THE USE OF EXACT EXPRESSIONS I N CALCULATING HYDROGEN ION CONCENTRATIONS1 E. R. NIGHTINGALE, JR. University of Nebraska, Lincoln, Nebraska
estimation of hydrogen ion concentration in aqueous solutions containing weak acids or weak bases and! or their salts often causes considerable difficulty to students because of the approximate and apparently intuitive methods frequently presented for such calculations. Even advanced students are often unaware of the assumptions which have been made in deriving a given expression and are uncertain of what simplificat,ions may be made to obtain a reasonable estimate of the hydrogen ion concentration, especially for a dilute solution of a polyprotic acid. It is the purpose of this paper to discuss a much neglected method, based upon the principle of electroneutrality, for the formulation of exact expressions for the hydrogen ion concentration in aqueous solutions of a weak acid, the salt of a weak acid, or mixtures of the acid and its salt, and to show how these exact derivations may be used, with proper approximations, to estimate the hydrogen ion concentration. Extension of these methods to solutions containing weak bases will become obvious. Similar discnssions based upon electroneutrality invariably present expanded expressionsZcubic, quartic, and higher order in hydrogen ion concentration, which are difficult to evaluate and generally impossible for students. If an exact expressiou for hydrogen.ion concentration is to be obtained, approximations in the derivation cannot be permitted. Approximations should be used to facilitate the desired answer, but only after the expressions have been derived in their mathematical entirety. For example, in calculating the pH of an aqueous solution containing a monoprotic weak acid such as acetic acid, the initial assumption that [H+]= [CHr COO-] should not be made, but rather, the initial formulation must be an exact expression of the conditions which actually exist in solution. These are best represented by the expression of electroneutrality for the solution. In the derivations which follow, it is assumed that the solutions are at constant ionic strength, and concentrations rather thau activities are indicated. T H E
AQUEOUS SOLUTION OF A MONOPROTIC WEAK ACID
Consider an aqueous solution of a monoprotic weak acid, HA. When the acid is dissolved in water, two equilibria are involved: HA=Ht+A-
(1)
'Presented in part before the Chemistry and Phyaics Section of the 66th Annual Meeting of the Nebraska Academy of Sciences, April, 1956. = I'mn, B., J. CHEM.EDIT., 30, 257 (1953).
VOLUME 34, NO. 6, JUNE, 1957
and H90 = H +
+ OH
The hydrogen ion concentration is given by the expression of electroneutrality for the solution, [H+] = [A-I
+ [OH-]
(3)
The equilibrium expressions for the dissociation for the weak acid and of water are K.
=
lH+l[A-I/lHAl
(4)
[Ht1[OH-I
(5)
and K,
=
where K. and K , are the respective equilibrium concentration constants. Substituting (4) and (5) into (3) :
+ Kw/[Ht1
[HC1= Kc-[HAIIIH+l
(ti)
or multiplying through by [H+]and extracting the square root, [H+l
=
+
~ K , [ H A I K,
(7)
Equation (7) is an exact expression for the hydrogen ion concentration in terms of the equilibrium constants K. and K , and the concentration of the undissociated acid HA. Generally, the product K,[HA] is much greater than K,, and if one may assume that the concentration of the undissociated acid HA is approximately equal to the stoichiometric ~oncentrat~ion C of the acid, where C = [HA] [A-1, eqnatioo (7) may be simplified to the more common form
+
[Ht] e d K C
(8)
This approach to equation (8) explicitly denotes the two assumptions necessary to simplify equation (7). It is easy for students to recognize that, numerically, K , is usually much smaller than the product K.[HA], but. the more discerning student should recognize that [HA] is never exactly equal to the stoichiometric concentration C. Under what conditions is this assumption permissible? Combining the equilibrium expression for the dissociation of the weak acid with the stoichiometric expression of concentration,
c=
[HA] + [A-]
=
[HA]
+ K.[HA]/[Ht]
(!I)
and CW+I [HA] = [H+l IC,
+
(10)
it is seen that [HA]is approximately equal F~~~ to the stoichiometric concentration. C only if K. is negligibly small compared with [H+]. 217
Since in (7) the exact concentration of acid HA is unknown, the preferred expression for hydrogen ion concentration in terms of the known stoichiometric concentration is obtained by substituting (10) into (7), and
SOLUTION CONTAINING THE SALT OF A MONOPROTIC WEAK ACID
The form in which eqnation (11) is presented is important. The square root of a quantity is comparatively easy to obtain whereas higher order roots are not. No attempt should be made to rationalize the equation, for even this relatively simple expression represents a cubic equation in hydrogen ion concentration which is not easily solved. More complex expressions give higher order equations for which there are no general solutions. Again assuming that K , is negligible, the serious student will realize that equation (8) can be obtained from (11) only if K. is negligibly small compared with [H+], the same criterion by which [HA] may be approximated by C in equation (10). Generally, this is the case. If, however, K. is not negligible compared with [H+], the numerical calculation becomes more difficult. The method of successive approximations is very useful for equations of this type, but it may be laborious. A simple graphical method is often the best. For example, it is desired to calculate the hydrogen ion concentration of a l.0.10-4 4111 solution of a weak acid with K, of lo-'. Using eqnation (£9, one might estimate that [H+] = M. However, the assumption by which (8) is obtained from ( l l ) , namely, that Ka is negligibly small compared with [H+], is not fulfilled, and this estimate undoubtedly is incorrect. However, by using an approximate value such as W 4 M for the hydrogen ion concentration in the right-hand side of equation ( l l ) , one calculates a corresponding value for the hydrogen ion on the left-hand side. Thus for assumed values of 10W4, and M hydrogen ion concentration in the right-hand side, one calculates values of 9.1,104, 7.1.10-5, and 3.0.10-6 M, respectively. Plotting the assumed values for the hydrogen ion concentration versus those calculated, the curve shown in Figure 1 is obtained. Obviously, the correct value lies a t the point where the assumed and calculated values are equal and for Figure 1, the hydrogen ion concentration is observed to be B.3.10-6 M. This value compares favorably with the 6.2.106 M computed using equation (11).
The stoichiometric concentration C of the salt is given
Consider an aqueous solution containing the salt MA of a strong base and weak monoprotic acid HA, in which the salt is assumed to be completely dissociated. The expression for electroneutrality is
C
=
[M+] = [A-I
+ [HA]
(13)
where [HA] is the concentration of acid formed by the reaction between A- and H20. Substituting (4), (5). and (13) into (lZ), and
It is frequently argued that if K, is negligibly small compared with [A--1and if it may be assumed that the concentration of salt [A-] is approximately equal to the stoichiometric concentration C, then (15) reduces to the more usual form
The preferable argument is as follows. From (4) and (13): K.C [A-I = C - [HA] = iHtI
+ K.
(17)
and the formulation of (15) in terms of the stoichiometric concentration is
Except for the salts of very weak acids, [H+] is negligibly small compared with KO,and (18) may be reduced to (19)
assumed (H*), rnole/l.
Comparing (15), (17), and (l9), [A-] is approximately equal to C if [H+] is negligible compared with K,, and the proper criterion for using equation (16) in place of (19) is that K, must be negligibly small compared with C. I t should be emphasized that this exact treatment does not utilize the concept of hydrolysis other than to recognize that some nndissociated acid is formed by the reaction between A- and HzO. The so-called "degree of hydrolysis" is mathematically ~ndefinable,~ and calculations based upon hydrolysis reactions are not to he encouraged. In considering the hydrogen ion concentration of an aqueous solution containing a mouoprotic weak acid together with its salt, the treatment is similar to that presented above for the salt alone except that the ratio [A-]/[HA] is not necessarily large, nor the ratio [H+]/ [M+] invariably small. The expression of electroneutralitv is aeain 112). and the stoichiometric relation is
araphicll Computation of[H + l for Solution of Monoprotic We& Arid. C = 1.0 10-'mola/L.. Ka = 1 . 0 . lo-'
R~ccr,J. E., "Hydrogen Ian Concentration," Princeton University Press, Princeton, N. J., 1952, pp. 55-8, 127.
~
Figulr 1.
.
~
a
JOURNAL OF CHEMICAL EDUCATION
+
+
C = C. C, = [A-1 [ H A ]2 [ M + ]in which C. and C8are the concentrations of added acid and salt, respectively. Substituting (5) and (7) into (12), and rearranging
from the exact expression (29) by recognizing that K,, and K., must be small compared with [ H + ] . If K,, is not small compared with [ H f ] , equation (29) becomes similar to (11) and may be evaluated hy one of the methods described therein. SALTS OF DIPROTIC WEAK ACIDS
where C , = [M+],or
The expression of electroneutrality for solutions containing the salts M H A or M 2 A of a strong base and the diprotic acid H 2 A is
If C , approaches zero, equation (20) reduces to ( l l ) , and as C , approaches C, the treatment becomes identical with (18). For intermediate values, 1 > [ M + ] / C > 0, the calculations may be evaluated using equation (20) and the graphical method described previously. If C,, = C8 = C/2, (20) may he rewritten as
and it is obvious that I(, is not identically equal to [ H + ] except when K, = This conclusion often is not appreciated if a student attempts to evaluate [H+]using only equation (4).
~IZ.
[MC1
+ [Ht]
=
[HA-] + 2[A--1 +[OH-]
(31)
In deriving expressious of this type for hydrogeu io11 concentration, it is most convenient to express the acid concentration in terms of that acid species which is present in the largest concentration in the solution. Thus for a solution of the salt M H A , the initial formulation will be in terms of the H A - anion. The stoichiometric concentration C for a solution containing M H A is
+ [HA-] + [A--1
C = [M+1= [H?A]
(82)
Substituting (23), (24), and (32) into (31),
DIPROTIC WEAK ACID
The expression of electroneutrality for a solution containing the diprotic weak acid H 2 A is [Ht] = [HA-] + 2[A--1 + [OH-] (22) The equilibrium concentration constants for the acid are K., = [Ht1[HA-I/[HzAl (23
Generally K,.K, is negligible and K,, is negligibly small compared with [ H A - ] , and the classical approximation is [Hi]
d\/K.,K.,
(:34)
From (23), (24), and (32),
and K.,
=
[Htl[A-VIHA-1
(24)
Substituting (23) and (24) into (22),
Omitting K,, and if 2K,,/[H+] is negligibly small compared with unity, (25) reduces to the more usual approximation The stoichiometric concentration C of the diprotic weak acid is given by C = [H,A]
+ [HA-] + LA--]
and the exact expression for hydrogen ion concentration in terms of the stoichiometric concentration C i s readily obtained by substituting (35) into (33)
For the salt MA, the stoichiometric concentratio~lis given by C = [Mt]/2 = [H,A] [HA-] [A--1 (37)
+
+
Substituting (23), (24), and (37) into (31), and expressing the acid in terms of the major component ['\--],
(27)
Combining (23) and (24) with (27), Usually [ H + ] and K,, are negligibly small comparcd with K.,, and (38) reduces to Substituting (28) into (25), the exact expression for the hydrogen ion concentration in terms of the stoichiometric concentration and the equilibrium constants is
From (23), (24), and (37),
Again neglecting the term containing K,, and if K,, and K., are negligible compared with [ H + ] ,(29) reduces to
and substituting (40) into (38), the exact expression in terms of the stoichiometric concentration is
(30) should not be "derived" from (26) by assuming that the concentration [ H 2 A ]is equal to C, but rather
An expression similar to (33) and (38) for a solution
VOLUME 34, NO. 6, JUNE, 1957
ront,aining the salt of a weak aeid and a meak base has b e ~ presented n by Swift,.' SUMMARY
A method for ealculat,ion of hydrogen ion eoncentrat,ion, based upon the expression of electroneutrality, is pmsented for solut.ions containing mono- and dibasic arids and their salts. The method is exact, does not require intuitive approximat,ions, and forces the student to ronsider all t,he eqnilihria involved and all the speries present in the solntion. Assumptions and simplifica'SWIFT, E. H., " I n t ~ ~ o d u ~ t oQuantitative r~ I'i.ent,iee-Hall, Ino., Re\\- I-"A, 1950, p. 22;.
Analysi-,"
tions are not introduced into the derivat,ions, and approximations are used to facilitate the calculation only after the proper expression for hydrogen ion concentration has been derived in its mathematical entirety. When the equations are expressed in terms of the stoiehiometrie concentration of the meak acid and the relevant equilibrium constants, the only unknon.n quantity is the hydrogen ion concentration itself. The method is advantageous in that it does not require any comparison of the stoiehiometric concentration with the c~ncent~rations of the several aeid species which may exist in solution. The concept of salt hydrolysis is not used; this concept is unnecessary and is not employed in t,he most rigorous treatment^.^
JOURNAL O F CHEMICAL EDUCATION
