An interactive model of the titration curve of a weak triprotic acid titrated by a strong base. For a triprotic acid, that is a quintic equation - too complex to evaluate by hand, but easy for a spreadsheet to handle. By using the sliders. you can change the acid concentration, acid volume, base concentrations, and the three acid constants of the acid (K1a, K2a, and K3a). (You can even change the Kw of water in cell J2). Assumptions: Activity effects are ignored. The pH is measured correctly without error at the ends of the pH range.
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(.txt)
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spreadsheet in OpenOffice format (.ods)
Note: to run these OpenOffice version of these spreadsheet, you have to first download the OpenOffice installer (download from OpenOffice), then install it (by double-clicking on the installer file that you just downloaded), and then download my spreadsheets from this page. Once OpenOffice is installed, you can run my spreadsheets just by double-clicking on them. Note 1: Don't use version 3.1. There is a bug in OpenOffice 3.1 that causes bad x-axis scaling on some of my graphs. The problems does not occur in version 3.0 or in the most recent version 3.2. Note 2: Downloading these files with Internet Explorer will change the file types from ".ods" to ".zip"; you will have to edit the file names and change the extensions back to ".ods" for them to work properly. This problem does not occur in Firefox or in Chrome.
The data entry version ("TriproticTitrationData") is similar to above except that it has a space to type in some experimental titration data (volume and pH). After performing a pH titration of a weak triprotic acid, students type their pH/titrant volume data. 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. The spreadsheet plots the experimental data on the same plot (as blue dots). The sliders can then be adjusted until the theoretical curve (red line) can be made to match the data (blue dots). The RMS error between the data and the model is also displayed.
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 K1Eliminating the variables H2A, HA, and OH, B, and CA between these equations and solving for Vb using a computer algebra program (e.g. Mathematica or Maple) yields:
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 titration
Vb = -((Va*(H^5 - Ainit*H^3*K1 + H^4*K1 - 2*Ainit*H^2*K1*K2 + H^3*K1*K2 -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.
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)))
To facilitate evaluating the above expression, the spreadsheets on this page already contain this equation, so you won't have to type it in. 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.