are completed. In addition, students then look at trends for different properties using KC?DISCOVERER and attempt to explain these trends using basic atomic theorv. The ma& objective of the lab is to observe periodicity in the trends of oro~ertiesof the elements. AS part of their laboratory report for the conductivity lah, students used KC?DISCOVEHER to examine four different periodic trends from a list we provided. The list included atomic radius, electronegativitv, first ionization enerev. oxidation numbers, and others. students used the GKPH, TABLE, and FIND options of the KC?DISCOVERER Drogram to look a t these-trends. In their report, students were asked to give a definition of the property they examined, give a brief description of the trend they observed, and give an explanation why the trends behave in the manner illustrated bv KC?DISCOVERER. Thev were allowed to use anv source i f information, including "KC?DISCOVERER, help them with their report. We also gave them an example of what their "trend reports" should look like. I t came as no surprise to us that explaining the whv of the trends would be the-most difficult (but most meanhgful) aspects of these reports for the students! Their explanation often required them todig into their lecture-course textbooks or some other source as a reference, thus reinforcing these lecture concepts. Because of the ease of use and variety of options in KC?DISCOVERER, many students spent extra time exploring properties and trends that wereinteresting to them. We believe that a threefold approach of (1)making measurements in the lab and observing periodicity after assembling these measurements, (2) seeing them with the graphics capability of KC?DISCOVERER, and (3) explaining the trends in a formal report is an excellent means of "packaging" the concept of periodic trends into a laboratory. Programs such as KC?DISCOVERER offer many untapped opportunities t o integrate good computer software into computer-based laboratories. ~
~
~
The equation for the hydrogen ion concentration in a mixture of monoprotic acids and bases is (7)
~
(1)
where the contribution from strong acids and bases is o = x[strong bases] - x[strong acids], Kbi is the dissociation constant of the ith weak base, Pi is the original concentration of the ith weak base, K,,, is the dissociation constant of the jth weak acid, aj is the original concentration of the jth weak acid, and K, is the dissociation constant for water. If we apply the definitions z = -In [H+],w = -In K,, bi = -In Kb,, and a, = -In K., eq 1can then be written as:
tb
The pH of Any Mixture of Monoprotic Acids and Bases D. P. Herman, K. K. Booth, 0. J. Parker, and G. L. Breneman
Eastern Washington University Cheney. WA 99004 The pH of any mixture of monoprotic weak and strong acids and bases can he calculated. A curve can also be plotted for the titration of the mixture by any monoprotic weak or strong acid or base. The required input is the stoichiometric concentrations of the acids and bases in the mixture before equilibrium, the dissociation constants for the weak acids and bases, and the volumes involved in the titration. The method applies t o a wide variety of mixtures including polyprotic materials with independent dissociation constants (not stepwise dependent) such as amino acids and polypeptides. Theory An innovative technique derived bv Rang (7) is used to solve the equations for given the v&mesand concentrations. The usual equilibrium, mass balance, and charge balance equations are rewritten in terms of hyperbolic trigonometric functions allowing a better behaved iterative solution than using the usual Newton-Raphson iteration on a polynomial in H+ concentration. For example, the program earlier described (8)for the titration of anv sinele ~olvoroticacid or base with stepwise dissociation c&&n& bGeaks down regularly if the number of equilibrium constants goes above eight. The method described here has no such limit on the number of acids and bases in the mixture.
where
and
Equation 4 is a corrected version of the one in the original reference (7). Letting X = z - w/2, eq 2 can be rearranged to give:
This equation allows an iterative solution to be obtained for X and thus [H+] using
where X,+I is the new value calculated from X,, the old or pH = 7). value. The initial value of X is zero ([Hf] = The iteration is continued until
The cut-off value of 10-6 in eq 7 is a compromise between speed of convergence and accuracy. I t results in pH values accurate to two places past the decimal point. Dlscusslon Figure 1shows an example of the program input and the titration curve of a mixture of two acids (one weak and one strong acid titrated with a strong base). The equilibrium constints can he input as K's or PK's. A strong aiid can be input a s a weak acid witha very large K (e.g., K = IW'or pK = -5) or in the separate stnmg arid category. The concentrations input are those in the fmal mixture beforeequilibri. urn has been established. The titration of a sinele u.eok acidby a weak base is another good example to runshowing that the break in DH at the end point mav not be verv sharn unless strong acids or bases are used foititrations instead df weak ones. There is no limit on the t w e of mixture beine titrated or what is used as titrant fromamong the choices 2 weak or strong acid or base, since if n acids and hases are being titratedihe pH during the titration is calculated for a mixture of n 1components with concentrations adiusted for the amount of titrant added.
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Volume 67 Number 6 June 1990
501
HOW MANY STRONG BASES ? 0
DO YOU WISH TO ENTER YOUR DISSOCIATION CONSTANTS FOR YOUR WEAK ACIDS AND BASES AS 1 = K'S OR 2 = PK'S
. -7
7
DO YOU WISH TO TITRATE THIS MIXTURE ? yes
ENTER THE NDIIEER OF WEAK ACIDS I N THE MIXTURE ? 1 INPUT THE ACID DISSOCIATION CMISTRNTS, PKA'S FOR EACH WEAK ACID ONE PER LINE ? 5 ENTER CONC OF HK ACIDS ONE PER LINE ? .05
ENTER A 1 IF YWR TITRRNT I S A WWK BASE ENTER A 2 I F I T I S A WEAK ACID ENTER A 3 I F A STROK ACID ENTER A 4 I F A STRONG BRSE ? 4 WHAT I S THE CONCENTRATION OF YOUR TITRRNT? ? .1 WHAT VOLUHE lMLl OF MIXTURE ARE YOU TITRATING? ? 50 WHAT TOTAL VOLUElE IKL) OF TITRRNT 0 0 YOU WISH TO ADD? ? 65 WHAT VOLUME INCREMENTS (ML) OF TITiWlT DO YOU WISH TO USE? ? 5
ENTER THE NDIIEER OF WEAK BASES I N THE MIXTURE ? 0 HOW M Y STRONG ACIDS ? 1 INPUT CONCENTRATIONS OF STRONG ACIDS I N THE MIX ONE PER LINE ? .05
----
A n o t h e r c a l c u l a t i o n ? l y or n l ? Figure 1. Titration of a 0.05 M strong acid and 0.05 M weak acid (p&
-
5 ) mixture using 0.1 M NeOH, showing all me input required.
Simple buffers can he handled by inputting the acid and its coniueate hase as two separate s ~ e c i ewith s the appropriate concentrations and eq;ilibriu& constant valu& complex buffers involving several acids and coniuaate bases can also he easily handled as lung as all species &ehonoprotic. Of course sinale . monoprotic acids and hases can he handied also. The program can also be used to titrate polyprotic acids if the dissociation constants are not for a stepwise dissociation and are thus independent of each other. This can be useful for obtaining titration curves of amino acids and polypeptides. If there are n dissociation constants, they are entered as n separate species of equal concentration. The upper limit on the number of components in the mixture has not been established. Runs with 100 weak acids have been successful and larger mixtures should prohahly be handled also. Of course the ideal proararn would handle a n y mixture of polyprotic acids and bases. Attempts have been made to extend this method to this most general case hut so far without success. The attempts will continue, hut in the meantime the program described here will find many uses for classroom demonstrations or student experimenting with complex acid-base mixtures and titrations.
ore
Technical Details The iteration runs into some round-off error problems using single-precision calculations. Attempts to use double precision IBM and GW BASIC failed because these two versions of BASIC incorrectly evaluate functions in the dou502
Journal of Chemical Education
ble-precision mode. The program runs correctly using True BASIC on an IBM or Macintosh, both of which do the calculations to 10 digits and function evaluation to 14 digits (all calculations are done to 16 digits on the IBM if a math coprocessor is present). Double-precision Microsoft Fortran (16 digits) also works on the IBM. Executable and source code files of all three of these versions of the program are available from SERAPHIM. The source code was included because users may wish to modify the programs to increase the dimensions above 100 acids and bases, to change the convergence limit thus changing the speed and accuracy, to custom tailor the input and output t o their own taste, or to make an attempt a t handling polyprotic mixtures.
Methods for Calculating the pH of Aqueous Solutions of Salts of Monoprotic Acids and Bases Edmund R. Malinowskl
Stevens Institute of Technology Hoboken. NJ 07030 For an aqueous solution of a salt of a weak monoprotic acid and a weak monoprotic base, the exact governing equais tion for determining the hydrogen ion activity, [H+],
