From Titration Data to Buffer Capacities: A Computer Experiment for the Chemistry Lab or Lecture Roy W. Clark, Gary D. White, Judith M. Bonicamp, and Exum D. Watts Middle Tennessee State University, Murfreesboro, TN 37132
This . DaDer . is for all chemistrv teachers who have b e-~ n ning students, computers, and spreadsheets, and would like these reactants to combine more effectivelv. Give vour students the titration data we present in ~ a h i e s1 and 2. Let them enter this data into their spreadsheets, plot it in several wavs that vou direct. and ~ r o d u c em a ~ h s In . the process, yo;r students will discovermany things about pH, the taking of derivatives, buffer capacity, and the way buffers behave upon dilution. If you need more data t h a n t h a t provided here, we s u g g e s t s e v e r a l a l t e r n a t e sources. Your s t u d e n t s might actually go into t h e laboratory and take titration data. Students have done this ( I ) , hut if your students do take titration data experimentally, they must acquire as much data as possible near inflection points. AlTable 1. Titration Data for 25.00-mL Samples of 0.100 M Hydrochloric Acid and 0.100 M Acetic Acid Solution mL Xtranta
pH, HCI Soln
pH, HOAc Soln -
ternatively, you can generate titration-curve data using a spreadsheet ( 2 4 , or you can write o r obtain a computer program to do this ( 5 ) .You might also t r y using t h e c h e m i s t r y s o f t w a r e p a c k a g e , E Q U I L (Micro~ a t h ' ) .EQUIL c a n g e n e r a t e a l m o s t a n y t i t r a t i o n data you can imagine. This is the way we got the data here. The advantage of t h e EQUIL software is t h a t you can t u r n on or off activity coefficients, choose different formulas for activity coefficients, and easily ent e r competing equilibria t h a t can affect pH and buffer dilutions. If none of t h e s e sources appeals to you, write to us. Mail u s a disk and we will send you more EQUIL-generated data tables. The remainder of t h i s paper contains instructions for the computer experiment. During the experiment t h e i n s t r u c t o r m a y w a n t to e x p l a i n t h e d a t a a n d 'Equil Version 2.11, Micromath Scientific Software, Salt Lake City, UT84121-3144. Table 2. Dilution Data for 0.500-mL Samples of 0.100 M HvdrochloricAcid Solution and 0.100 M Acetic Acid10.100 M Sodium Acetate Solution mL Water
Added
HCI pH of HOAcINaOAc pH of Soln Soln
Formal Concentration M
0.100
"4)
746
48.0 12.412 50.0 12.436 tiltrant is 0.100 M NaOH: (-) titrant is 0.100 M HCI. Journal of Chemical Education
20
30
40
rnL of titrant
Figure 1. Plots of pH vs. mLof 0.100 M NaOH added to (a)25.0 mL of 0.100 M HCI and (b) 25.0 rnLof 0.100 M HOAc; the reverse titration region is for mL of 0.100 M HCI added to 25.0 mLof 0.100 M HOAc.
Figure 2. Plots of mL of 0.100 M NaOH added vs. pH for (a)the HCI sample and (b) the sample of HOAc. The HOAc plot includes the results of the reverse titration with HCI.
graphs produced. We provide here some suggested commentary.
the weak acid-strong base titration. Therefore the indicator is less critical in the strong acid-strong base titration, but quite carefully chosen for the weak acid-strong base titration. There are regions in which the pH is least dependent upon mL of titrant. I n the HCI titration there are two of these r e ~ o n s one : a t low DH and the other a t high DH. These are called the pseuddbufYer regions. These solutions are resisting a change of pH, but not due to some equilibrium buffer action. Instead, these regions (below pH 2 and above pH 12, rtrsist a change in pH due to the logarithmic d t h i t i o n ofoH. Addition of im:ill amounts of ~ H'~ and ~ - ~014 --can have little effect on the pH because [Htl or [ O K ] is already very large and small additions make little difference (6). The region of moderate slope before the inflection point in the weak acid curve is often marked off and labeled the buffer region in textbooks, even though this slope is not a s small a s the slope in the pseudobuffering regions of strongacid titration curves. In the buffer region for the acetic acid titration, some base has been added to neutralize some of the weak acid, producing the salt of that weak acid (sodium acetate in this case). Therefore, these solutions contain the weak acid plus its salt, the requirements for a n equilibrium buffer. The equilibrium established in this region acts to regulate pH (recall the Henderson-Hasselbach equation.) Buffering is best a t the half-way to nentralization point, and the pH there is the pK, of the weak acid being titrated (pK, = 4.76 for acetic acid).
Plots of Data on Spreadsheets Instructions for the Student The data in Table 1describe the titrations of two acid solutions: 25.0 mL of 0.100 M hydrochloric acid and 25.0 mL of 0.100 M acetic acid. The titrant volumes from 0-50 mL are for titrations of the acids with 0.100 M sodium hydroxide solution. The (-1 mL entries correspond to the volumes of 0.100 M HCI you would need to add to the original acetic acid sample to lower its pH to that in the table. We will call the addition of acid to a n acidic solution a reverse titration. The data in Table 2 describe dilutions of two acid samples with water. Enter the data from 'hbles 1 and 2 in appropriately labeled columns in your spreadsheet. Plot curves of pH vs. mL titration for the two acids titrated, with pH on the y axis and mL titrant on the x axis. The y-axis range should be 0-14. Include the data for the sodium hydroxide solution and the data for the reverse titration with HCI. The reason for the reverse titration with HC1 will become apparent later. Instructor Comments Notice the inflection points (Fig. 1). This is where the slope of the line ceases to increase and begins to decrease. The inflection regions are the pH ranges within which we want a colored indicator to change. This pH range is large for the strong acid-strong base titration, but smaller for
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F g ~ e 3I p L ) \,pH) as a ILnctfonof pH for (a)lne hC sampleano (bj lne samp e of dOAc ,oafs lor lne rer erse t Ira1 on 1s also p one0
Figure 4. Buffer capacity as a function of pH for (a)the HCI sample and (b) the HOAcsolution. The buffer capacity wing to the left in b was acquired using the reverse titration data.
Reverse the Variables Instructions for the Student
to accommodate the new y values.
Now reverse the variables, thus plottingmL of titrant vs. pH (mL on the y axis, pH on the x axis). On most spreadsheets this is a simple task involving the choice of x block and y block. The new plots are instructive. Instructor Comments Now that the variables are reversed, the steep slopes have become small slopes, and the best buffer regions are now steep regions (Fig. 2). The reason for the reverse titration with HC1 becomes apparent. We wished to hack into the low pH range in order to show the steep pseudohuffering region in the weak acid plot more convincingly. In each plot there is a steep pseudohuffering region a t high pH also. I n the weak acid plot there is a third moderately steep region due to the equilibrium buffer of the weak acid and its salt.
fined. You will need to increase they range in your graphs Instructor Comments The slopes of the plots in Figure 3, are A(mL base)/A(pH). Regions where the values of A(mL baseUA(pH) are large indicate that the pH of the solution does not change very much for changes in mL of acid or base added. The solutions in these regions have a larger buffer capacity than those elsewhere. The usual definition of the huffer capacity is centered around the pH change caused by the addition of small amounts of acid or hase to a given solution. The quantitative relationship for buffer capacity was first introduced by Van Slyke in 1922 (7). The differential ratio dCddpH is the huffer capacity P; it expresses the relationship between the increment of strong base added, in gram equivalents per liter, and the resulting increment in pH. The relationship is
Plot the Derivative Instructions for the Student The spreadsheet can easily plot the derivative of the mL of titrant vs. pH. By constructing the formula for the Ay over Ax values, and copying this down the column, you obtain the derivative values that should be plotted versus averages of the pH values used to obtain them. You may have to figure out the correct formulas to enter, or your instructor may tell you explicitly. In either case, the resulting graphs are instructive because they are proportional to (but not equal to) the huffer capacity a s it is usually de-
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where Cb and C, are the number of moles of monoprotic, strong hase, or acid added per liter (8).In either case, P is always positive. Near the pK, o r pKb, Cb a n d Ca are the number of equivalents per liter needed to produce a unit change in pH. If measurable changes like those in the tables are substituted for infinitesimal changes, then the approximate huffer value is P = ACdApH. Students can obtain approximate p values from the slopes in Figure 3.
6
k i s i o i s j o j s i ~ k k mL of Water
Figure 5. Plots of pH vs. mL of water added to (a)0.500 mL of 0.100 M HCI and (b) 0.500 mL of a solution 0.100 M in both HOAc and NaOAc.
Figure 6. Plots of -log of the formal concentration vs. pH for (a)0.500 mL of 0.100 M HCI and (b) 0.500 rnL of a solution 0.100 M in both HOAc and NaOAc.
Plots of an Extensive Property vs. pH Instructions to the Student
An Unexpected Equilibrium Instructions for the Student
T h e y-axis values from your previous plots, A(mL base)lA(pH), must he multiplied by the concentration of the titrant solution and divided by 1000 and by the liters of solution. Themolarity of the titrant (0.100 M for the NaOH) times mL gives millimoles of hase, which divided hy 1000 gives moles of hase. Then divide by the total liters of solution a t that point in the titration (0.025 L + L of titrant added). Use 0.100 M for the HC1 concentration in the reverse titration also. After such formulas are added to create columns of buffer capacity, P, set the y-axis range to 0 to 0.12, and plot the P values vs. pH.
Table 2 contains dilution data for two solutions: 0.500 mL of 0.100 M HC1 solution, and 0.500 mL of a solution that is 0.100 M in both acetic acid and sodium acetate. The pH values in the table are for dilutions of these two solutions with water. For the dilution data, plot pH vs. mL for 0 up to 50 mL of water added. Use 0-7 a s the y-axis range.
Instructor Comments
Point out that p is a n extensive property, not a n intensive property of some solution concentration. The high buffer capacity regions called pseudobuffering are always in a titration cuwe a t the high-pH end if the titrant is a strong base, and a t the low-pH end if the titrant is a strong acid. If the species titrated is strong, both high-pH and low-pH regions show a pseudohuffering wing (Fig. 4a). Because the titrant was a strong hase and one of the titrated species is weak, we artificially hack-titrated it with a strong acid in order to get both of these regions on one plot (Fig. 4h). The buffer capacity hump in Figure 4h is the result of real equilibrium buffering by the mixture of a weak acid and its salt.
Instructor Comments
Although the low pH region in the hydrochloric acid titration had a pseudobuffering capacity to the addition of small amounts of acid and hase, this solution has no ahility to keep the pH constant upon dilution of the solution (Fig. 5a). The ahility to resist pH changes upon dilution is an important property for buffers because they are usually mixed with other reagents in a n experiment. Often commercial buffers are shipped concentrated and are expected to buffer though diluted 10 to 100 times. Strone electrolvte solution^ cilnnot do this. The equil~br~um b u k r dot:s'iln excelleut iob of resisting DH rhnnae uDon dllut~on(Fie. - %I. . This curve would h e even flaker k e p t for the small change a t the start. This small DH change is due to a n equilibrium not expected by stud& who hi~vc,mcmori~ed that sodium hydroxido is n strong - hase. I t IS the cuuilit). rium (9) Nai(aq) + OK(aq)
~ a ~ ~ ( a q K) = 0.20
In the EQUIL-generated data presented here this equilibrium was one of the simultaneous equations that the Volume 72
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dilution. At extremelv" high dilution everv solution must necessarily approach a pH of 7, and one can see this trend a t the right side of each plot in Figure 6 . First Derivative of the Dilution Data lnstructions for the Student
Finally, plot the first derivative of this dilution data by first constructing t h e appropriate formulas for &(-log (concn))/A(pH),and copying the formulas down the columns of the spreadsheet. Inspect the derivative values to find the appropriate y-axis range, and then plot the derivative vs. pH a s before. Instructor Comments
The previous definition of buffer capacity will not do for dilution because we are not adding a strong acid or base. The quantity calculated, A(-log (concn))/A(pH),is a s good a definition of buffer capacity with respect to dilution as any. I t will be large if ApH is small for a given dilution. I t also has the advantage of being an intensive property of the solution; i t does not depend upon the sample size. This new buffer capacity, Ddil, is quite large for the equilihrium buffe r over four decades of dilution. For the strong electrolyte i t is a constant value of 1due to the identity between the molar concentration of a strong acid and its hydrogen ion concentration. Summary
Figure 7. Buffer capacity upon dilution, pdii. as a functionof pH for (a) 0.500 mL of 0.100 M HCI and (b) 0.500 mL of a solution 0.100 M in both HOAc and NaOAc. software was solving. In real solutions this equilibrium presumably has the same result, which could be avoided by the use of another strong base with lesser urge to associate. One of the advantages of EQUIL is that this competing equilibrium may he removed and the data recalculated to show what effect the equilihrium had on the pH. We leave the effect i n our data because otherwise the slope of this line is close to zero initially, and the inverse of this slope (buffer capacity) will he close to infinity. Extreme Dilutions Instructions for the Student
Again we wish to reverse the axes and plot pH for dilution on the x-axis. However, in order to see the effect of extreme dilutions, we need to plot -log (concn) vs. pH. So incorporate data tabulated for the very dilute solutions into these two plots. After constructing a column of -log (concn),inspect i t to find the proper y-axis range. Instructor Comments
Looking a t Figure 6, and remembering that the divisions on the y-axis are decades of concentration, we see that equilibrium buffers do quite well a t keeping a constant pH despite dilution. If we remove the effect of the NaOH(aq) equilibrium, the line would be even steeper. The hydrochloric acid solution i s not a good buffer upon dilution, changing pH by 1for each decade, or 0.303 for each double
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Much more can be learned by the manipulation of titration data besides the location of the equivalence point. Turning titration plots sideways and taking the first derivative displays buffer capacities a s a function of pH. These plots show clearly why there is pseudobuffering a t high and low pH values, and they show a true buffering maximum a t the pK, or pKb of the weak electrolyte species. The ability of a buffer solution to keep a constant pH upon dilution is important, and equilibrium buffers have this property to a much greater extent than nonequilihrium buffers. A convenient definition of buffer capacity with respect to dilution is P d i ~where Pd" =
d(-log eoncn) d(pH)
Other definitions of the buffer capacity upon dilution have been suggested; the most prominent is from Bates (101.His buffer capacity upon dilution is defined as the pH change upon 2-fold dilution. I t has the disadvantage of being specific to a 2-fold dilution and is large for small buffer capacities. We prefer our Pdil definition of buffer capacity upon dilution and have not been able to find it suggested in the literature. The program EQUIL is a valuable source of easily simulated titration data, in lieu of the real thing. Literature Cited 1. Lynch. J. A : Nsrramom. J. D. J. C l i i r ? .E:din.. 1990.47,533-636.
2 Fvcirer. H . Ccoreurs ond Coiririnriririx ,ti Aiinivliml Chrrnislrv: A Sixodrlhel Apimwcli. 3rd d.:CRC Press: Bocl Rntim. 1092: Chrptar 8. 3. Cumio,J. 0.;Whitelev. R. V.J. C l w m Edrrr. 1991.68, 923-926. 4. Bicncman. G. L.: P W ~ CD.~ J. , il C1ii.m Elliic. 1992.69.46-47. 5 . Kiny. D. W.: D e s t r ~D. R. J C h i n . liii~rr.1990. Gi.032-933. 6. Robect~;,H K .I. Cilnm Edoi 1966.13 108. 7. Van Slvke. D. D. J. Bin1 Chcnx. 1922.52. SF-570. R. Hap& D. Buo,irirn,!ce Chen~icolA,~oli.ris. 3d cd.: W H. Freeman: NewYork, 1991: p 202.
9. Havris. D. Bun,ili!n,ir~eClicrniioi Aiiolyrir, 3lil ed.:\U. H. Freeman: New York, 1991; p 100.
lo
Bates. R. G. Drcrr,iino,io,, o f p H ; Tllmi:,, "iKi P,nr,lie. 2nd e d ; John wiiey r n d sons:h'ew Y"7.k. 1978: p 111.
