426 PRIDEAUX AND WARD : CALCULATIONS ON THE

LI.-Calculations on the Xeutralisation of &fixtures of Acids, and a Universal Buger Mixture.

By EDMUND BRYDGES RUDHALL PRIDEAUX and ALFRED THOMAS WARD.

THE buffer solutions at present in use are each applicable only over a limited range of hydrogen-ion concentration, as they are not governed by a sufficient number of acid dissociation constants, which also bear too high a ratio to one another, so that the stronger acid is completely neutralised, or nearly so, before the neutralisation of the weaker acid commences. If, however, a series of acids be chosen, differing by relatively short stages in their dissociation constants, the effects of the successive acids overlap during their neutralisation, and a buffer solution of much wider applicability is produced. Such a buffer solution containing a mixture of phos- phoric, acetic, and boric acids was proposed and calibrated by one of us (Proc. Roy. Soc., 1916, [A] , 92, 463). In that case, the neutral- isation curve may be regarded as determined by five monobasic acids. In the present paper, we describe a similar mixture which covers the range between p E = 2 and pH = 12, and possesses some advantage over the earlier one. Phosphoric acid has been retained, and the other acids have been chosen so that their constants are the geometric means between the first and second, and the second and third constants of phosphoric acid respectively.

Volatile, oxidisable, and reducible acids, and those which are unduly expensive being excluded, the most suitable are phenyl- acetic and boric acids. The mixture is made up so that the final solution is 0-02N with respect to each hydrion and the whole is 0.1 N .

The Constants of the Acids.-Starting at the acid end of the curve, the first hydrion of phosphoric acid has a dissociation con- stant K = 1-1 x 10-2 corresponding to a hydrogen-ion concen- tration of 1-03 x When this acid is half neutral- ised, we have a mixture of 0-O1N-acid of K = 1.1 x together with a 0-O1N-solution of its salt. If the ionisation of the acid be

or p H = 1-99.

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NEUTRALISATION O F MIXTURES OF ACIDS, ETC. 427

x, and the salt be considered as completely ionised, the relationship will hold : x(1 + x)/(l - x)V = K , from which the hydrogen-ion concentration x / V = 4.3 x A slight additional acidity amounting to 2 x 10-4 will accrue from the phenylacetic acid and the total concentration of hydrogen-ion will thus be 4.5 x or

The constants of phosphoric and boric acids have been critically examined in other papers, and i t has been shown that they vary to some extent with degree of neutralisation and total concentration.

PH 1 2-35.

FIG. 1. Phosphoric Phenylacetic Phosphoric Boric Phosphoric

1st stage 2nd stage 3rd stage

Per cent. neutralisation.

Using the fact that each constant acts most powerfully at the degree of neutralisation at which h = I<, and taking into account the concentrations present in the mixture, we have selected the values given in the table below.

Methods of Calculation.-Points intermediate between those at which h = K , and h = l / K I K , are calculated b y the general equations of dibasic acids (Proc. Roy. SOC., 1915, [A] , 91, 537).

. . . (1) ’ 1 + 2KJh + Kw/h x [NaHA]

1 + K,/h + h / K , R =

or

The third terms in the numerators are required only at low total concentrations and high alkalinities.

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428 PRIDEAUX ABD WARD: CALCULATIONS ON THE

Considering each pair of successive constants, either (1) or (2) may be used in the immediate neighbourhood of 73 = dK,K,, whilst at lower values of p , (1) alone, a t higher values (2) alone, gives rational results.

A specimen calculation is as follows :- At h = 1 x 10-4, by means of (2), R = 1-34 equivalents of alkali

to 1 mol. of the dibasic acid (KK,). The degree of neutralisation of this acid is therefore 0.67 and the percentage neutralisation of the whole is 40 x 0.67 = 26.8. The same point should also be calculable from the acids (K,K,) if these have been correctly chosen to give a rectilinear graph. The acid (K,K,) is 40 x 0-176 = 7-04 per cent. neutralised, therefore the percentage neutralisation of the whole is 27.04. When the value of pH has fallen to about 9.5, the hydroxyl-ion concentration begins to be comparable with the total salt concentration, and special treatment is required. Thus at 12 = 1 x 10-l1, p H = 11, we see from the general trend of the graph that lOOx is about 85. Therefore 85-60 equivalents of alkali have been added to acid (K3K4), and R = 1.25. If x = total Na,P04 and 1 - x = NaH,BO, (without hydrolysis), then x = (B - 1)0.25 and C = 0.25 x 0.02 - [OH] = 0.004. From equation (1) the corrected value of R is 1.27 and lOOx = 85.4. I n the case of trisodium phosphate a t C = 0.02 the ordinary hydrolysis equation leads to p H = 11.91, whilst the more exact equation for high degrees of hydrolysis, [OHI2KA = CK, - [OHIK,, leads t o p , = 11-80.

Equation (1) gives R = 0-353.

Acid. Phosphoric. Phenylacetic. Phosphoric. Boric. Phosphoric. K Kl K2 k'3 K4

1.1 x 10-2 5.4 x 10-5 1-41 x 10-7 6 x 10-10 3 x 10-1' P k = 1.96 4-2 7 6.85 9.22 11.52

Values of pH during neutralisation, calculated from the constants.

PR. 1.99 2-35 3.1 1 4.00 4-27 5.0 5.56 5.0 6.5 6.85 7.0 7.5

% neutral- isation.

0 10 20 26.9 30 37.1 40 42.1 46.0 50 51.8 56.7

Nature of formula. Pa.

Free acid K 8.0 Acid K and its anion dzz, 8-5

8-03

I<, K,: and (1) or (2)

K1, K 2 , and (1)

9.0 9.22 9.5

10.0

Kl

% neutral- Nature of isation. formula.

59.8 K, , K,, and (1) 60 dK,Ki 63.0 K,, K,, and (2) 67.4 K,, K,, and (2) 70 K, 73-41 K,, K,, and (1) 7 7 - 8 1 full equation

85.4 Special equation

100 SpeciaI equation

so d m 90 K ,

Experimental Calibration.-The boric acid had been recrystallised

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NEUTRALISATION OF MIXTURES Ob ACIDS, ETC. 429

according to the procedure of Sorensen, the other ingredients were the purest A.R. chemicals supplied by the British Drug Houses, and each was checked separately by titration against alkali free from carbonate. The same alkali, suitably diluted, was used in preparing the solutions. [Note.-The acid mixture is supplied as a powder by the British Drug Houses.] The measurements were made with a hydrogen electrode and potentiometer which had been used many times for similar purposes. The other half-cells were hydrogen electrodes in 0-O1N-hydrochloric acid ( p , = 2.022) or in standard acetate (pn = 4-63), or saturated potassium chloride calomel electrodes.

The checks of these electrodes against one another agreed satis- factorily with the accepted values. Temperatures varied from 10" t o 16", and the value of each pH was calculated separately by the usual formula, which corrects it to 18".

The experimental curve consists of three parts and these together &th the minor undulations in the long middle part are closely reproduced by the calculated curve.

Use as Buffer Hixture.-The mixture supplies, as was intended, a buffer which gives all p E values between 2.5 and 11.5 by means of two solutions only, namely, the mixture itself and alkali (strong acid) of the same normality. We have calculated an interpolation formula by the method of least squares, which, of course, treats the deviations as experimental errors, whereas they are characteristic of the constants.

The rectilinear formula p H = 0.773 + 0-1 185( 1OOx) is convenient and sufficiently accurate for many purposes. It holds between lOOx = 15 and 90. The colours with the B.D.H. universal mixed indicator have also been noted, with the results which are included in the following table.

PE by Pa by 01 10 p~ linear % p a linear

0 1-99 - 55 - 7.28 sap green 5 2.13 - 60 7.91 7.88

10 - - 66 8.62 8.48 green 15 2.G5 2.65 '10 9-11 9-07 greenish- 20 3-10 3.14 blue 25 3.73 3-74 75 - 9.66 30 4.21 4.33 red SO 10-21 10.25 violet 35 4.80 4.91 yellowish- 85 - 10.85

red 90 11-41 11.45 reddish- 40 6-48 5-51 orange- violet

50 0.84 6.70 yellowish-

UNIVERSITY COLLEGE, NOTTINGHAM.

neutraln. expt. formula. Colour. neutraln. expt. formula. Colour.

yellow 95 - - 45 6.30 6.105 100 11.94 -

green

[Received, September 28th, 1923.1

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