James A. Goldman
Polytechnic Institute of Brooklyn Brooklyn, N e w York
Potentiometric Titration Curves for Simple Ion Combination Reactioas
A
simple ion combination reaction is one in which the slightly dissociated (either slightly ionized or slightly soluble) product of the chemical reaction is composed of two symmetrical charge type ions. The rigorously derived (from considerations of electronentrality and/or material balance) equation which theoretically characterizes a potentiometric titration curve obtained for a simple ion combination reaction is a quadratic with respect to the concentration of either ion involved in the reaction ( 1 , 2 ) . Obviously, an exact solution for the concentration of interest, C, may he obtained via the quadratic formula; subsequently, the negative logarithm of the concentration, pC, may then be obtained ( 2 ) . Alternatively, one may obtain pC directly, by recognizing that the logarithm of the solution to any quadratic equation may be expressed in terms of the inverse hyperbolic sine of a "titration function" which is dependent upon the fraction titrated, the initial concentration of the species to be titrated, and the dissociation constant characteristic of the reaction product which results from the ion combination. Account may be easily taken of the dilution which occurs during the titration. This form of the exact equation is not really new, although it does not appear to have ever been considered as providing a relatively easy method for the rigorous calculation of these titration curves, under any conditions. I t lms, however, been used as a basis for determining endpoints, by Cavanagh (3) and Fortuin (4). Investigation of this equation allows one to determine the region, in terms of the "fraction titrated" parameter, j, wherein the exact equation must be used. Outside of this region, the rigorously derived equation reduces to the more commonly used expressions. I t should he remembered that alternatively, a rigorous calculation of a titration curve can always be accomplished by considering the concentration of one of the ions, rather than the fraction titrated, to be the independent variable. Such a procedure is always applicable, and especially useful for titrations which result in the formation of unsymmetrical reaction products, as has been recently emphasized by Bishop (5). The equation which will be discussed in this article is applicable only to the formation of a symmetrical reaction product. The Exact Equation
Consider the simple ion combination reaction: M+X=MX
charges being omitted. MIX is the slightly dissociated product for which: K = [MI [XI (2) where the brackets represent molar concentrations of the indicated species, and K is a concentration (or formal) equilibrium constant, rather than a true thermodynamic one. However, under the usual conditions, this concentration constant remains approximately constant, a t least in the vicinity of the equivalence point, because of the approximate constancy of the ionic strength in this region. The smaller the amount of dilution that occurs, the more constant will be the ionic strength. The variation of ionic strength during the entire titration can always be virtually eliminated by use of a large concentration of inert electrolyte. The complete chemical equation for the titration may be written as: MB+CX=MX+CB where B and C are the accompanying anion and cation respectively, of M and X. Therefore, the condition of electroneutrality may be written as: [MI
+ [Cl
=
[XI
+ [Bl
(3)
If we consider that we are titrating Vo milliliters of a solution of CX of original molar concentration Co, then the molar concentrations of B and C, at any fraction titrated, f,are: [B]
=
C"VO V" + v f
[C] =
C0V" V' + v
where we assume the volumes to be additive, and where V is the volume, in milliliters, of the added titrant, at the corresponding fraction titrated, j. By definition: f = V/q (6) where q is the volume of titrant necessary to reach the equivalence point, at which: C0V" = C T ~ (7) where C , is the molar concentration of the titrant. Rewriting Vo/(V Vo) as 1/(1 V/VO) and recognizing that V/Vo = (q/Vo) (V/q), we then may solve equation (7) for q/Vo and write:
+
+
(1)
where M and X represent a cation and anion, respectively; the equal in magnitude, but opposite in sign,
if we define r as: 1. =
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CD/C~
40,Number 10,
October 1963
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519
where r is an auxiliary parameter; - f nmy be considered to be the primary parameter. Substitution of equation (8) and of [XI from equation (2) into equation (3), yields: C" K Caf [MI + ---- = - -(10) l+rf
[MI
+Ifrf
Two comments should be made concerning equation (10). First, it applies for all values off (including zero) only for acid-base neutralizations, in which case MX represents HOH; for precipitation titrations, equation (10) is valid only for a nonzero value of f because equation (2) does not apply until some precipitate is formed. Second, if we ueglect dilution (i.e., r = 0) we obtain the usual approximate equation for electroneutrality. Our definition of the auxiliary parameter, r, is thus logical because the greater the concentration of titrant relative to the concentration of the species being titrated, the smaller is the avlue of r. Rearrangement of equation (10) to: [Mi KII2
Approximate Forms of the Exact Equation
K1'% C y f - 1) [MI - K'12(l r f )
+
and recognition of the following identity :'
'4 -
KU1
[MI
=
=
- exp (-In
exp (~n
2 sinh
slightly asymmetrical, and therefore the endpoint and equivalence point will not coincide (6). Equations (11) and (12) are equivalent, but prior to the equivalence point it is more convenient to use the latter. From either equation (11) or (12), it is easily seen that pM = (+)pK a t f = 1, because sinh-' 0 = 0. It is also obvious (from either equation (10) or (12)) that the result of taking the dilution into account is to yield a t any particular value of f prior to equivalence, a smaller value of pRI than if dilution is neglected, while (from equation (10) or (11)) a t any value off beyond equivalence the value of pM is larger than it would be if dilution is neglected. I n other words, the calculated curve which neglects dilution intersects with the one which does not a t f = 1,as it must, because pM at the equivalence point is independent of dilution and dependent only upon the magnitude of the equilibrium constant.
9%)
( In g)
For values of F such that F2 >> 1, i.e., F2 much greater than unity: sinh-I F = In 2F, so that equation (12) may be written as:
for values off such that:
yields :
(
CYf ( - 1) '121+rf)
or:
Equation (13) is the usual approximate equation that applies before the equivalence point. If equation (11) is written as:
where sinh-1 represents the inverse hyperbolic sine. After conversion from natural logarithms we obtain?
then for GZ >> 1: sinh-' G (11) may be writtenas:
where, as usual, pM = -log [MI and pK = -log K. Making use of the identity: -sinh-'x = sinh-'-x, we may also write equation (11) as:
for values off such that:
2 sinh In ' " ' ) = K
= In 2 G, so that
equation
Equation (15) is the familiar expression for the region beyond the equivalence point. Therefore the exact equation (either equation (11) or (12)) must be used in the region:
or:
where the titration function, F, is equal to Go ( 1
- f)/2KVl
+rf)
It is important to emphasize that pM is symmetrical with respect to F, but P i s not ~ymmet~rical with respect to f unless dilution is neglected (v = 0); thus, in general, pM is not symmetrical with respect to f. The significance of this is that the theoretically calculated titration curve, which takes account of dilution, is I
1 Exp z = ex, ainh I = - [exp (z)- exp ( - % ) I . 2
For the reverse titration, Lo., titration of MB by C X , we obtain: 1 P M = ~ ~ K - - sinh-1 C"(1 - f ) 2.303 3KL'l(l ~ f )
+
520 / Journal of Chemical Education
It is seen that the effect of a nonzero value of r is to slightly extend the region within which we must use the exact equation; clso, the portion of the region subsequent to equivalence is some~rhatlarger in magnitude than that prior to equivalence, unless r = 0. The magnitude of the region defined by equation (17) depends on the value of the equilibrium constant K, the initial concentration Co,and the dilution parameter r. Indeed, it is the ratio K"'/Co, which is of interest rather than the absolute value of either term comprising the r a t i o . V h e region within which we can use the customary approximate expressions is increased when
I1
Note that KIi1 is the concentration of X at f = 1 while Co is its initial concentration, at f = 0 . J
the initial concentration is increased, the equilibrium constant is decreased, and, or, the dilution parameter is decreased. Error in Using the Approximate Instead of the Exact Equation
Of considerable interest is the error in p M , A p M , which is incurred by the use of the approximate rather than the exact equations. Prior to equivalence, the value of pM calculated from equation (12) will be greater than that obtained from equation (13); subsequent to f = 1,the rigorously calculated value of pM (from equation (11)) will be smaller than that obtained from the approximate formula (equation (15)).4 At f = 1, neither of the approximateequationsapplies but pM = (1/2) pK; thus, a t the equivalence point, the theoretical curve calculated from the approximat,e equations intersects with that calculated from the exact equations. Let us consider the region prior to equivalence. Defining ApM so as to be positive, we may subtract equation (13) from equation (12), to obtain:
+ r f)/C0(1 - f ) = K'I'/2F
so by substitution of K1/*/2F into the argument of the natural logarithm, we obtain: ApM
=
This can be easily seen because 1 + 42% - - ...
consideration of equation (10) leads to the same conclusion.
1
Values of
Minimum Values of Fa For Selected Values of
APM
from which it is easy to calculate the error in pM corresponding to any selected value of F. More enlightening is the investigation of the minimum value, a, that F must possess so that ApM does not exceed some selected value, b. As equation (18) cannot be solved explicitly for F one must substitute various values for F until one attains the desired value of A p M . An extensive table of inverse hyperbolic sines is useful for this purpose (7). The results of these calculations are presented in Table 1 for some representative values of AplLI.
Table 2.
where G = Co (j- 1)/2K"a (1 r f). Use of the values of a found in Table 1, for G, will result in the same values of b listed there except prefixed by a negative sign. Table 1.
1 - [sinh-I F - In 2F] 2.303
sinh-I z = In 22
which is derived directly from F 2 a. Thus, to make an error of b (or less) in the calculation of p M , we cannot use the approximate equation for any value of f greater than that obtained from equation (19). Until now we have only considered the region prior to equivalence. However, as previously discussed, pM is symmetrical with respect to F. Therefore, subsequent to equivalence, if ApM is still defined as the difference between the approximate (equation (15)) and the exact (equation (11)) equations:
+
However, K(1
It is interesting to note that though it is usually thought that F must exceed unity in order to transform the exact equation into the more usual approximate one; in reality, to make an error no greater than 0.10 in the calculation of p M , F need only equal or exceed 0.875. Of course, for values of F less than unity, sinh-' F-F is a better approximation, and indeed A p M is then reduced to 0.04 for F = 0.875. To find the value off corresponding to auy value of a in Table 1, we use:
fi
I n other words, the exact equation must be used in the region:
if ApM is not to exceed + b; the corresponding value of a being obtained from Table 1 or a similar type of table. For convenience, we may write equation (20) as: f . > f 2fl (21) Values of fi and fz for various values of K112/C0 are presented in Table 2 ; r = 0 corresponds to neglect of
and f2Defined by Equation (21)
Volume 40, Number 7 0, October 1963
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521
dilution; r = 1 applies to a titration in which the concentration of the titrant is equal to the original concentration of the species being titrated. It is evident that for r 5 1,theapproximate equation can be used, with an error not exceeding 0.0100 in pM, even at 0.1% titrated prior or subsequent to equivalence, as long as C o / K L ' 2exceeds lo4. For acid-base neutralizations, this corresponds to a pH difference of 4.00 units between the initial pH and that a t equivalence; thus Comust equal or exceed 1.00 x 10Wa At! It should be emphasized that because pM is logarithmically related to [MI, a relatively large error in [MI, results however in a considerably smaller error in pM. This somewhat mathematically fortuitous result does not excuse a student from being able to calculate [MI, to as great an accuracy as may be desired, by use of the appropriate formula. However, recognition of this relationship may save considerable time if only a calculation of a titration curve (pM versusj) is required. A brief consideration of this relationship follows. Prior to equivalence, the approximate value of [MI from equation (13) is greater than that calculated from the exact equation (12); i.e., pM,., < pM,.., Thus, if t,he error in the calculation of [MI is loo EM%:
From equation (1:3):
+
K (l rf) R"* - C 0 ( l - f) 2F ~
=
KV2 exp [-sinh-I F ]
so that equation (22) may be written as:
In Table 3 are presented the minimum values that F must possess in order that E M does not exceed the indicated error. Through the use of equation (19) one may estimate the region of j values wherein the exact equatioll must be employed to obtain [hI] to any desired accuracy! From equation (22), we obtain: A ~ M= log ( 1
+ EM)
(24)
which reflects the lesser dependence on EM,of the accuracy of phI than that of [AI]. A ten per cent error in [MI, i.?., E M = 0.10, results in only an error 0.04 units in phI; a five per cent error in [ X I yields only an error of 0.02 in pM, Thus to calculate pM to within 0.10 unit, it is only required that EN 5 0.26.
1 - sinh- 1 C'/21Z1f2=lag C"/K'la 2.303 far Co/Kfl' >> 1; K V r J a= 1.00 X 1 0 ' at 25' C. 6
=
For values of f>l:E'u=
+
-Ex - where E ' u 1 EM is defined by equation (22), but now only applies subsequent to equivalence, and EX is ohtained from Table 3. 522
/
F
EM
Summary
Examination has been made of the properties of the rigorously derived equation which is applicable to the calculation of a potentiometric titration curve for a simple ion combination reaction. Under appropriate conditions, expressed in terms of the fraction titrated, the customary approximate equations may be used. The presentation of the exact equation emphasizes the principle that the complete titration curve may he represented by a single function, i.e., the curve is continuous. Under conditions where the approximate equations cannot he accurately used, the calculation of phI is nevertheless relatively easy as compared to first ohtaining [hl] by use of the quadratic formula. The use of tables of the inverse hyperbolic sine should present no difficultiesand allows the student to become familiar with some properties of hyperbolic functions. Finally, a brief consideration has been made of the relationship betweeu the error in [MI and in pM.
Journal o f Chemical Education
A N D STENOER, V. A,, 'Volumetric Analysis," 2nd rev. ed., Interscience Publishers, Inc., New York, 1942, Chap. 2 and Chap. 3. ( 2 ) Rrccr, J o n ~E., "Hydrogen Ion Concentration," Princeton University Press, Princeton, 1952, Chap. 9. H ,J . C h a . Soc., 1928, 857. ( 3 ) C A ~ A N A GB., J . M . H., Anal. Chim. A d a , 24, 175-91 (1961). ( 4 ) FORTUIN, (6) DIEHOP,E., Anal. Chim. A d a , 22, 101-5 (1960). ( 6 ) ROLLER, P . S., J. Am. Chem. Soe., 54, 3498 (1932). ( 7 ) WARMUS,M., "Tables of Elementary Functions," Pergamon Press, Kew York, 1960: COMRIE,L. J., "Chamber's Six Figure Math~matirslTables," D. Van Kostrand New Ynrk, 1949, Vol. 2.
(1)
and fromequation (12):
~ H j - 0- p H f = 1
Minimum Values for F for Selected Values of Ex
Literature Cited =
6
Table 3.
KOLTHOFF, I. M.,
