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143

SHORT COMMUNICATIONS

The calculation and use of a coefficients

Side-reaction coefficients-usually called r coefficients--are nowadays widelyused in analytical chemistry to simplify equilibrium calculations. The concept of the3~coefficient was originally introduced by Schwarzcnbach), but several authors havelater extended its USC. Ringborn’ has susgcsted that the influence of practically alloccurring side reactions on a certain main reaction should be taken into account bymeuns of such coefficients The net influence of several side reactions is given by an“over-all Q cocflicicnt”. which is often approximately equal to the sum of the individualu cacfficicnts.

If a metal ion M and a ligand L form a complex ML. hydrogen or hydroxideions may affect the reaction in three ways. The reactions of the ligand with hydrogenions are taken into account by the cocffrcicnt aJ. Jf3r the reactions of M ions withhydroxide ion by the coefficient %M(crt) and the reactions of ML with hydrogen andhydroxide ions by the coefficient c~+~r,,~~, n). Analogously, reactions between L andinterfering cations, Mt. M,,. . . ., arc taken into account by the coefficients CQ.,~,,,%M,,). - ** and reactions bctwcen M and interfering liyands. A,, A,,, . ,., by the coef-ficicnts zh)(&)~ zh((,,,,). . . . The interfering species responsible for a side reaction isalways given in parentheses in the subscript of ~1.

The calculation of x cocfficicnts is, of course, based on equilibrium constants.For instance.

Iv

where /In is the cumulative protonation constant and N is the maximum number ofbound protons.

Similar expressions can bc written for other cy coefficients. On the basis of thesecoefficients, conditional stability constants of numerous complexes can be cal-culated’-*.

An Q coefficient is thus the sum of a number of exponential terms, and its cai-culation is actualiy a very simple mathematical operation. For calculating cc coef-ficients, (c~.{j. I.(II) as a function of pH for a polybasic acid), modern desk calculatorsand computers arc particularly appropriate and are commonly used.

Nevertheless, the analyst often meets problems demanding cc coefficients whicharc not tabulated in any textbook, and suitable calculators or computers are notalways immedi~ltely available. Moreover, if many side reactions occur, computersdo not always represent the most rapid and practical means of estimating the valuesof individual and over-all cx cocffrcients. The mason is that, as a rule, the final resultis affected by only a few of the numerous side equilibria.

We have noticed that many chemists waste time by performing calculationswithct coefficients in a far from rational way and arc not aware of the fact that additions

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144 SHORT COMMUNKCATIONS

for subtr~~ctions~ of cxponcntiaf terms can bc performed by rcfcrcnce to short tablesoften included in tables of logarithms.

Tables I and II arc given in a form particularly appropriate for chemists, whogenerally arc not intcrcstcd in logarithmic x values accurate to more than twodccimul pluccs. The tub& hardly need any explanation, but a few comments may beappropriate. If the sum of two cxponcntial terms or their diffcrencc is to be cal-culated with ;t precision of two dccimnls in the exponent. the smafler term can beneglected if the difference between the exponents exceeds 2. When the exponents differby more than one unit. tflc exponent of the larger term will change at most 0.04. Thismeans that scvcral terms can bc neglected in calculations of individual or over-alIx cocfficicnts.

The USC of the tnblcs can best be illustrated by means of a few cxamplcs.xmnple 1. Calculate al.oit for the complexing agent TTHA at pl-i 6.50. The

cumulative protonation constants are’ log /j, = IO. 19. log [j, = 19.59. log /jX = 25.75,log /f4 = 29.9 I ., log Ii 5 = 32.86, log &, = 35.28.

The complete expression for ~~~~~~~ s

~ ,.,,, )= 1 + IO-‘>.” ~~lO.l“+ ,o- 13.0 I(J“‘.““+ lo- IO.5 102575~

+ *()- 26.0 1()2‘1.‘~1 ,_ IO-- 32.5 1()32.HC’+ jo- 3‘J.O 10”“.2H(2)

It is immediately seen that the third term, lO”*“‘. and the fourth term. 10f’.25,arc the two largest ones. The difference bctwcen their cxponcnts is 0.34 and. accord-

‘1’AIJL.E I

ADl~l‘l IOS 01: IiSI’ONliN’l’lAt. ‘I‘I<I<.btS t 0” .C t 0” =: to“ ” I 2 h)

0.00 1

0 2

0.30.4

0.5Oh0.70.80.9

1.01.11.21.3t .4

1.51.G1.7I.8

0.301 0.296 0.29 I0.254 0.250 0.2450.212 0.209 0.20s0. I76 0.173 0.1700.146 0.143 0.140

0.286 0.28 10.24 1 0.2370.201 0.1970.167 0.lG.l0. I 37 0. I35

0.1 12 0.1100.09 1 0.0900.074 0.0730.060 0.0590.048 0.047

0.277 0.272 0.2670.232 0.228 0.2240.194 0.190 0.1870.160 0.157 0.1540.132 0.129 0.127

0.119 0.117 0.1150.097 0.095 0.0930.079 0.077 0.0760.064 0.063 0.06 10.05z 0.050 0.049

0.108 0.106 0.1040.0X8 0.086 0.0840.07 1 0.070 0.0680.057 0.056 0.0550.046 0.045 0.044

0.041 0.040 0.040 0.039 0.038 0.037 0.036 0.0350.033 0.032 0.032 0.03 1 0.030 0.030 0.029 0.0280.027 0.026 0.025 0.025 0.024 0.024 0.023 0.0230.02 1 0.02 I 0.020 0.020 0.0 19 0.019 0.019 0.0180.017 0.017 0.016 0.016 0.015 0.015 0.015 0.014

0.0 140.0110.0090.007

1.90,- 1.98

0.013 0.0130.011 0.0100.008 0.0080.007 0.007

0.005 1.99-2.09

0.013 0.012 0.012 0.0120.010 0.010 0.010 0.0090.008 0.008 0.008 0.0070.006 0.006 0.006 0.006

0.004 2.10-2.23 0.003 2.24 .2.46

0.0120.0090.0070.006

0.002

0.263 0.25X0.220 0.2fb0.183 0.1800.151 0. I480.124 0.122

0.101 0.0990.082 0.08 10.067 0.0650.054 0.0530.043 0.042

0.035 0.0340.028 0.0270.022 0.0220.018 0.0170.014 0.014

0.01 P 0.0110.009 0.0090.007 0.0070.006 0.006

2.47-2.94 0.001~II-___--.---_____

.-_-_-_-____

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SHORT COMMUNlC TIONS 45

TABLE II

(I - 12 f

0 1 2 3 4 5 6 7 8 9._.. _-......__ ___.._.._ .._.__._.^ _..._.. .-..__.._-_ -_. __-.--. ------.----- -_--- .- --.- - .___.__. -_--__--__ _

0.6870.4330.3020.220

0.5 0.1650.6 0.1260.7 0.0070.8 0.0750.5, 0.0%

I .o1.11.21.31.4

1.51.61.71.8I. )

0.0460.0360.0280.0320.018

0.0140.0 1 I0.00’)0.0070.006

1.91, I.98__.

1.643 1.347 1.176 1 OSh0.6SO 0.6 17 0.587 0.5600.416 0.401 0.386 0.3720.292 0.283 0.274 0.2650.2 14 0.208 0.202 0.196

0.160 0.156 0.152 O.l480.122 0.119 0.116 0.1 If0.094 O.OY2 0.0x’) 0.0870.073 0.07 1 0.070 O.O6S0.057 0.056 0.054 0.053

0.045 0.044 0.04.7 0.0-120.03s 0.034 0.030 0.033o.o2I( 0.027 0.026 O.WG0.022 0.02 I 0.02 I 0.0200.017 0.017 0.016 0.016

0.0 14 0.0 1 3 0.013 0.0130.011 0.01 1 0.010 0.0100.009 0.008 0.00x o.Oot(0.007 0.007 0.006 0.006

0.005 l.YY 2.w 0 004 2.10~224 0.003 2.25 2.46 0.002 2.47,.2.94 0.001

0.964 0.8X90.535 0.5110.359 0.3460.257 0 149.0.190 0.1X5

0.144 0.1400.110 0.1070.0x5 0.0830.066 0.0650.052 0.050

0.04 I0.032O.0250.0200.016

0.0400.03 I0.0250.0190.015

0.0 12 0.0120.0 1 O 0.0100.008 0.0080.006 0.006

0.827 0.7740.490 0.4690.334 0.3’30.242 0.234O.lpIO 0.175

0.136 0.133O.lo4 0.1020.08 1 0.0790.063 0.06 10.049 0.048

0,039 0.0380.030 0.0300.024 0.0230.019 O.QlX .0.015 0.015

0.012 0.0120.009 0.0090.007 0.0070.006 0.006

0.7280.45 10.3120.2270.170

0.129o.cm0.0770.0600.047

0.0370.0290.0230.0180.014

0.01 I0.0090.0070.006

ing to Table I. tho sum of the two terms is 106~5‘~+o*1~‘= 10f’.7s. The other five terms

of eqn. (2) haveexponents more than 2 units smaller than thcexponent of this sum andcan thus be neglected.

Esuntple 2. What is the conditional constant of the zinc-DCTA complex utpH 10 in the presence of free ammonia, [NH,] =O. 1 M, and sodium ions, [Na] =0.2 = 1O-“.7o M‘? Calculate pZn,,, if rt 10s3 M zinc salt solution is titrated.

The answer requires knowledge of the over-all coefficients. The expressionsfor the individu~ll coeff 5ents arc given below : the values of the constants needed weretaken from the collection in ref. 2, except KNu,,= 104*40 and fir = KI1,_= 10’3.‘7 ofDCTA (H,L). which are from a recent paper by Carr and Swartzfagcr’, Kz,,t_ correct-ed by noting their new value of K,,,. is 1020.*4.

TX,(,,, = 1 + lo- IO 101_3.17+ lo-20 1()1‘).37 + 10-30 1022.97 +

+ lo-40 102S.48 = 103.17.

al_(p&,f = 1 f lo- O*‘O los.40 = 103~70.

~X nGufi,i =f + I()-” 1 102.27 + 10-2 104.“‘+ lo- 3 107.0’+ 10-J 1()9.06= 105.10

.

a,,,,,,, = 1 -t- 1o-4 104.j+ Io-‘2 1014.j+ lo-*6 10’s*5= 102v4.

No constants of basic or mixed ammonia ZnL complexes are given in theliterature.

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146 SHORT COMMUNICX-IIONS

ft follows that the over-ail a coefficient Eg+{f#,N;r)= 03.’ ’ + 103.‘” - 1, i.e., ac-cording to Table 1, ai.= 103.70+0.1 ’ = l()“-x1. The over-all coefficient azn(Nfg,.o)g)=10”. I”. The conditional constant log Kc:;:‘-’ = 20. I - 3.8 - 5.1 = I 1.2, and if a 1 - M

solution is titrated, pZn,, = 4 (3 + 1 1.2) -t 5.1= 12.2. If the transition point of the used

indicator, pZnrrzlnu, is known, the attainable accuracy expressed as per cent error canbe directly read from an error diagram (ref. 2, p_ 80).Norc~. If the equilibrium constants valid at ~~=O.I wcrc corrected to values

valid at the ionic strength of the solution (about 0.2-0.25). the pZn,,, value would be

about 0.1 unit lower.

It may also

tion ofexponcntiulbc useful :

lo”+ 10”.---_-- =lo”-- IO”

occur that other muthcmatical operations including the subtrac-terms huvc to be pcrformcd. The following formula may sometimes

, 0” + s--_.._ =10” .- f

lo”+’ (3)

In a papct’ from this laboratorya, the stability constants of themononuclear silver---TTHR complexes wcrc dctcrmined from pAg values. The vulue ofthe stability constant at pH 4.60 when pAg=2.72 (given in Table IX, Part II) wascorrected by means of the expression :

4 CL - CA61Correction factor = -B C,+ EAsl

10-2.30- 10-2.72 _ 1o_“‘_%5= I0_->mgjj.-2.72 -

cr-h=0.42 and, according to Tables I nd II, x=0.140 and y=O.208. Conscqucntly,we obtain IO-(O.14+0.21)= l()-().JS.

A chemist familiar with the USC of Tables I and II, ci\n easily and mentallymake such corrections in a few moments.

I G. SCWWARZENIIACI-I. D i e k o ~ ~ r ~ ) f l ~ so t r r er r i s c ~~ l ~ i / r c r t i o n . Ferdinand Enkc, Stuttgart. 1955: c/I Ci. SC’IIWAK-ZBNOACII AND H. FLASCI~KA, C ot r r p l ~ x on t et r i c T i r r t r r i o t w. nd Enyl. Ed., Mcthwn Co., London, 1969.

2 A. RINGIKIM. C b n r p f esc r t i o n b r A n c r i _ vr i c u l l w t r r i st r y . Wiley-Intcrscicncc, New York, 1963.3 L. HARJU AND A. Rwciimhl. ArmI. C m. Acm, 49 (t970) 205.4 A. RINCitlOhl A N D L. HARJU. Attcil. C iri?. Actit, 59 (1972) 33. 49.5 T. A. E~O~IWIAN AND A. C. MARTELI.. i m w ~ g , C l t c n r. , 4 1965) f 264,6 J . D. UAlw AN11 I>. G. SWAI~W.VAGI~I~. rltxl. C ’ / J C W 42 (1970) 1238.

(Received 12th October 1971)

Awl. Ci~ivr. Acta, 59 (1972)