There is somthing wrong with the data you have been given. In the
absence of any dissolved CaCO3 (i.e., the pure CO2-H2O system) the pH
of a solution in equilibrium with a partial pressure of CO2 =
3.162*10^-4 that one would calculate using the data you are given
would be 5.65. Addition of CaCO3 would only serve to *increase* the
pH (make the solution less acidic).
The calculation of the pH for the CO2-H2O system goes as follows:
A) [H+][OH-] = K_w = 10^-14
B) [H2CO3]/PCO2 = K_CO2 = 3.388*10^-2 (given, where K_CO3 is the
Henry's Law coefficient for CO2 and water)
C) [H+][HCO3-]/[H2CO3] = K_a1 = 10^-6.33 (given, where K_a1 is the
first acidity constant for H2CO3)
D) [H+][CO3-]/[HCO3--] = K_a2 = 10^-10.33 (this is something one can
look up. It turns out it's not needed here, but I include it for
completeness)
E) [H+] - [OH-] - [HCO3-] - 2*[CO3--] = 0 (charge balance equation,
i.e., the sum of the charges on the positive ions equals the sum of
the charges on the negative ions)
Begin by ignoring the second dissociation constant of H2CO3 (this is
only important at high pHs, and we can show that this approximation is
fine after we have run through the calculation). This assumption
implies that [CO3--] = 0. Making this assumption, and using equation
(A) to eliminate [OH-] in equation (E) gives:
[H+] - 10^-14/[H+] = [HCO3-]
Substitute this into equation (C), and use equation (B) to eliminate
[H2CO3] to obtain:
[H+]*([H+] - 10^-14/[H+])/(3.388*10^-2 * 3.162*10^-4) = 10^-6.33
[H+]^2 = 10^-6.33 * (3.388*10^-2 * 3.162*10^-4) + 10^-14 = 2.236 *
10^-6 M (only the positive root has any physical meaning)
pH = -log[H+] = -log(2.236 * 10^-6) = 5.65. This is the pH of a
solution in equilibrium with the specified P_CO2 and the given Henry's
Law and dissociation constants.
Using this value of [H+] in combination with equations (B) and (C), we
can calculate [HCO3-]:
[HCO3-] = 10^-6.33 * 3.388*10^-2 * 3.162*10^-4/(2.236 * 10^-6) = 2.241*10^-6 M
To see that we were justified in neglecting the concentration of
[CO3--], we can substutute these values for [H+] and [HCO3-] into
equation (D) to see what the approximate concentration of [CO3--]
would be:
[CO3--] = 10^-10.33 * 2.241*10^-6/2.236*10^-6 = 4.69 * 10^-11 M, which
is vanishingly small, so our approximation is ok.
-------------------------------------------
For grins, let's crank through the calculations for the original
problem, assuming everything is ok. We'll end up with an
inconsistency, which will serve to show that there is a problem with
the data.
The chemical equilibria relevant to this system are:
1. H2O <-> H+ + OH-
2. CO2(g) + H2O(l) <-> H2CO2(aq)
3. H2CO3(aq) <-> H+ + HCO3-
4. HCO3- <-> H+ + CO3--
Because the problem does not specify that this solution is in
equilibrium with solid CaCO3, the dissolution reaction of calcite
(CaCO3) does not add any additional equilibrium constraints relative
to the problem solved above.
For each of these reactions, we can write an equilibrium constant:
1a. [H+][OH-] = K_w = 10^-14
2a. [H2CO3]/PCO2 = K_CO2 = 3.388*10^-2
3a. [H+][HCO3-]/[H2CO3] = K_a1 = 10^-6.33
4a. [H+][CO3-]/[HCO3--] = K_a2 = 10^-10.33
Plus a charge-balance equation:
5a. [H+] + 2*[Ca++] = [OH-] + [HCO3-] + 2*[CO3--] Note that now we
are admitting the possibility that there are Ca++ ions in solution.
We are told that the pH = 5.1, so [H+] = 10^-5.1 M. This immediately
(from 1a) gives that [OH-] = 10^-14/10^-5.1 = 10^-8.9 M.
Equation 2a and the fact that PCO2 = 3.162*10^-4 gives us that [H2CO3]
= 3.162*10^-4 * 3.388*10^-2 = 1.071*10^-5 M
Using this result, and the value for [H+] found above in equation 3a
gets us [HCO3-]:
[HCO3-] = 10^-6.33*[H2CO3]/[H+] = 10^-6.33 * 1.071*10^-5/10^-5.1 = 5.011*10^-12 M
Because we are at a pH of 5.1, we can neglect the second dissociation
constant of H2CO3, just as we did above. The concentration of CO3--
at acidic pH's is vanishingly small. (To see this, plug in values for
[H+] and [HCO3-] in equation 4a above.)
We now know the concentrations of all the charged species in this
solution except for [Ca++]. TO find the concentration of this ion,
plug the other values into the charge-balance equation (equation 5a)
and solve for [Ca++]:
[Ca++] = 1/2*([OH-] + 2*[CO3--] + [HCO3] - [H]) = -3.971 * 10^-6 M, a
negative quantity, which doesn't make any sense physically! |