chemiccrl principle/ revbited
Edited by MVRIELBOYD BISHOP Clemson Universny Clernson. SC 29631
Ill. A Few Math Tricks Adon A. Gordus The University of Michigan, Ann Arbor, MI 48109
In this article in the series1 on chemical equilibrium we consider a few math "tricks" that are useful in equilibrium calculations and approximations.
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Estlmatlnct Lw Values The prerisedefiniton of pH is -logall-, but,if thesolution is assumed to be ideal. then DH = -loa IH71. If vou know the value of [H+], then you can estimateth; d u e of pH to within f 0.1 pH unit without the aid of a calculator simply by remembering thevalue of log2 (it is = 0.3 or, to be precise, 0.30103). You then know the logs of 4,8, and 5 since log 4 = log22=210g2=0.6,10g8=log23=31~g2=0.9,andlog5= log (1012) = log 10 -log 2 = 1.0 0.3 = 0.7. The logs of other values can he mentally interpolated from these values.
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Slgnlflcant Flgures In pH and Other log Terms The number of significnat figures in [H'] determines the number of decimal olaces in DH. This reauirement follows (approximately) from the propagation-of-error expression for a loaarithrn"ut can also be shown by chanaina the last significant figure in [H+] and observing which digitis affectM, pH ed in the pH term3. For instance, if [Hf] = 6.38 X = 4.195; if [H+] = 6.36 X M, pH = 4.197. The exponent in the IH+1 term determines the value of integers that precede the dicimal point (the characteristic) whereas the i o n exponential portion of IH+l determines the decimal portion (the mantissa) of the logarithm.
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The previous articles in this series are: Gordus. A. A. J. Chem. Educ. 1991, 66.138-140 and 215-217. For the equation: pH = -log [HP],propagation-of-errorresults in up,,= a[~+~1(2.303[H+]), where the sigmavalues are the (infinite-size sample) standard deviations. Thus, the uncertainty in pH is equal to the fractional uncertainty in [H+] divided by 2.303. For instance, if [Hf] is known to f4 % , which corresponds to two significant figures in [Ht], the uncertainty in pH is 0,0412,303 = 0.02, which allows writing the pH to two decimal places. This method of determining the proper number of significant figures, while not as precise as the propagation-of-error method, will apply to any mathematical function. (A similar, but more complicated, procedure has been discussed by Schwartz. L. M. J. Chem. Educ. 1985, 62, 693-6977 For example, if you have x6 = y, and if x = 2.539, then y = 267.90215. if you change xto 2.537 then y = 2.5376 = 266.63846. The change in y occurs in the digit just to the left of the decimal Doint indicatina that onlv three sianificantfioures are iustified in the vaiue of v. Thus.. 1 . ~ 3 =.268. 9 ~ in &a this &ocedure to- check ~, - -~ significant figures i t ,sbes! lo lry it a few limes oy moditying the last og 1 first oy one unit an0 then again oy no more than 2 or 3 unitsto see where the typical change occurs in the answer. ~
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The reverse also andies. The number of dieits in an antilogarithm is determihed by the number of deiimal places in the loearithm. Thus. a DH = 11.71 corres~ondsto rH+1 = 1.9 X M (two significant figures) and a p = ~0.6064corresponds to [H+] = 0.9854 M (four significant figures). Although this last example may appear to have far too many significant figures, the need for four significant figures can be shown by altering the last digit in the pH. If p H = 0.0065, [H+] = 0.9851 M. Neglecting the Water Equlllbrlum I t is relatively easy to describe the conditions under which the water equilihrium can be neglected in calculating the pH of a solution. However. we must limit our exam~lesto those involving species that result only in the production of H- or only in the production of OH-, but not both. Thus, solutions of acids, mixtures of acids, bases, or mixtures of bases can be considered. Solutions of suhsrances such as HPOI- that ionize t o yield H+ and hydrolyze to yield OH- are excluded as are solutions of salts of weak acids and weak bases such as NH4F because both ions hydrolyze; NHa+ yields H+ and Fyields OH-. However, a solution of a weak acid and its salt, HA NaA, would not he excluded, even though the HA ionization can yield H+ and the A- hydrolysis can yield OH-. The reason is this is an "either/or" situation. Either the reaction to establish equilibrium can be written as the ionization of HA (Ht ions are oroduced) or the reaction can be written as the hydrolysis of'^- (OH-ions are produced) and the principal reaction depends on the relative concentrations of HA and NaA as well as the equilibrium constant for the acid. Consider first the case in which the acidic substance or substances added to water result in the production of a moles per liter of H+. Because the water ionization also produces H+ and OH-, hut in equal concentrations = w moles per liter, the total [Hf] = a w and [OH-] = w. If we neglect the water equilihrium, we asume that w 100w. But a = [H+], and w = [OH-], so the inequality is [H+] > 100[OH-1. Multiplying both sides by [Hf] results in [H+]2 > 100[H+][OH-I. However, K , = [Hf][OH-] = 1.00 X so the inequality is now [H+j2> 1.00 X 10-12. Taking the square root of both sides results in M; taking the negative log of each side [H+] > 1.00 X results in pH < 6.000. If a basic substance or substances added to water results in
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the production of b moles per liter of OH-, a similar set of calculations results in the requirement that DH > 8.000. Therefore, within a maximum error of 1%, the waterequilibrium can be nerlected when DH < 6.000 or > 8.000. If the DH is between 6.060 and 8.000,the water equilihrium m u s t he included in the calculation. [Similarly, for a 5% maximum error, the water equilibrium can be neglected when 6.349 > p H > 7.651. For a 10% maximum error, the corresponding ranges are 6.500 > pH > 7.500.1 In practice, the simplest approach is usually to neglect the water eouilibrium. introduce anv other a~oroximations. and . .. calculate the pH. Then, using the 19 maximum-error criterion. we recalculate the oH onlv" if theorieinallvcalculated DH is betwen 6.000 and 8.000.
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Neutral Polnt In Acld-Base Tltrallons Students sometimes ask a t what stane in the titration of a weak acid HA with a strong bare the solution changes from acidic to basic or at what stane in the titration of a weak base with a strong acid the solution changes from basic to acidic. I t is easy to derive. Define a as the fraction of weak acid (or weak base) that has been titrated so that a = 0.0 a t the beginning of the titration, cr = 0.5 half-way to the equivalence & ooint. and . a = 1.0 a t the eouivalence ooint. Consider the titration of HA, weak acid, with NaOH. If C. is the remaininn concentration of HA (corrected for dilution) and C, is theconcentration of the salt NaA a t point or, then the total concentration of acid plus salt is CT = Ca C3, and C, = ~ C and T C, = (1 - a)CT. At a pH = 7.00, the [H+] and [OH-] can he described as arising solely from the dissociation of water. Therefore, no ionization of HA (to produce additional H+) nor hydrolysis of A (to produce additional OH-) is needed to describe the equilihrium system and, neglecting any nonideality, [HA] = C, and [A-] = C, precisen l ( l - a). ly. Thus, K. = [H+][A-]/[HA] = 10-7C,/C. = This can he rearranged to:
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For the titration of a weak base having a constant of Kb with a strong acid such as HCl, the solution will change from basic to acidic when:
Figure 1. Fubrum-lever anaicgy for an acid-salt mlxture. Shown Is a solutlon that is prepared by mixing 0.500 mol NaH.P04 and 0.200 mol Na.PO, and diluting to 1.00 L. The overall composition, indicated by lhe positlon of the required fulcrum corresponds to a solution where HzP04- and HP04>- an,the principal species In solution.
Solving Compllcated Equalions In equilibrium calculations i t is sometimes necessary to solve fairly complicated equations. However, if you have a fairly good idea of the approximate value of the unknown variable, the equation can usually he solved by sucessive eraohical aomoximations. We will describe here a . method fiiohtaining the answer that can often require less work. Given in the second article' in this series is a verv comolicated equation for the [H+] of the solution that resilts &en V. mL of a C. M H3X is titrated with Vb mL of a Cb M ROH. Consider the case where 25.00 mL of 0.100 M H3P04 is titrated with 0.100 M NaOH. We could ask for the value of [H+]when 50.00 mL of NaOH was added. This corresponds to the second equivalence point in the titration. Using these V, C, and K values, together with Kb = a large number = 100,000 in eq 1of article 111,we have an equation that is fifth power in [H+].I t is5:
is titrated For instance, if acetic acid (K. = 1.8 X with NaOH, the solution will change from acidic to basic when a = 0.9945. i.e.. when 99.45% of HAc is titratedd,! The titration of hyp&hlorous acid, HClO with K, = 3.0 X lW', will chanee fromacidic to basic when n = 0.23, i.e.. when 234 of HClO Ts titrated.
We take the complicated equation and, as was done in eq 3, write part of i t on left side of the equals sign (call that part A) and the remainder on the right side of the equals sign (call that part B).Thus,
Fulcrum-Lever Analogy for Weak Aclds The fulcrum-lever principle of physics can he applied to a solution that is a mixture of two different species of the same polyprotic acid. For instance, a solution can be prepared by dissolving 0.500 mol of NaH2P04 and also 0.200 mol of NaaPOa and diluting to 1.00 L. The two initial phosphate concentrations (0.500 M H2POa- and 0.200 M PO&) can he imagined as weights placed on the opposite ends of a lever that mans the HqPOa- to POa3- renion of the phos~horic acld tGratlon diairam. This is showkin Figure 1; where the eeneral sham of the H I P O ~tltration curve is displaved. The position of'the fulcrum that will allow the lever t o be in balance will require that the lever arms be in an inverse ratio to the "weights". Thus, the ratio of the HzPOa- lever arm to theP0a3- lever arm must he in the ratio of O.ZOO/0.5OO = 215, so that, if the length of the lever is 2 5 = 7, the fulcrum is 21 (2 5) = 0.286 of the distance between the H2P04- and POa3- equivalence points. As is shown in Figure 1,this places the fulcrum in the HzP04-1HPOF region so that these two ions are the principal phosphate species in solution.
What we do is construct agraph where we plot both A and B on t h e y axis versus [H+] on the r axis. We then choose a small range of [H+] values in the vieinit9 of the calculated,
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'Because K, for acetic acid Is glven here to only two significant figures, it might appear as If we are using too many significant figures in the answer. Using the procedure described in footnote 3, we can, for example, chanse - K.- to 1.9 X lo-? the result is a = 0.9948. and four significant figures are Indeed justified. Another way of showing this is to calculate ( I - or) = 10-7/(10-7 K,) = (1.00 X 10-')/(1.8 X 10W) = 0.0055, which has only two signfflcant figures;thus. a = 1
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As in all calculations, extra digits are carried in the intermediate calculations to avoid rounding-offerrors in the final value. Often, an approximate value can be easily calculated using a simplified form of the equation that can be derived using various assumptions. For instance, if it is assumed that the water equilibrium can be neglected and that there is negligible dissociation of a C.M solution of a weak acid. HA, then: [H+] = (C,K,)1'2. The exact equation (for the ideal solution) is a cubic in [H'] as given by eq 2 of the second paper in this series.'
Figure 3. Values of A: 0 and B: 0 for a vicinity of the expected solution of eq. 3. Figure 2. Values of A: 0 and 6: 0 far a moderate range of [Ht] values in me vicinity of the expected solution of eq. 3. approximate value of [H+]. An approximate%quation for the p H a t this second equivalence point is pH = 1/2(pKz pK3).T h e K values for H3P04areK1 = 7.50 X 10-3, Kp = 6.20 X 10-8, and K3 = 4.80 X 10-13, so that the estimated pH = 9.763 a t the second eguivalence point. This corresponds to [H+] = 1.73 X 10-lo M and represents our approximation of the [H+] for the solution a t the second equivalence point. If the range chosen for the graph includes the solution to the eq 3, then the curve for A will intersect the curve for B at the point where A = B, which is the value of [H+] that corresponds to the solution of eq 3. If the two curves do not intersect, we extend the range to [H+] values where intersection of A and B occurs. Normally, complicated equations such as eq 3 will show some curvature as is shown in Figure 2
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small range of [Ht]in the
for a moderate range of [Hf]. If we had a better idea of the value of [H+] that is the solution to eq 3, or if we use Figure 2 as a guide, we could choose only two or three more closely spaced pairs of [H+]values, as in Figure 3. Here, we see very little curvature, as happens when you examine a very small range of any equation, and the intersection is found to be [Ht] = 2.20 X 10-lo M, the solution to the equation. This corresponds to pH = 9.658. The usual procedure, therefore, is tocalculate, if possible, only two or three pairsof values for A and B in the immediate vicinity of the expected solution of the equation, to plot them, and to determine graphically the point of intersection of A and B7. 'Another example, Involving an equation with log x and xtt2, is given in Gordus, A. A. Schaum's Outline of Analyfical Chemishy; McGraw-Hill: New York. 1985: Chapter 1.
Summer Workshop for High School Chemistry Teachers H o ~ College e in Holland. Michigan, will host a five-week Summer Workshob and leaders hi^ Activities Proiect of notlonk srope'irom ~ u n s 2 toJul~?fi, 4 1991.ThisproLTam, supponed by the ~aiionaiSrirnrr ~.d.indation, ir d&ed for teachers of h t - y e a r honor*, accond-year. and Advanced I'lnrement chemiirrv courses. Six srrnvstcr h w r ~of Lvadunte rredn will be awarded. The purposr $6thc Project i s to assw high school trachrrs in m n k ~ n alaborswry nn integral part of their instructional participants will perform experiments, receive enhaneedsubject matter t v exchanee of ideas of interest to chemistrv backmound relatine to these emerirnents. be orovided the o.~.~ o r t u n ifor teacher, and he e&lpped ro e&agr m follow:op act~vitwsdirected 1; other tcilrl;krs in their regums. .irl&ds an; allowance support tor books, room, board, and travel will be prm~ded.'The program is cantmgenr un flop^. C'olle
