The Many-Acid Problem KAROL J. MYSELS' Harvard University, Cambridge, Massachusetts HERE 1s a relation between the degree of neutralizabon . 'and the pH of a solution containing known concentrations - of several adds of known dissociation ~h~ problem is to find this relation and a method of calculatine the deeree of neutralization when the pH is given, or inversely the pH when the degree of neutralization is given. Michaelis2 points out the importance of this problem and gives a formula for the pH of a solution containing two acids, namely
T -
-
-
This result can be obtained by the following obvious substitutions
C&=p=y="b A
i
i
A
A
A
In other words the problem of calculating the degree of neutralization of a mixture of acids is reduced to the problem of calculating the degree of nentralization of I L) KdAa - L ) H+ = K ~ A each acid. This is a s i m ~ l~roblem e when the DH of the ZL solution is given. A, - L ) + K9(A2- L)]' K I K ~ A I Aa - L ) This theorem is general and independent of the naL 2L ture of the acids present. We can simplify the problem still further when all where A1 and Ae are the two acid concentrations, K L and Kz the corresponding dissociation constants, and L the acids involved are monobasic. We have then (if the acids undergo only ionization) : the amoutit of base added. When the general problem is approached in a similar way the resulting equations cannot be solved for H because their degree, with respect to this unknown, is equal to the number of acids present. It is nevertheless or solving for b; possible, as will be shown, to calculate easily the degree of neutralization of any such mixture, when the pH is given. Once this is done for several values conveni- combining (1) with (2) we obtain ently chosen, then the pH value corresponding to a given degree of neutralization can be obtained with any desired precision. Let us consider a solution containing a total concentration A (equivalents) of acid composed of concentra- which is the desired relation. tions A , of the different acids. That is In this formula we have A , A;, K; which are determined by the conditions of each experiment. The A = CAi fraction in the parenthesis can be calculated in adi vance, as a function of pK - pH or H/K. This has Let the concentration of alkali be called B. We can been done for pK - pH, which is in general more write as a first approximation useful, in the following table. B = Cei
+
+
-7
+
+
i
where a; represents the concentration of the ionized part of the acid whose total concentration is A;. Let us introduce the degree of neutralization b for the wholemixtureand biforeachindividual acid. Then we have by definition b = B/A a n d bi
=
ai
If we calculate the weighted average of the degrees of neutralization of all the different acids present, we see that it is equal to the degree of neutralization of the whole mixture:
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1 Present address: 6 2 1 S t o c k t o n Street. S a n Francisco, California. MICHnEL1s, ion concentration." by P E R I . ~ ~W IO i l l ,i a m s a n d Wilkins Co., Baltimore. Md.. 1926,
D. 46.
nwuesof -I- = 1 + IOsK-sH 1
**
*K0.00 -
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.00 0.65 n n ,
+ _HK
0.500 0.471 0.442 0.415 0.387 0.380 0.334 0.309 0.285 0.262 0.240 0.220 0.201 0.183
n inn
*K0.80 -
-
b for varying pK
0.85 0.90 0.95
0.137 0.124 0.112 0.101
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40
0.091 0.082 0.074 0.0~8 0,059 0.053 0.047 0.043 0.038
1 . -. --G
n n u
-
OK1.55
0,027 0.025 0022 0.020 0.017 0.010 0.014 0.012 0.011
1.00 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.25 2.50 2.75 2
- pH)
o.010 0008 0,003 0.002 n nn*
nn
0.75 0.151 1.50 0.031 Nole: T o obtain b values when pK p H is negative, subtract from 1 the b value cornspanding to the absolute value of PK PH, ~ h i e his PH - PK.
-
-
Thus, with the help of this table, the calculation of the degree of neutralization of a mixture of any number of monobasic acids necessitates only one division and one multiplication per acid present when the pH is known. If the degree of neutralization is known, the
pH value can then be easily calculated by any approximation or graphic method. The table gives also directly the degree of neutralization of a single monobasic acid for any pH.
