the computer bulletin b o d condition for the solution. In formal terms the proton condition' is a statement roneernmp, the t~alanceof proton rxress and d~ficienryin a system, based upon some ini. tial reference condition. In the above case the initial reference condition is the addition of sodium ethanoate to water. A fraction of ethanoate ions will combine with protons to form ethanoic acid. Every mole of ethanoic acid so formed represents a mole of proton excess. The self-ionization of water yielding protons and hydroxide ions gives rise to proton excess and deficiency too. Proton balances are useful for the analysis of polybasic acids. Figure 2 shows species distribution as a function of pH for a 0.05 M solution of phosphoric acid. We may determine the nH of a 0.05 M solution of Na.HPO4 by writing theappropriate proton condition for the solution: [Ht]
log concentrotion
0
1
3
2
5
4
7
6
8
9 1 0 1 1 1 2 1 3 1 4
pH
Figure 2. Log(speciesconcenlralion)vs. pH f a a 0.05 M solution of phosphoric acid. loo concentration
+ [H,PO,-] + 2[H3P0,] Proton excess
+ [PO:-] Proton deficiency
= [OH-]
The coefficient of 2 in front of the H3P01 concentration indicates that the H3POd molecule contains two rotons in excess of the reference ion, HPo!-. The intersection of the H2P04- lines and the PO:- roughly approximates to the above proton condition. The approximation is not entirely satisfactory since the OH- concentration is
0
1
2
3
1
5
6
8
7
9
10
11
12
1314
PH F'igue 3. Log (species m c e m i o n ) vs. pH for 0.05 M solutions of emanolc acM and ammonium ions.
only slightly lower than the concentration of PO?-. This problem is overcome on the spriadsheet by computing the composite line of log ([OH-] + [PO:-]!.The pH at the intersection of this cumoosrte line with [he H2POI- line is identified as the pH of the solution. More than one acid may be shown on the equilibrium graph. In Figure 3 acidbase equilibria for 0.05 M solutions of ethanolie acid and the ammonium ion are shown. The pH of a 0.05 M solution of ammonium ethanoate may be determined by identifying the appropriate proton condition: [CH,COOH]
+ [HC]= [NH,] + [OH-]
The intersection of the ethanoie acid and ammonia lines at a pH of 6.9 approximates to this condition. The Theoretlcai Course of a Titration Any stage in the addition of a base like NaOH to an acid (HA), is characterized by an electroneutrality ~ondition.~" [~a']
+ [H']
= [A-]
+ [OH]
(3)
The concentrations of [Na+] and [A-] are given hy the expressions: ma'] = Chase * Vhase Vacid Vbase
+
[ A ] = Cacid * Vacid * FrA Vacid + Vbase where Cacid and Cbase are the molar concentrations of acid and base. Vacid and Vhase are the volumes of add-
log concentration
~preadsheei:Acid-Base Systems Stephen Leharne Schwl of Blol~glcaiScleHealth Thames Polyiechnlc Wellington Steet LondonSE186PF United Kingdom
Acid-Base Equilibria The species distrihution of any acid H,A can he formulated as a function of pH by considering equilibrium descriptions of the system in terms of proton addition to the anion A"-. (Charges have been omitted for clarity.)
The fraction of unassociated anion (FrA) is given by [A1
+ [HA] + [H&] + ... [H.A]
Dividing top and bottom by [A] and suhstituting for the resultant fractions yields FrA =
.
.
.
.
.
.
.
.
.
.
.
.
...
...
...
and Environmemal
In a previous issue of this Jourml Atkinson et al.' outlined a number of uses far plots of pH vs. log concentration for species in acid-base equilibria. These included the ability to display trends as a function of pH and the solving of problems typically encountered in undergraduate chemistry courses. A numher of useful texts have been written on the subject of such graphical disp l a y ~ . ~In. ~practice the object of such graphs is to show how the concentration of an aqueous chemical species varies with some master variable such as pH or ligand concentration. Once the relationship between concentration and the selected master variable is formulated, one can [hen use a spreadsheet to compute speries roncentrstion as a function of this master variahle and show the variation
[A]
o
1 1+ X~K~[H]'
where K, is the prounation constant for sperieai. [Alisgiven by FrA' Cacid (theacid concentration), The fraction of species i, where i is the numher of associated protons, is given by Kt* [HIi* FrA. The spreadsheet may be set up to perform these computations and the resulting species distribution plotted as a function of pH. Figure 1shows a plot of log (species concentration) vs. pH for a 0.05 M solution of ethanoic acid. The graph may he used to determine the DH of the solution. To do this we formulate &I electroneutrality expression. Such an expression equates the molar concentrations
.. -8
..
'
0
1
..
..
1
2
3
4
\
1
5
6
7
8
9
1
0
1
1
: N
8
1
: I
2
1
3
L
1
4
pH
Figure I. Log (species concentation)M. pH for a 0.05 M solution of elhanoic acid. of positive and negative charges. For 0.05 M ethanoic acid the equation is [Ht] = [A-]
+ [OH-]
If we consider Figure 1, it is apparent that this electroneutrality condition is approaimated to by the intersection of the 11- and A- lines since at this point the coneentration of OH- is very low. The pH of this intersection is 3. We may alsonseFigure 1todetermine the endpoint pH in the neutralization of 0.05 M solution of ethanoic acid by a strong base like NaOH. At the endpoint the solution is essentially 0.05 M sodium ethanoate. The charge halance for this solution is
[Na']
+ [Ht] = [A-1 + [OH-]
(1)
Given that some ethanoate ions will hydrolyze to form ethanoic acid we may write the following mass balance: [Nat] = [A-]
+ [HA]
and thus eq 1may be rewritten as [HA]
+ [Hf] = [OH-]
(2)
On the graph this is approximated to by the intersection of the HA and OH-, lines. The endpoint pH is thus about 8.7. Equation 2 is a statement of the proton (Continued on page A2401
ed acid and baseand FrA is the fraction of A. By substitutron in and rearrangement of eq 3 it ran he shcmn that
+
Vbase = Vacid * Cacid*FrA [OH] - [HI Cbase [HI - [OH]
+
(4)
Willis'argues that Vbase may be considered the dependent variable, and thus various values for the proton concentration are used to compute the hydroxide ion and conjugate base concentrations. These are then suhstituted into eq 4 to compute the volume of base that would need to he added to give that proton concentration. I t should he noted that same proton eoncentrations will give negative values for base volume added. These values may, however, be erased from the spreadsheet allowing only positive values to be plotted.
Summary The above procedures have been used in undergraduate classes to good effect. The diagrams have proved useful in enabling students to "visualize" acid-base systems and the changes in such systems during titrations. While the production of the diagrams and development of charge, proton, and mass balances has necessitated the development of a real working grasp of the chemistry of the systems.
'
Atkinson. G. F.; Doadt, E. G.: Reil. C. J. Chem. Educ 1986.63.841. Slumm. W.; Morgan. J. J. Aqustic Chemistly: An inwuction Emphasizing Chemicd Equiiibria in NBhKBl Waters 2nd d . : Wlley: New York. 1981. Sillen. L. G. Qaphiml Presentation of Equiiib ria Data. In Treatise on Anaiytimi Chemistv; Kollb off. I. M.; Elvlng. P. J.: Eds.: Interscience: New Ywk. 1959: Part 1. Vol. 2. 4Wllls,C. J. J. Chem. E&c. 1981, 58.659.
Solving Equilibrium Constant Expressions Using Spreadsheets Clyde Metz and Henry Donato, Jr. The College ol Charleston Charleston. SC 29424 The concepts of chemical equilibrium, the equilihrium constant, and the equilibrium constant expression are presented in virtuall" all introduetorv calleee ehemistrv coursesfor both majorsand nonmajon. Stu. dentsare usually asked to solve equilibrium problems both ro test their understanding of the underlying chemical principles and to illustrate the utility of these principles. We have observed that the solution to these problems can he divided into two steps: 1. Students must desrrihe a chemical aituation with an algebraic equation. 2. The algebraic equation must be solved to find a physirally meaningful root.
The first step requires some understanding of chemical principles to identify the chemieal reaction involved and initial conditions for the chemical reaction, describe the net changes that must take place to reach equilibrium, and write down the equilihrium constaht expression in terms of a single unknown quantity. The second step involves finding the unknown quantity. In general,
this may require solving a quadratic, cubic, or higher order polynomial equation. The mathematical complexities of finding roots of cubic or higher order polynomial equations has led to the situation in which only problems leading to quadratic equations are considered in many introductory chemistry courses. Breneman (5) has addressed this lack of completeness by investigating the use of spreadsheets for finding r w t s of polynomial equations arising from equilibrium problems. He has shown that a spreadsheet can easily be constructed to find the root of a fourth-degree polynomial using an iterative procedure, the Newton-Raphson method. Joshi ( 6 ) has described a spreadsheet template which can be used to find the roots of any polynomial of order three, againusing the Newton-Raphson formula. While quick and convenient, these methods require from the student the ahilitv to d a c e the eauation in pdynomial form,a knowledgeofdiffermtial calculus, and enuugh chemiral intuition to start the iterative procedure reasonably close to the physically meaningful root. In this paper, we describe a general numerical method for obtaining a chemically relevant root to polynomial expressions generated from a consideration of chemical equilihrium. This method is based on a trial and error procedure tur findma the physically relevant root and rs convenrently executed on a personal romputer runnmg a spread sheet program
volved in solving the equilibrium constant expression. Suppuse that at a temperature of 400 'C a mixture of 3 parts H&J and 1 part NAg) is placed in a container with a total pressure of 10 bar and the parrial pressures of H2. N2. and NHx at equilibrium are desired. The reartion involved is
The initial partial pressures of Np and Hp can be extracted from information given in l the nrahlem. If onelets x he t h e ~ a r t i aureasure of the nitrop;en gar that reacts in order t t reach ~ equilibrium, then the equrlibrium partial pressures of the gases are given below:
. .
Description and Application of the Method Consider a reaction that is usually not discussed in introductory chemistry courses because of the mathematical difficulties in-
The equilibrium constant expression then hecomes
Remmd at me 10th Blsnnlal Conference on Chsrn!cal Educatoon. Purdue Unlverslly. A~gust 1988.
(Continued on page A242)
