Triprotic Titration Data Analysis
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After performing a pH titration of a weak polyprotic acid, students type their pH/titrant volume data into a spreadsheet that contains an exact algebraic model of the titration curve. For a triprotic acid, that is a quintic equation - too complex to evaluate by hand but easy for a spreadsheet to handle. By adjusting the parameters of the model and observing graphically the fit between the experimental data (circles) and the calculated model (line) they can estimate the unknown parameters, such as the pKs of the acid.
Download links:
TriproticTitration.wkz;
TriproticTitration.hqx
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The algebraic description of a triprotic titration (neglecting activities, as usual) is completely specified by three equations for the three ionization steps (K1, K2, and K3), the equation for water ionization (Kw), equations for the mass and charge balance, and in addition two equations to account for dilution of the acid and base concentrations during the titration.
Using the usual "computer algebra" notation, with H representing the hydronium ion and CA representing the total concentration of acid in all forms:
H*H2A/H3A==K1 expression for K1 H*HA/H2A==K2 expression for K2 H*A/HA==K3 expression for K3 H*OH==Kw water CA==H3A+H2A+HA+A mass balance for total concentration of acid H+B==H2A+2*HA+3*A+OH charge balance (B = the base cation) B==Vb Binit/(Va+Vb) concentration of base during titration CA==Va Ainit/(Va+Vb) concentration of acid during titrationEliminating the variables H2A, HA, and OH, B, and CA between these equations and solving for Vb using a computer algrebra program (e.g. Mathematica) yields:
Vb = -((Va*(H^5 - Ainit*H^3*K1 + H^4*K1 - 2*Ainit*H^2*K1*K2 + H^3*K1*K2 - 3*Ainit*H*K1*K2*K3 + H^2*K1*K2*K3 - H^3*Kw - H^2*K1*Kw - H*K1*K2*Kw - K1*K2*K3*Kw)) / ((Binit*H + H^2 - Kw)*(H^3 + H^2*K1 + H*K1*K2 + K1*K2*K3)))This expression gives the volume of base Vb as a function of hydrogen ion concentration H, the three Ks, Kw, the volume of acid Va, and the initial concentrations of acid and base Ainit and Binit. It is completely general and works for all concentrations and for Ks. The titration curve is obtained by plotting Vb on the x-axis and pH (=-log(H)) on the y-axis. This is referred to as an inverse solution, because we usually think of Vb as the independent variable and H as the dependent variable. In fact, it is in principle possible to solve this expression directly for H as a function of Vb, but the solution is extremely complex. Fortunately, this is not necessary for our purposes because we simply want to know how closely the theoretical expression above describes the experimental data. Our titration data will consist of pairs of experimentally measured volumes and pHs, and we could use either an expression for H as a function of Vb (to predict the volume Vb at each measured H) or an expression for Vb as a function of H (to predict the volume Vb at each measured H). We will do the latter.
To facilitate evaluating the above expression, I have set up a computer spreadsheet that already contains this equation, so you won't have to type it in. It is on the Fileserver, in the Wingz Spreadsheet/titration data analysis folder. A screen shot is attached. Follow the instructions in the scrolling text field at the bottom of the screen. Adjust the variable parameters to get the best possible fit between your experimental data and the theoretical curve (line). Get a print-out of the screen and submit with your lab report.