The usual equilibrium

H₂CO₃ + H₂O < => H₃O⁺ + HCO₃^- K₁ = 4.45 x 10^-7

should be replaced by

H₂CO₃ + H₂O < = > H₃O⁺ + HCO₃^- K₁ = 1.5 x 10^-4

and

CO₂(aq) + H₂O < = > H₂CO₃ K = 2.8 x 10^-3

(see also ref 1 p 92).

With respect to our previous text the structural equations become:

For the solubility of barium carbonate

[Ba][CO₃^2-] = K_sp (1)

(2) with K₂ = 4.7 x 10^-4

(3) with K₁ = 1.5 x 10^-4

(4) with K = 2.8 x 10^-3

[Ba²⁺] = [CO₃^2-] + [HCO₃^-] + [H₂CO₃] + [CO₂(aq)] (5)

2[Ba²⁺] + [H₃O⁺] = 2[CO₃^2-] + [HCO₃^-] + [OH^-] (6)

[H₃O⁺] [OH^-] = 1.0 x 10^-14(7)

Solubility = [Ba²⁺]

For the solubility of silver carbonate

[Ag⁺]² [CO₃^2-] = K_sp(8)

(9)

(10)

(11)

(12) [Ag⁺] = [CO₃^2-] + [HCO₃^-] + [H₂CO₃] + [CO₂(aq)]

(13) [Ag⁺] + [H₃O⁺] = 2[CO₃^2-] + [HCO₃^-] + [OH^-]

(14) H₃O⁺] [OH^-] = 1.00 x 10^-14

Solubility = ¹/₂[Ag⁺]

The problem now results in one more equation, leading to seven equations and seven unknowns. The solution is given in Table 1. As expected, the error only appears in the H₂CO₃ concentration.

We still want to emphasize that the true contribution of our paper was to report on the efficient computation of solubilities using a symbolic mathematics package such as Mathematica. However, for application to the carbonates, the form of Skoog et al. (2) was a less recommendable choice, given the chemical features Bader pointed out very correctly.

In the adjusted and more complex form above, Mathematica confirmed its effectiveness. On a SUN-Sparc station computation took about three seconds for the Ba²⁺ case and about eight seconds for the more complex Ag⁺ case. Thus it supports David (3), who also insisted on the importance of the tool in chemical education.

Literature Cited

1. Harris, D. C. Quantitative Chemical Analysis, 3rd ed.; Freeman: New York, 1991.

2. Skoog, D. A.; West, D. M.; Holler, F. J. Analytical Chemistry, 6th ed.; Saunders: Fort Worth, 1992; pp 175-181.

3. David, C. W. J. Chem. Educ. 1995, 72, 995-997.