The Hydrolysis of Salts Derived from a Weak Monoprotic Acid and a Weak Monoprotic Base Some Critical Remarks Mario Emilio Cardinali, Claudio Giomini, and Giancarlo Marrosu "La Sapienza" University, Via del Castro Laurenziano, 7, 00161 Rome, Italy Most general chemistry textbooks do not give a detailed discussion of the hydrolysis of salts derived from a weakacid and a weak base because the mathematics involved is difficult to deal with, a t least a t an elementary level. Recently, a simplified approach to this subject was proposed~by Aguirre-Ode1. In the present paper, the limitations as well as the conditions under which this approach can yield almost exact results are discussed. For a solution of a salt of a weak monoorotic acid and a weak monoprotic base (henceforth designed as a w-w salt), whose ionization constants are K, and Kh. resoectivelv, the exact equation determining the [H+] value caibe written in the form
as "a simple, general, and almost exact solution of the problem" of the pH calculation for salt solutions "for a wide range of conditions". He remarks that "the values of pH arising from eq 4 deviate less than *0.001 when compared with those calculated bv means of eo 2 in the ao~licable .range", and that "when eq 4 is introduced in the definitions of rr and B (eqs 3), it is realized that both factors are close to Table 1. Llmltlng Values ot the [H+] and (a.8 ) Ranges Accerolble to a Solutlon of a Salt Derived from a Weak Acld and a Weak Base as a Functlon of K. and Kh'
where c is the analytical concentration of the salt, and K, is the auto-ionization constant of water. Aguirre-Ode rearranges eq 1as
where After stating that "a careful examination of the factor n. B shows that i t is close to unity in the range pK. < p H < pKb", he proposes the formula
' Agulrre-Ode, F. J. Chem. Educ. 1987, 64,957-958.
. ..
.
..
'In this fable K.
.
> K.
..
. ..
1.111 2.020
1.064 1.210
.
and K. Ka 2 1 1O2K. Fw me meaning of a and 8, see text.
Volume 67 Number 3 March 1990
221
.
unity whenever K. Kb >> K,". From eq 4, Aguime-Ode concludes, the formulas for the pH calculations of solutions of salts deriving from monoprotic strong acids and weak bases (or vice versa) can be deduced as particular cases. In the present paper, it is shown that: (1) the range of [Ht] values accessible to a solution of a w-w salt is not K, > [Ht] > Kb, 88 Aguirre-Ode implies; (2) in the range of [HC]values accessible, and wbenK.. Kb >> KK., eq 4 is, strictly speaking, almost exact only for c -; (3) regardless of the analytical concentration of the salt,eq 4 is valid only if the conditionK, .Kb>> Kwis replaced by more restrictive ones, concerning the individual values of K. and Kh. In demonstrating the above points, the discussion will be restricted to the case of an "acid-hydrolysis" w-w salt (K. > Kb); the symmetrical case (K. < Kb), and the special case when K. = Kb, will be left to the reader. Point 1. From eq 2, it follows that
-
Table 2.
.
.
c
[Htlm
[HtIap
(~OIIL)
(~OIIL)
(~OIIL)
5.099. lo-'
9.000. lo-'
Kb
this table. K.
> K, and K..
.
Ka2 1 1O2K.. but Ka is not much higher than
marked with an asterisk (see tent).
222
According to the values of K, and Kb, range 7 may be internal to, external to, or partly overla ping with the range Ka-Kb. only for K, > K F and^^ 5 K$, and provided that K.. Kb Z Kw,will the [Hf] values accessible to the solution be completely internal to the range K.-Kb. On the contrary, when bothK, and Kb are greater thanK?', and such that (K, KwlKb)112< Kb, the [H+]values will always fall out of the above range; this last statement will always be true when bothK, and Kb are less thanK:'. Should these be the cases, any consideration about the factor a @ in the range K. > [H+]> Kb would be quite meaningless.
Exact and Approxlmated (Eq 4) [H+] and pH Values Calculated for Low-, Madlum-, and High-Concenbatlon Solutions 01 a Salt Derlvlng from a Weak Acid and a Weak Base, as a Function of K. and Kbe
K,
a h
Hence, the range of [H+] values accessible to a solution of a w-w salt is
Journal of
Chemical Education
9.153. lo-'
K?'.
AIHtl
P%
P h
6.046
6.038
(%I
1.69
1a.B)
1.211
Values of A[Hi](%) and (cr. 8) are also shown for comparison's sake. Absurd values are
Points 2 and 3. Taking eq 6 into account, the exact eq 2 can be rewritten as
.
value of (a p)o depends on that of Kb according to the approximate relation
.
In Table 2, K, and Kb were chosen such that K.. Kb 2 1 1O2K,, but Kb was always lower than (or equal to) 1.10-$ in order that one of the inequalities in eq 12 was not satisfied. For each pair of K, and Kb values, the following three [H+] values were considered:
where
From the above equation, via eqs 5 and 6, it follows that
and
Therefore, as c varies from 0 to m, (or, as [HI] increases from [HfIoto [Hf],), the product rr B decreases from ( a . B)o (as the upper limit) to ( a . 8)- (as the lower limit). This can be deduced hy examining the derivative of rr . B with respect to [H+].Consequently,since (rr B)o > ( a .B), > 1, thecondition K.. Kb >> Kw.set by Aguirre-Ode, ensures that a B is close to unity only when c m. In a more general case, in order to obtain an u. Bproduct not toodifferent from unity, the more restrictive conditions
.
-
.
.
must both be satisfied (see eq 10). The statements made in points 2 and 3 are given a practical test in Tables 1and 2. Table 1reports the values of [H+]o, [H+],, (a.B)o, and (a 8)- for solutions of w-w salts whose K, and Kb values are in the range of practical interest (1 10-2-1 10-lo), and such in order to satisfy the condition K.. that Ka. Kb 2 1 1O2KW, Kb >> K wIt is evident that (a. B)o divergesfrom unity if Kb is sufficiently small, as deduced from eq 10 by noting that the
.
.
.
as representative of very diluted, very concentrated, and intermediate-concentration solutions of the salt, respectively. In correspondence with the above [H'] values, the values cd, c,, and ci were calculated for the analytical concentration of the salt, by numerical inversion of eq 2, i.e., by means of the following exact equation (see eq 8):
c=
l-(Ey \LA.
1-1
.-'@
(14)
Last, hy introducing the c values into eq 4, the approximated [El+]., values were obtained. In the table, the corresponding pH., values are compared to the exact pH., values. From Table 2, it is evident that, for cd and ci, eq 4 yields [HC]., values that may he grossly in excess with respect to the exact ones. In some extreme cases, marked with an asterisk in the table, the [H+], are even absurd because they are greater than the stoichiometric values (c Ky2)that would be obtained if the cation hydrolysis were complete m ) and the anion hydrolysis absent (K,IK, 0). It is clear, therefore, that both of the two requirements exmessed bv inequalities in eq 12 must heaatisfied, in order that eq 4 can yield reliable results also for diluced solutions.
+
Volume 67
Number 3
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March 1990
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