computer mrief. 111
edited by Arizona State University. Tempe. JAMESA P. z 85281 BlRK
BUFCALC: A Program for the Calculation of Buffers of Specified pH, ionic Strength, and Buffer Capacity William J. Lambert Drug Delivery RBD-Sterile
Products, The Upjohn Company. Kalamazoo. MI 49001
Many solution state properties are influenced by pH, ionic strenerh ( I ) . or buffer concentration. Examples include reaction rate and mechanism, solubility, and Many of these properties are of crucial importance in applied chemistry, creating the need for stringent buffer control. Unfortunately, the preparation of a buffer of specified composition is difficult due to the laborious calculations involved. For this reason, various authors have either listed aqueous buffer recipes (1-5)or presented programs for the calculation of buffer recipes (6). Investigators are limited using listings of recipes, since the combinations of buffers, pH, ionic strength, and buffer concentration are finite. The algorithm in the program developed by Perrin and Dempsey (6) uses assumptions that limit the pH range and buffer concentration and does not allow for overlapping ionization processes. A Turbo Pascal (Borland International) program named BUFCALC has been written for the IBM PC for the calculation of buffer "recipes" given the appropriate thermodynamic dissociation constant(s).' The ionic strength corrections included are valid up to an ionic strength of 0.5 M. The program has several advantages over other recipe programs in the literature. First, no simplifying assumptions that would limit the pH range or buffer concentration were used. Second, the program works with mono- through triprotic weak electrolytes, including cases with overlapping ionization processes. Finally, a data file of buffer capacity as a function of pH can be created. The basic concepts of the algorithms used in BUFCALC are presented in this report, providing a fundamental understanding of weak electrolyte equilibria. Theoretical
The thermodynamic dissociation constants used in BUFCALC must be corrected for ionic strength in order to calculate species concentrations and buffer capacity. Activity coefficients ( y j ) are calculated using either the Debye-Huckel equation,
or Davies equation,
'For a copy of me compiled version of BUFCALC, please send a postage-wid self-addressed envelope and an IBM PC f m t t e d 3.5- or 5.25.in. diskette to the author. 150
Journal of Chemical Education
s interest, A and B where z is the ionic charee of the s ~ e c i e of are functions of temperature and the dielectric constant of the solvent, a is the effective size of the hydrated ion, and b is an empirical constant. The user may specify the values of A, a, B, and b. For ionic strengths of less than 0.5 M, the activity coefficients in an aqueous medium a t room temperature can be calculated by Davies equation with A equal to 0.509 and b eoual to 0.2 (7). to follow. , ,, as was done in the exam~les Buffer capacity (O), sometimes referred to as buffer value or buffer unit (6,8), is a quantitative measure of the buffering ability of the buffer solution. Van Slyke defined O as dB1 ~ D Hwhere . B is the eram eauivalent Der liter of strong base abded to a buffer solution Van derivation &'3 for a monooroticacid titrated with NaOH is unnecessaril~complicated, so a different approach follows. consider the charge balance equation for a monoprotic acid with sodium as the counter ion: -
~
~
~
A ~
~
~
~
~
~
(6. like's
[Ht] + [Nail = [OH-] + [A-]
(3)
First, the derivative of the charge balance equation is taken with respect to [HI giving where ( 1 signifies activity. Since Van Slyke found the buffer capacity for a monoprotic acid to be
8 = 2.303(K.A,,,lHI/(K, +(HI)' + [HI + [OH])
(6)
As one would expect intuitively, eq 6 holds for a monoprotic base as well. Van Slyke also derived buffer rapacity equations for polyorotic acids ( 8 ) .He used the assumption that the pK,'s did hot overlap, which is equivalent to hiving two or more monoprotic electrolytes in solution a t the same concentration. This is a trivial case since buffer capacity is simply the sum of 6 for each buffer. In reality, overlapping ionization processes are frequently seen in buffer systems (e.g., citric acid). Buffer capacity has been solved for di- and triprotic compounds by Perrin and Dempsey (6); however, their equation for dibasic acids is apparently missing a term, and the equation for tribasic acids is missing a square superscript. In addition, their equations are solved with the unusual convention of pK,, > pK,z. For a diprotic weak electrolyte, the appropriate charge balance equations are solved in a manner similar to the monoprotic acid, giving
BUFCALC lnput lor Monoprotlc Acid Example Resmnse
Ouew
PKw Equation A
For a triprotic weak electrolyte
0.509# 0.2008 1 (Manoprotic) 4.756 1 (Acid) 0' -1' la(e.g.,NaOH, KOH)
b
where
Choice pKal Type
-
IHA LA
r (Titrant)
The major assumptions used in deriving these equations are that the strow base (or acid) is 100% dissociated and the assumptions used in Debye-Huckel theory. A schematicof the algorithm used for BLTCALC isshtwn in Figure 1. The user must specify the thermodynamic dissociation constant(s), the ionic charge of the various buffer and titrant snecies., and.. finallv. the desired uH. ionic strendh. and total biffer concentration. The apparent dissociatGn constants are determined usin8 I and either Davies eouation or the Debye-Huckel equation. 17 the desired 6is input, rather than the total buffer concentration, Am must be calculated iteratively using eqs 6, 7, or 8. This root-solving step uses the bisection method (Turbo Pascal Numerical Methods Toolbox), stopping when the relative change in @ is less than 1 X lo@. From the KaAp(s),the various buffer species concentrations may be calculated. Next the charge balance equation is rearranged in order to solve for the concentration of titrant (strong acid or base) to adjust pH. If At,, was initially input, 6is solved using eqs. 6,7, or 8. Finally, the amount of NaCl needed to adjust ionic strength is reported. The user is given the option of creating a data file of @ as a function of pH. This calculation uses the values of AM,and I
I
amount
.
01
-~
Figure I.Algwithrn for BUFCALC
---
Monoprotic Acid
17
... .
NaCl to adjust
- --
~
~etaultvalue.
Examples
lnput desired pH and I
(aclua lte
a
5.0 0.2 1 (Total Cam.)$ 0.02648
from above. The program finally loops back to the point where the user may input new values of pH, I, and buffer concentration or capacity.
I
lnput KaT(s)and Zi(s)
N
Other ions pH I Specify ~ d aconc. l
- -
Miller and Golder (2) report that a pH 5.0 buffer of 0.2 ionic strength can be made by adding 72 mL of a 5 M NaCl solution, 20 rnL of a 2 M sodium acetate solution, and 3.7 mL of a 3.5 M acetic acid solution together to make 2 L of buffer. This final solution represents 0.02 M sodium acetate, 0.00648 M acetic acid, and 0.18 M NaCl (10.5 g m L of NaC1). How does this compare with the BUFCALC calculation (pK, for acetic acid is 4.756)? Note that the total concentration of acid is 0.02648 M. There are two common ways of making buffers of the acid HA. First, the buffer can be made using acetic acid with pH adjustment using strong base (e.g., NaOH). The program queries and responses for this example are listed in the table. The concentration of titrant (e.g., titrant counterion charge of 1for NaOH) necessary to bring the pH to 5 is calculated. When this is done, the screen output shown in Figure 2 is constructed. The program reports that a titrant concentration of 0.01869 M is necessary to adjust pH, off approximately 6.5% from the value Miller and Golder report for sodium (from the sodium salt of the acid). It should be noted that these authors do not report the pK, used, or the ionic strength corrections used, limiting the usefulness of their recipe tables. BUFCALC also calculates that 10.6 g of NaCl is necessary for ionic strength adjustment, in good agreement with the literature value of 10.5 g. The program also reports a buffer capacity of 0.01268. As a check, a buffer capacity of 0.01268 could be entered initially (rather than concentration). When this done, BUFCALC calVolume 67
Number 2
February 1990
151
SPECIES CONCENTRATION I I ) AT pH IHZAl-
IHA1'
4.900
PKW
= 14.000
LlllE-03 5794E-03
I*,= ,.4iZS-OJ TOTAL CONCENTRATION IN, oa BWESR CAPACITY: 5 8 Z I B - 0 3
CONCENTRATION In,
OF
auaree:
TirRInT
~ ~ Z O E - O ~
lroarc CHIROB OF
IONIC STRENGTH DESIRED: FROM BVBBBR:
1,
1
b
:0 . 5 0 9 i
ozoo
To m u m nu: s s o 4 s - o a
1.0008-02
1.ODJB-02
.........
CORRECTION WEC655ARY: -900E-05
CONCENTRATION OP BOBBER TOO HIOH! ! ! ! PLOT BUZPER CAPACITY VERSUS pH (BUFFER CONCENTRATION ill, i 8 . 5 2 0 6 - 0 5 ) ? N
F i g m 3. BUFCALC screen output for a succinic acid butter.
. GrnC I
0 000E .-' ta
-
'io:
PH Figure 5. Butter capacity f a a phydmxybenzoic acid buffer.
Figure 4. Buffer capacity fw a succinic acid buffer
culates that the total buffer concentration necessary is 0.02648 ~ a . expected. ? The second method of making the buffer involves utilizing the free acid (HA) and salt form (e.g., NaA) of the buffer, as Miller and Golder suggest. T o mimic this using BUFCALC, all entries would he the &e as in the table except for the "other ions"query. After responding yes, enter an ion charge of 1and a concentration of 0.02 M (both for the Na+ from sodium acetate). Respond no to the "other ions" query the second time. The outout will be identical to Figure 2 except that an additional lind will report the ionic strength contrihtion for other ions (0.00, and the concentration necessary to adjust pH will he reduced representing the fact that sodium acetate was utilized rather than sodium hydroxide. In this case, the latter value is slightly negative, representing the 6.5% difference between the calculated value for Na+ and the value from Miller and Golder. Dipmtic Acid with Overlapping Ionization
Calculate the concentration (M) of NaOH and the amount of NaCl (g/L) necessary to make a pH 4.9,0.01 M ionic strength huffer using 8.323-3 M succinic acid (pK,'s equal 4.229 and 5.649). Also.. . d o t the buffer cavacitv. . . Perrin (51 ha. suggested that this roncenrration of succinic acid will vield a 0.01 M ionic strenrth buffer at pH 4.9. uithouf the addition of any NaC1. Upon entering the appropriate parameters, BUFCALC suggests adding enough NaOH tomake a 8.6043-3 M solution (Fig. 3), in close agreement with Perrin (8.583-3 M). The ionic strength calculated by BUFCALC is within 0.3% of the 0.01 M value suggested by Perrin. No NaCl ~
~
152
Journal of Chemical Education
Figwe 6. Butter capacity f a a cibic acid buffer,
has to be added. A plot of buffer capacity is shown in Figure 4. At the extremes of the pH range plotted, the huffer capacity is due primarily to water. Two maxima are seen at pH 4.18 and 5.51, which represent the apparent pK.'s at an ionic strength of 0.01. Diprotic Acid withwt Overlapping ionization
Plot huffer capacity versus pH for 0.1 M p-hydroxybenzoic acid (pK.'s equal 4.60 and 9.31) at an ionic strength of 1.0 M. The plot of buffer capacity for p-hydroxybenzoic acid is shown in Figure 5. In contrast to succinic acid, there is virtually no overlap of the ionization processes for p-hydroxybenzoic acid. Triprotic Acid
Plot buffer capacity versus pH for 0.1 M citric acid (pKis equal 3.14,4.77, and 6.39) at an ionic strength of 1.0 M. The buffer capacity for citric acid is shown in Figure 6, demonstrating the use of BUFCALC for a triprotic acid with overlapping ionization processes.
pUlc
8.060
pEllbPP=
SPECIES CONCENIRLIION [H11= 5 9 1 8 Z - 0 2 [A].
DAYIES BPUATION
8.172
( ~ 1AT DH
8.000
4.0228-02
TOTAL CONCBNTRATION in, or BUFFER: 1.0006-01 BOrssR CAPACITY: 5.5386-02
C O N C B N T R ~ I O N.1i
OF TITRANT
IONIC STRENGTH DSBIRED: FROM BUFFER:
mx = A
=
I~.OOD
0 50'1
b = ozoo
(IONIC CHARGE OF -11 TO ADJUST
DH-
5.sm~-02
IOOOP-or 591BE-02
.. . . .... .
CORRECTION NECESSARY: I O 2 Z E - 0 2
C W L OW
buffer, made with 0.1 moVL of tris(hydroxymethyl)aminomethane (Tris), a weak base with a pK. of 8.06. Note that by responding that the buffer is a base, the proper ionic charges are automatically supplied for all species. The program will report a titrant concentration of 5.9783-2 M and 2.35 g/L (4,023-2 M) NaCl (Fig. 7). Note that the same buffer could be made by adding enough of the HCI salt of the base to make 5.983-2 M, 4.023-2 M of the free base, and 2.35 g/L of NaCI.
N a C l HEEDED: 2.350581
.
PLOT BUFFER CIPACITY VERSUS pH (BUFFER CONCENTRATION ( M i 1.000E-011" W
Figure 7 . BUFCALC screen output for a Tris buffer.
Momprotic Base
amount of the concentration (M) of HCI and NaCl (g/L) necessary to make a pH 8.0, 0.1 M ionic strength
Lnerature Clted I. 2. 3. I. 5. 6.
Rower,V. E.: Batos.RG. J . R s s . N d . B u r . Stand. (U.S.1 1355.55.197-200. Mi1ler.G. L.:Guidor,R. H.A?ch.Biochem. 1950,29,42C423. E1ving.P. J.:Markowits.J. M.; R o ~ n t h dIAnai. , Chsm. 1956.28.11791180. Rates, R. G. Ann. N.Y. Acad. S o . 1961,92,S41-356. Perrin.D.D.Aud. J.Chem. 1963,16,572-578. Perrin, D. D.: Dempsey. B. Bu//em /or pH and Melo1 Ion Confml; Hslstsnd: New York, 1974; chapter 5. 7. Butler. J. N. lmk EpuilibRum: A Mafhamoucol Appmoch: Addipon-Wesley: Reading, MA, 196%Chapter 3. 8. v a ~ s l y k D.D. ~ , J ~ i o chsm. i 19zz.6~~2~70.
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Number 2
February 1990
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