On the Preparation of Buffer Solutions Bruce M. Thomson Department of Civil Engineering, The University of New Mexico. Albuquerque, NM 87131 Michael A. Kessick Physical Sciences Branch. Alberta Research Council, 11315 87th Avenue, Edmonton, Alberta, Canada T6G 2C2 When dealing with both biological and chemical aqueous systems in the laboratory, two very important variables are pH and ionic strength (I).These parameters are generally interdependent due to the presence of acid-base mixtures. A common method of controlling pH is through the use of buffers. Such solutions are often prepared with little regard to their effect on the ionic behavior of the solution which is commonly controlled by an inert swamping electrolyte. A method is presented here which allows calculation of the pH and ionic strength of a mixed solution of salts of an acid, or conversely calculation of the appropriate mixture of salts of an acid to produce a solution of desired pH and I. The calculations are quite useful and of didactic value in that they are relatively straightforward and can be performed easily on many popular programmable calculators. Basic Equations For an arbitrary, triprotic acid HBAand a mixture of its alkali metal salts (MH2A, M2HA, and M3A) a t equilihrium, several equations are needed to define the equilihrium conditions. First are the equations descrihing the stepwise dissociation of the acid and the dissociation of water
H z 0 = H+ + OH-
k,
= IHtllOH-I
(4)
Next are mass balances on the alkali metal and acid species and an electroneutrality expression.
[Mt] t [Ht]
=
[HzA-]
+ 2[HAZ-] + 3[A3-] + [OH-]
Figure 1 . Dependence of phosphoric acid species on pH as shown by a logarithmic concentration versus pH diagram.
sulting expressions can be readily solved either algebraically or graphically (2). Figure 1shows the species distribution for M solution of phosphoric acid using this approach. At a concentrations typical of huffer solutions, difficulties are introduced by the non-linearity of the activity coefficients, so approximations are generally made to solve for conditions a t equilibrium ( 3 , 4). However, these equations can be solved numerically yielding a method of preparing solutions for use in laboratory systems. The above equations are first simplified by defining the ionization fractions.
(7)
Note that the dissociation equations are written in terms of activities ([ill while the mass halances and electroneutrality condition are written in terms of concentrations ([i]).To relate concentration and activity, individual ion activity coefficients are calculated using the Davies approximation to the Dehye-Huckel theory ( 1 ).
log f ; = -Azi2
(- 4
1 + JI
0.31)
(9)
Here zi is the charge on the ion. The ionic strength is defined by: 1
1
2
2
I = - X[i]Zjz = I I M t ]
+ [H2A-] t 4[HAZ-]
+ 9[A3-] + [OH-])
fo is the activity coefficient of the uncharged species H3A, f l is the coefficient of H2A-, and so on. The hydrogen ion activity is used instead of its concentration as this is approximately what a pH electrode measures ( 5 ) .In a similar manner, the concentration of the alkali metal cation (generally Na+ or K+) can he expressed as a ratio of its concentration to the total concentration of the A species, which is designated here as the B ratio. B , !?%
(10)
At low concentrations (of the order of M or lower) the activity coefficients are nearly one and activities in the first four equations can he replaced by concentrations. The re-
CAT
(1.5)
Thus, it is possible to express the electronentrality condition in terms of R, CA.",/H+),the individual ion activity coefficients, and the dissociation constants. Volume 58
Number 9
September 1981
743
The individual ibn activity coefficients are related to one another and to the ionic strength as follows. log f i = -0.509
(-
Jr
1 + JI - 0.31)
(17)
log f n = 4 l W f l log f a = 9 l o g f ,
Note that since HxA is non-ionic, its activity coefficient, fo, is one. Thus, the expression for ionic strength is
Combining (16). (17), (18), (19), and (201, and recognizing that
Set all I,
= 1
Calculate eqn. (23)' Calculate I
results in the following expression Figure 2. Flow chatt of calculations for determining Band Ifor specified values of C, and pH.
where
The solution is thus defined hy the three eqns. (16), (17), and (23), which contain a total of five unknowns. Therefore, two of the following five parameters must be specified in order to characterize completely a buffer solution: I, f l , (H+J,A, CAT. Discussion
Calculations made with these equations, using dissociation constants from Sillen and Martell (6, 7) and comparing these results with standard buffer solutions resented hv Bates (5) -~~~~~~~ . . are summarized in the table. They were made by selecting CA, and (H+Icorresponding to the standard huffer, and avalue of I3 was calculated for comparison with the published values. An iterative numerical solution is required, a flow chart for which is presented in Figure 2. All cases converge rapidly, generally within six iterations, and excellent agreement is found between the calculated A and puhlishedB ratios. ~
~
~
Calculate a , eqns (12) (13) (14)
a Ceiculate CAT eqn. (22)
Comparison of Calculated and True B Ratios for Standard Buffers Buffer
Totai Concentration
Primary Standards 0.05 rn KHCaH404 0.05
0.025 0.025
m KH~POI rn Na2HP04
0.008695m KH2P04 0.03043 rn Na2HPOh
0.05
oH
Btrue
%ale
m
4.008 1.000 1.000
0.0534
m
6.865 1.50
1.497
0.0997
7.413 1.778 1.778
0.1000
0.039125m
Figure 3.Flow ch& of pH and I.
Secondary Standards
0.01 0.09
MHCl MKCl
0.1 0.1
M
2.07
0.90
0.8898 0.1000
MCHsCOOH 0.2 MCH3COO-Na+
M
4.64
0.50
0.4968 0.0994
0.01 0.01
MCH&OOH 0.02 M CH~COOO-NB+
M
4.70
0.50
0.4920 0.00966
0.025 0.025
MNaHCOl MNa2C03
M
10.00
1.50
1.5070 0.1006
0.10
0.05
m denotes molaliv ~ d e n o l e smolarihl
744
I
'calc
Journal o f Chemical Education
Csleulate B eqn. (16)
I
of calculations for determining B and CA, for specified values
To prepare a huffer solution of specified pH and I, &and B are calculated directly by first finding fl, solving for a l , then determining CA, and B (see Fig. 3). For example, if a phosphate buffer is desired at pH 7.00 with ionic strength of 0.01 M, and B would be 1.459. Hence, a huffer solution with these M NaH2P04and properties might he made from 2.821 X M Na2HP04.Alternatively, the solutions could 2.393 X he made from H3P04 and Na2HP04, Na3P04, or NaOH as long as the criteria for C Aand ~ B are met. Two points of interest are the huffer intensity, or slope of
Figure 4. Dependence of the Buffer Intensity on pH for two different concentrations of phosphate.
Figure 6. Dependenceof the lonic lntensityand lonic Strength on pH for a solution of 0.1 Mphosphate.
A similar calculation is carried out for the change in ionic strength with pH, which is called ionic intensity here.
~ to strength near neutral pH for different values of C A due departures from ideality (i.e., activity coefficients not equal to 1). and K,, are plotted versus pH for a 0.05 M phthalic acid solution in Figure 5. ~ y and , I are plotted versus pH for a phosphate system in Figure 6. Note that the calculations are only valid above the pH corresponding to a solution of pure acid. Values below this must he ohtained hy addition of a strong acid which is not accounted for by the present electroneutrality condition. Figure5 Dependence of the Buffer Intensity and lonic Intensityon pH for a 0.5 M~olution of phthalic acid.
the titration curve, and the change in ionic strength as pH varies. The buffer intensity is defined as:
Limitations
When using calculations of this nature, one must he aware of the limitations of their applicability. 11 In d w i v ~ n d~ h ~ s e r q u : i t ~ m ; , ,m t pk~e d~..t . ~ > t ~ . >, ni t h v GII. a..wnctl, h m c r .prr>risue h :tr h l . 1 - .md \I.\--.>re nut d
