George M. Fleck
Smith College Northompton, Mosrachusetts
Student Research Experiment
With Multiple Equilibria
A n open ended multi-equilibria experiment with "research" character is being used as an important part of a junior course in quantitative chemistry a t Smith College. The student is given freedom to take initiative in experimental design, and has choice in methods of data treatment. Time is available for preliminary data to be evaluated and for a second r e fined set of experiments to be performed. The amount of lead in an aqueous solution in equilibrium with PbCl? depends on the total chloride in solution. The problem is to find the functional form of this dependence and to interpret the data in terms of plausible chemical equilibria. Extensive classroom work on aqueous equilibria prior to the experiment provides the basic theory and mathematics. More than one reasonable conclusion is consistent with a set of good data; the absence of one clearly "right" answer can teach much about relationships between experiment and theory. The following directions are given the students: Prepare a set of solutions of vsrying total chloride concentration, each in equilibrium with solid PbCL, and determine the total lead and total chloride wncentratiions in each solution. Choose some temperature far the measurements. A thermostated 30°C water bath is available, or a 0" or 100' bath can be constructed. The aqueous and solid phasea can be brought to equilibrium either by shaking solid PbCh with a solution, or by precipitating PbCb. The additional variable amount of chloride must wme fromsomewhere. Supernatant liquid may be removed for analysis with a pipet fitted with glass wool, or with a sinteredpolyethylene or sintered-glass filter stick. After completing a preliminary set of experiments, try to analyze the data. in terms of same set of chemical equilibria, starting with the simplest case first. Obtain numerical values within upper and lower bounds for the relevant equilibrium con-stants. Results of the experiment are to be reported in a concise paper written in good scholarly form. Consult current chemical journals for tvoied format and stvle. Distineuish cerefullv between experimental reeults and interpretive conclusions. "A
An Opportunity for Creativity
Practical alternative procedures are many in this experiment, and many decisions have to be made by each student before beginning and (perhaps more significantly) as the experiment progresses. Initially the student must decide how many solutions to analyze, what range of total chloride concentration to try, and what analytical methods to use. The suggestion is made that if the solid is stoichiometric PbCI,, only one analysis need be made for each solution, and the other analytical concentration can be calculated if the solution has been quantitatively prepared; not all students believe this to be safe. A n EDTA titration for lead is encouraged and students are referred to the library for instructions (I, 2). Several 106
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Journal of Chemicol Education
EDTA methods for lead are found in the standard references and a choice must be made. Since interfering cations are absent, masking agents which affect the indicator can be safely omitted. Some procedural choices may be better than others, but in most cases different procedures serve to turn the investigations in slightly different directions. Preparation of saturated solutions and analysis for lead are not very time-consuming once techniques have been worked out, and so there is ample opportunity for the creative student to try to approach equilibrium from two directions, to check on ionic strength dependence, to see if there are any specific effects from the non-lead cation or the non-chloride anion, to extend the concentration range, or to compare data taken a t a second temperature. Since not all facets of the problem can be studied at one time, the one which appears to be most important or most interesting to the student is studied first. Mathematical Methods
The experimental data consist of quantitative information on how the saturated solutions were prepared, together with one or more analyses of each solution. Initial calculations convert these data into values for the two quantities which characterize each solution: [CI], the concentration of all chloride-containingspecies, in moles of chloride per liter; and [Pb], the concentration of all lead-containing species, in moles of lead per liter. A chemical model is postulated, and from this model a mathematical relationship between [Pb] and [Cl] is derived. Graphical or numerical methods are then used to find out if the experimental data are consistent with this chemical model. If not, a diierent or a modified chemical model is tried. The simplest chemical model consistent with the observed white precipitate is PhClp
$
Pb++
+ 2c1-
At equilibrium
where the a's are activities, quantities in ( ) are molar concentrations of species in solution, K,, is the thermodynamic equilibrium constant, and K.,' is an apparent equilibrium quotient which contains concentrationdependent activity coefficients. Since Pb++ and C1are the only lead and chloride species in solution, Equations 2 and 3 yield
Comparison of data with Equation (4) is straightfonvard. If there is lack of agreement, three possibilitics exist: data are inaccurate, it is not safe to assume constant activity coefficientsin the concentration range employed, and/or additional equilibria are important. The following discussion will be limited to the case of coniplex formation. Iriclusion of a single conlplexation step gives this chrn~icalniodel: PhClve Pb+++ 2C1K,,' = (Pbt+)(C1-j2 (5)
and the assulned value of KI' must be consistent with the observed values of slope and intercept. Studen! Results
Figures 1, 2, and 3 are graphs based on calculations of student data. These graphs arid calculations illustrat,e t,he sorts of decisioris which must, be made as the student tries to find a chemical model and a set of data which are in agreement. Calculations from preliminary experiments are sufficient to show that the data are inconsistent with a chemical model including only a precipitation reaction. Clearly the product [Pb][C1I2 (plotted versus [Cl]in Fig. 1) is not constant as required by equation (4). The next st,eo mav -" well be to trv eauatiori (12). ,, and the resulting plot for K,' = 1 is shown in kigure 2. The data do not fit equation (12) over the entire concentration range, and no value of K1'will n~akethis plot linear. Possible activit,y coefficient variation can explain substant~ialdeviations from equation (12), arid so a second set of experiments at constant ionic strength may be called for. Students with a flair for algebra may be more inclined to look for a set of additional equilibria which will be consistent with the data. The more species assumed, the easier it becomes t,o fit data to an equation, but the better the experiments have to be for a clear test to be made between alternative schemes. The algebra of course becomes increasingly challenging. At low chloride concentrations, higher complexes inay he present in only insignificant amounts. Figure 3 shows an attempt to fit the low-chloride data to equation (12), and here possibilities for success look promising. For a trial value K,' = 1, the intercept is negative. Neither K,' = 100 nor K,' = 50 yields slope and intercept consistent with the trial K,' value, but both are close. Since both slope and intercept have large uncertainties arising from data scatter, better experimental technique is needed; precision and accuracy required of measurements are thus internally controlled. Probably the fit will be best at lowest concentrations of chloride, suggesting experiments in which excess lead is added. ~
Combining equations
(T,),
(G), and (7), there results
Froiii equat,ions ( 5 ) , (G), and (8) comes (CI-)z = [CI](Cl-) - K,'K,,'
(10) Restricting the derivation to the physically-meaningful, real, positive roots of siniultaneous equations (9) and (lo), (C1-)%is equated in the two equations wit,h the rrsult
+
K1'1Pbl I [Pb][Cl]- KI'K.,'
(cl-) = Ka,,'ll
Substitution of equation (11) into either equation (9) or (10) yields an equation which contains only the apparent equilibrium constants and the two experimental observables, [Pb] and [Cl]. This final equation ran he written as IPb]lCl]Z = K,*'+ K,'SK.,'* + KI'K.~'IZ[P~I [Cli - KI'ILP~IICII- [Phlzll (12)
+
+
Plots of [Pb][C1Izversus 2[Ph] [CI] - KI1([Pb] [Cl] - [PbI2) car1 be constructed for different trial values of K,'. There are two criteria for consistency among the assumed value of K I r , the experimental values of [Pb] and [CI], and equation (12) : the plot must he linear within the limits of experimental uncertainty,
Figure 1.
formotion.
Test of equation 14) for o chemicol model xithovt complex
Figure 2. K,' = 1.
~
~
L~
Ted of equation (1 21 for
"
0
A
chemical model with PbClt complex.
Volume 42, Number 2, February 1965
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107
Comparison of results with data in the chemical literature is particularly instructive. A good entrance into the literature on chloride complexes with lead is the bibliography given in the Chemical Society "Stability Constants" (3). Students who wish to postulate many complexes can find encouragement when they see tentative suggestions for species up to PbC16-s from ion
exchange experiments (4). However, some lead chloride solubility data has been explained in terms of the single complexation step (5). PbCh(.,
+ C1- =
PbCI,-
Although not easily extended to higher concentrations, there is a detailed description of how activity coefficients can be used in a careful determination of K1 (6). The single way to interpret lead chloride solubility data over a wide concentration range is not found in the chemical literature, nor is it to be found in this student experiment. A chemistry major, with a simple set of experiments, has an opportunity here for some insight into the relationships between experiment and theory in chemistry, insight into the nature of a chemist's proof of the existence of chemical species, insight into what is meant by an explanation of physical-chemical data, and insight into how the creative chemist can and does put the mark of his individuality on his research. Literature Cited (1) S C ~ A R Z E N B AG., C H"Complexometric , Titrations," Interscience Publishers, New York, 1957, p. 92. (2) WELCHER,F. J., "The Analytical Uses of Ethylenediamine
Figure 3. Test of equdion ( I 2) ot low concentrations of chloride.
Table 1.
Calculation of K,' ond K,.' from Graph in Fiaure 3
(1954).
K.,' X 101, calculated from slope and assumed K,' &', calculated from intercept and KaZ..'
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Tetraacetic Acid," D. van Nostrand, Princeton, N. J., 1957, p. 189. (3) BJERRUM, J., SCHWARZENBACH, G., AND SILLEN,L. G., "St%bility Constants of Metal-Ion Complexes, with Solubility Products of Inorganic Substances," Chemical Society, London, 1957, Part 11, p. 107. (4) NELSON,F., and Knnus, K. A., J. Am. Chem. Sue., 76, 5916
305
negative
lourno1 o f Chemicol Education
6.3 42
3.6 86
(5) GARRETT,A. B., NOBEL, M. V., AND MILLER, S., T m s JOURNAL, 19,485 (1942). (6) BIGGS,A. I., PARTON, H. N., AND ROBINSON, R. A,, J. Am. Chem. Soe., 77,5844 (1955).
