A General Approach for Calculating Polyprotic Acid
Speciation and Buffer Capacity D. Whitney King Colby College, Watewille, ME 04901 Dana R. Kester Graduate School of Oceanography. University of Rhode Island, Narragansett, RI 02882 The speciation of polyprotic acids is a common problem faced by many chemists. Applications requiring these types of calc"1ations range frompH control in simple laboratory experiments to determination of alkalinity and CaC03 solubilitv" in ~-~the deen oceans. This article describes a set of general equations for calculating the speciation of polyprotic acids. The eauations can be incornorated into a simple comto calculate the d&ee of protolysis of each puter acid species (aj), the concentration of each acid species, and the degree of dissociation (7) of the acid as a function of pH. This formulation also provides a general equation for the buffer capacity of polyprotic acids. The polyprotc acid H,A can undergo n dissociations to form n 1species ~
+
H,A, H,-,A1-, H,_,A",
.. . H,,A'-
(1)
where the order of each dissociation is denoted by the index j, which ranges from 0 ton. The dissociation constants of the acid are defined forj=O
Kj = 1
(2)
The brackets in eq 3 denote free species concentration. Therefore, Kj is a stoichiometric dissociation constant. Alternatively, species activity and thermodynamic dissociation constants could be used. A dummy variable, u, can be defined for the general polyprotic acid. I" J
Using the dummy variable, the concentration of each acid speciescan be related to [H+]and the dissociation constants.
0.0
0.5 1.0 1.5 Degree of iss so cia ti on
Figure 1. solutlon pH as a hrnctlon of me dsgree of dlssociatlan of carbonic acid in seawater. The solid line Is for a solution at 25 ' C and atmospheric presswe. The dashed line is the same solution compressed to 1000 aim pressure. The avows indicate the values of the first and second dkrochtion wnstants.
where the degree of protolysis for species j is defined as the ratio of the concentration of the species to the total concentration. In an analogous manner the concentration ratio of any two species or the ratio of any species to the total charge of the system can be calculated. A particularly useful parameter is the degree of dissociation, q, of the acid which is equal to the ratio of TC to TM. q=
The total concentration of the acid, TM, is equal to the sum of each weak acid species.
-
..
The total charge carried by the weak acid, TC, is equal to the sum of each acid species multiplied by its charge.
The dummy variable incorporated into eqs 5-7 can be eliminated by taking appropriate ratios and canceling u. For example, the degree of protolysis for the jth species, orj, can be calculated from the ratio of [H,-jAj-]lu to TMIu 932
Journal of Chemical Education
2.0
TCITM
(9)
A plot of q as a function of pH provides the dissociation curve of the ~olvvroticacid. Under conditions where the polyprotic acib is the only acid in the system (H+ and OHcontribute less than 0.1% of the charge), q becomes a conservative parameter and will be independent of solution temperature and pressure. This ~rovidesa useful tool for determining the effect of tempeiature or pressure on the acid dissociation constants. As an example, q for carbonic acid can be calculated using eqs 6 , 7 , and 9.
Since q is conservative, any pressure or temperature induced changes in the dissociation constants will result in a measurable ihift in solution pH. Figure 1shows the pH change as a function of q associated with compressing an artificial sea-
water solution containing carbonic acid as the only weak acid. Culberson et al.' performed this experiment and by measuring the pH change ([H+] change) as a function of pressure at two or more q values were able to determine the pressure dependence of K1 and K2 for carbonic acid in seawater. A final parameter that can be calculated using the above equations is the buffer capacity, 8, of the system. The buffer capacity is defined as the amount of acid, C., required to change the pH of the system by a given amount.
This relationship can also he expressed as function of the ,~ is equal to the BNC, base neutralizing c a p a ~ i t y which equivalent sum of all acids that can be titrated to an equivalence point with a strong base. Expanding the derivative of BNC for a general polyprotic acid, the buffer capacity can be rewritten in terms of each acid species. Flgure 2. Buffer capacity and degree of promlysis as a function of pH fw carbonic add. The solid lines are fw eo,CY,,and orn. The dashed line Is for @.
The buffer capacity of the hydrogen ion, -d[H+]/dpH, and the hydroxide ion, d[OH-]IdpH, is a function of the proton concentration.
OH+ = -d[Ht]/dpH
= 2.303[H+]
(13)
Taking the derivative of eq 8 with respect to pH provides a general equation for da,ldpH.
Substituting eqs 13-15 into eq 12 provides the general equation for the buffer capacity of a polyprotic acid
dissociation constants, total concentration of the weak acid, and solution pH. The program will compute [H,-,AA-j]Iu for each acid species and store the results in a one-dimensional array. TMIu and TCIU can be calculated by appropriate summation of the array components. aj can be computed from the ratio of each array component to TMIu. Finally, 8 can be determined from the nested summation of the a j terms and the total concentration of the weak acid. Acknowledgment The authors thank Jie Lin for assistance with Chinese translations. Appendix. Example Calculatlons for Carbonic Acid Acid species: H&O, HCO;, CO:n=2
;=o
j=,
which can be condensed to eq 173. @ = pHt
+ pOH-+ 2.303 T
n j-l
M
1~0' - i)'ojai
(17)
=,.i=o
As an example, the Appendix codtams all of the equations described here worked out for adiprotic acid (carbonicacid). Figure 2 is a plot of uo, ul, az, and 8 as a function of pH for carbonic acid. Notice that 8 has localized maxima when uo = a', and a1 = as. Of course, these are the points where the pH is equal to pK, (pK, = 6.00 and pK2 = 9.11 for carbonic acid inseawater at 25 OC). While the equations have been derived for acid dissociation reactions, this approach is also valid for any homogeneous ligand dissociation process. The above equations can be conveniently incorporated into a computer program. Program inputs include the acid
-
' Culberson, C.: Kester, D. R.; Pytkowlcz, R.M. Science 1967, 157,
59-61. Stumm. W.:
Moraan. - . J. J. Aouatic Chemistry. 2nd ed.;Wiley: New
YO^ 19815p iss.
Zaofan, 2.; Pihong, 2. HWUB Tongbao 1984, 11.43-45,
Volume 67
Number 11 November 1990
933
