George E. Knudson and Dale Nimrod Luther College Decwah, Iowa 52101
Exact Equation for Calculating Titration Curves for Dibasic Salts
Several computer programs are available for the calculation of curves for the titration of dibasic acids using a successive approximation technique to solve higher-order equations for the hydrogen ion concentration as a function of volume of titrant.'J Fleck derived an equation in which the volume of titrant can be calculated exactlv as a function of the hydrogen ion c~ncentration.~ The method of Fleck has been found to be equally applicable to the titration of salts. I t is possible to make the calculations using a hand held calculator, but a simple Basic program has been written to simplify the process. The equations apply to the following cases
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1) H2A NsOH 2) HNaA NaOH 3) NaHA HCI
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calculations for the titration o f a monobasic acid or saltcan also he done if Kz is taken to be equal to zero. Undergraduate students are generally able to write the charge balance equation, the mass balance equation, and the three equations for the ionic equilibria. In addition there are equations relating the concentration of sodium ion, chloride ion, and acid anion to the volume and concentration of titrant and solution. Since seven equations must he combined for cases 1and 2, and eight equations for cases 3 and 4, algebraic manipulation is extensive. The find equat i o n are similar enough for all cases so that a neneral equation may be written which has the following form
where Vt = volume of titrant, Ct = molarity of titrant, V , = original volume of solution being titrated, C. = original molarity of solution being titrated, B = H4 H3K1 H2(K1Kp - K,) - HKIK, - K,K2Kw(H is the hydrogen ion concentration), D = H3 H2K1 HKlK2, a = +1 for cases 1and 2, a = -1 for cases 3 and 4, F = H2K1 2HKlKz for case 1, F = -H3 HKlK2 for case 2,3, and F = -2H3 - 2H2K1for case 4. We have written an interactive computer program4 which asks the user to choose the case number. to inout values for V,, C., Ct, K1, and Kz, and to choose the pH interval and limits. Outout is in the form of a table " eivine V+and DH.The program also presents an opportunity to refine the p~ interval and limits. A listing of the program and a sample run is available from the authors, hut one of the points of this note is that programming these derived equations is extremely straightforward and, in fact, computation is not impossible On a
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Presented at the annual meeting of the Iowa Academy of Science held at Ames, Iowa in April of 1975. IBreneman, G. L., J. CHEM. EDUC., 51,812 (1974). 2Ellison,H. R., J. CHEM. EDUC., 51,738 (1974). 3Fleck, G. M., "Equilibrium in Solution," Holt, Rinehart and
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Winston. New Ynrk.. 1966. o, R5. ~,~ "'rugram H2A BASK', 1:iStl words in length, interactive, output t o 10 cps terminal iatisfactory,minimal additional dorumentatim required, prwram written on HPLOtX)F/Accei+, tramponability not ~
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certified.
Volume 54, Number 6, June 1977 1 351
