SPECIES PRESENT IN COMPLEX BUFFERS EDWARD L. KING University of Wisconsin, Madison, Wisconsin
THE
calculation of the concentrations of the species present in a solution containing three or more weak acids and their conjugate bases appears formidable a t first sight. This is true not only for the undergraduate student of analytical rhemistry or physical chemistry hut also for the graduate student who is involved in a research problem in ~5-hichbuffered solutions are used. The simple approach of assuming that the predominant equilibrium in the solution is Acid,
+ Rase, 2 Aeidz + Base,
is useful if only two weak acids and their conjugate bases are present.' While equations have been derived for the calculation of the concentration of hydrogen ion in a solution containing two weak acids and/or their conjugate bases,z the corresponding equations for systems involving three or more weak acids would undoubtedly be so complicated that their usefulness to those interested in complex buffer solutions is questionable. There is, however, a relatively simple approach by which the composition of even the most complicated buffer solutions can be calculated. I n this method, one carries out the hypothetical processes of first removing the acidic hydrogen from all of the acids in the solution and then allowing it t o react with these several bbases which are present. The hydrogen ion preferentially
'
EASTMAN, E. D., and G. K. ROLLBFSON, "Phy~icalChemistry; eon cent ill ~~~k co.,I ~ ~ N~~ . ,yark, ,947, pp. 367-8, DEFORD, D. D., Anal. Chim. Ada, 5,34&51,352-6 (1951).
unites with the strongest base. Successive reaction of the hydrogen ion with weaker and ~veakerbases occurs until the hydrogen ion is used up. The composition of this resulting solution is the composition which is sought. It is believed that this method is of value in clarifying the subject of acid-base equilibria to the student and that, in addition, it provides a rapid and straightforward method for solving of the type - problems .. being discussed. A Specific Problem. The method will first he discussed in terms of a s~ecific~roblem:what are the concentrations of the species present in a solution with the following stoichiometric composition: 0.050 M NaH2PO&,0.030 M NazHPOa, 0.080 M HOC1, and 0.040 M NaOAc? If all of the acidic hydrogen were removed from the acids in this solution, there would be 0.210 mol of hydrogen ion per liter of solution; in addition there would he 0.080 mol of 0.080 mo1 of OC1-, and 0.040 mol of OAc-. If we now carry out the hypothetical process of adding this amount of hydrogen ion to these bases OC1-, and OAc-), it is not difficult t o determine how much of the hydrogen ion is used up by each base, and if this is known correctly, the hydrogen ion concentration of the final solution is also known. The determination of the amount of hydrogen ion which is tied up by each base is accomplished h s the use of a grauh which will now be described. - ~any t given-hidrogen ion concentration, a definite fraction of any particular base is converted to its conjugate acid. This is shown in the figure, in which the
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JOURNAL OF CHEMICAL EDUCATION
value of (-log [H+]) for a solution in which q mols of hydrogen ion have combined with one mol of each particular base is plotted versus the value of 7. Thus, Curve 1 corresponds to the conversion of PO4-%into HPO4--; a t the right-hand side (q = O), there is no conversion of PO4C3 t o HP04--, while a t the left-hand side (q = I), the conversion (to HPOI--) is complete. Each curve corresponds to the conversion of a particular hase to its conjugate acid, and it is to be noted that for a base which can react with more than one mol of hydrogen ion per mol of base, there are the appropriate number of curves. Curve 5 corresponds to the conversion of HP04-- to H2P04-, and Curve 8 oorresponds to the conversion of HaPOI- to H3P04. The curves for monobasic acids do not cross the q = 0 and 7 = 1 axis; a t no realizable concentration of hydrogen ion is an acid which exists in only two forms, HX and X-, present exclusively in one of these forms. While it is also trne that polyhasic acids do not exist exclusively in one form a t any concentration of hydrogen ion, i t is nevertheless trne that the average number of hydrogen ions tied up per mol of polybasic acid does have inte-
"
valvn of u
.
Function of th. conc.ntltion of Hydropn Jon
The bases for which the curves are drawn and the acid dissociation quotients for the corresponding conjugate acids are: 1. PO> I [OH-] - [H+]j is valid unless (H+) is very small, and thus the equation (HC) =
x Cm
(2')
twtion of hydrogen ion in the region where 7 > 1, if the existence of A- - is ignored), and in addition, to how close an approximation in the region q < 1 may one assume that q = ?", where
(i. e., horn correct is the q value a t a particular coucentration of hydrogen ion in the region where q < 1 if the existeuce of H,A is completely ignored)? One may derive the equations
I
may be used. .In the graphical approach, one decides by inspection coupled with simple arithmetical calcu- and lations, upon the value of the hydrogen ion concentration at which the total hydrogen ion available is combined with the several bases. The Construction of the Graph. For a monobasic acid In the region in mhich these questions are of interest the value of 7, mhich is equal to the fraction of the base (q E I), it is generally true that hl > 1 B that is present as the acid HB, is given by the equa- and therefore these equations simplify to give: tion and The drawing of the graphs is rather simple if one uses a template with the form h / ( l h) versus log h with the values of h ranging from low3to lo3. For a polybasic acid, the form of the curves is slightly different and depends upon the relative value of the successive acid dissociation quotients. The curves are generally the same as for a mouobasic acid over a large range of q values. Thus, in the case of Curve 2 (the conversion of C03- -to HC03-), q is within 0.2 per cent of what it would be in the range q = 0.00 to = 0.945 if H2C03did not exist; in the case of Curve G (the couversion of HCO1- to H2COI), 7 is within 0.2 per cent of what i t would be in the range 7 = 0.055 to 7 = 1.00 if C03- - did not exist.5 The equations used in the calculations leading to this statement are of interest. For a dibasic acid HIA, the value of q is
+
Equations (7) or (7') and (8) or (8') enable one to calculate, rapidly, the extent of the error in 7 introduced by neglecting those forms of a weak acid mhich are not directly involved in the principal buffer equilibrium which exists a t the hydrogen ion concentration of interest. Powell's Method of Solving Complex Buffer Problems. The graphical method of solving complex buffer problems which has just been outlined involves the construction of graphs and therefore might be viewed as inconvenient, particularly if only one such problem were to be solved. If such is the case, the approvimation method devised by Powells is convenient. In this method one first makes a guess a t the value of [H+]; it is then possible to calculate the roncentrations of all of the species in the solution. A general equation for the concentration of the species H,BJ is
The question of interest is this: to how close an approximation in the region q > 1 may one assume that q = q', where The numerator of the fractiou is the ratio of the concentrations of HkB' and H,W, and the denominator is the ratio of the sum of the concentrations of all species (i. e., how correct is the n value a t a articular concen- containing the base B' to the concentration of H,BJ. q t i8 to be noted that some ambiguity exists in the definition This fraction is therefore the fraction of the base W of n for polgbasie acids. In this statement regarding carbonic which is present as H,Bi. One then calculates the net acid, the convention is that adopted in the fylre where 7 has only electrical charge on the solution from the concentrations the range 0.00 to 1.00. The two stages in the conversion of the base COI-- to t,he acid H2COJare considered sepsrately (i. e., 7 and charges on all of the dissolved species. Barring a goes from 0.00 to 1.00 as CO1-- is converted to HCO1- and also minor miracle, the first approximation of the hydrogen goes from0.00 to 1.00 as HC0.-is converted to HICO~). On the ion concentration will be incorrect and the calculated other hand, it is often more convenient to use the eonvention net elect&l charge will not be zero, This approximawhich allows n the values 0.00 tom, the maximum number of hydrogen ions which can be asaoeiated with the base BJ. VOWEIJ., R. E., private communication (1948). ~
~
~~~~
~
~
~~
~~~
~
~~
~
APRIL, 1954
tion serves as a guide for the second approximation, however, since a calculated positive charge on the solution indicates that the assumed value of [H+] is too great and a calculated negative charge on the solution indicates that the assumed value of .IH+1.is too small. (This follows from the fact that the concentrations of more negatively charged species are greater the lower the concentration of hydrogen ion.) One proceeds to make guesses of the concentration of hydrogen ion until the calculated charge on the solution is small compared t o the concentration of any buffer constituent in the solution which exerts control over the hydrogen ion concentration of the solution. Unless one's first guess a t the hydrogen ion concentration is very poor, four or five approximations are sufficient to yield the answer to the problem. A semiquantitative application of the approach outlined in this paper (i. e., looking upon the solution as being made ndof the appropriate amount of hydrogen iou and the various bases with their acidic hydrogen removed) should make possible a very good first guess at the value of [H+], thus reducing the number of successive tries required. Regarding Conventional Titration Curves. The curves given in the figure are the same as conventional titration curves in the hydrogen ion concentration range such that I [OH-] - [H+]I is small compared to the concentration of the acid or bme being titrated. The
187
conventional titration curve is a plot of (-log [H+])as the ordinate versus ( - (H+)/CJ), which is equal to
. . as the abscissa. ..
(-
7'
+
([OH-] - [H+l)
Ci
)
The construction of a titration curve for a polybasic acid (with m acidic hydrogens) in the hydrogen ion concentration ranges in which more than two forms of the acid are important, and for any acid, regardless of the number of acidic hydrogens, in the case where / [OH-] - [H+]I is not small compared to CJ, is greatly simplified if one calculates the value of vJ as a function of the hydrogen ion concentration rather than the reverse. This can be done with an equation which results from the combination of equations (1) and (9) (with K2 defined as equal t o 1) :
It is clear that the calculation of v1 from the hydrogen ion concentration is easier than the reverse calculation. Once vJ is known, the stoichiometric composition of the solution may be calculated using equation (2), for the solution's composition is defined by C, and (H+).
