oa lHtl
the NGF(M) value. The new value of [Bsl is obtained by multiplying its old value by the NGF(q) value. Another way of producing new values for the next iteration is to divide the old value of [H2Blby NGF(q) and to multiply the old value for [B"l by NGITM). The procedure in Table 3 will produce results for all points from 0% to 300% and above. The two points that usually require the most iterations are a t 0% and ZOO%, especially when pKal is greater than 3 and pKd is less than 11. Under these circumstances [Bsl a t 0% and [HBI a t 200% become extremely small compared to the values of
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Journal of Chemical Education
Conclusion The NGF method of calculatinz e~uilibriumconcentrations has the advantage of being s&le mathematically. It brines most calculations for ~ o i n t son titration curves withyn reach of the pocket cal&lator. It opens up the study of wmplex chemical equilibria to students who do not intend to pursue advanced mathematics or develop computer programming skills. When personal computers are available spreadsheets are more easily programmed due to the use of simple mathematical expressions. Literature Cited 1.Chasten, S.J Chsm. Edue. in press. 2. Breneman, G. L.; Parker, 0.J.J. Chem Edue. 1992.69.4647
