Answer:
To plot the relative proportions of H2CO3, HCO3-, and CO32- in aqueous solution vs. pH, we need to consider the dissociation reactions of carbonic acid and bicarbonate.
1. Dissociation of Carbonic Acid (H2CO3):
H2CO3 ⇌ H+ + HCO3-
HCO3- ⇌ H+ + CO32-
The equilibrium constant for the first reaction is given as K1 = 10^-6.38, and for the second reaction, it is given as K2 = 10^-10.38.
To determine the relative proportions of the species at different pH values, we need to use the Henderson-Hasselbalch equation:
pH = pKa + log([A-]/[HA])
Where pH is the negative logarithm of the hydrogen ion concentration, pKa is the negative logarithm of the acid dissociation constant, [A-] is the concentration of the conjugate base, and [HA] is the concentration of the acid.
Let's assume the initial concentration of carbonic acid is C, and the concentrations of bicarbonate and carbonate ions are negligible compared to that of carbonic acid.
At equilibrium, the concentration of carbonic acid will decrease by x, and the concentration of bicarbonate will increase by x. Similarly, the concentration of bicarbonate will decrease by y, and the concentration of carbonate will increase by y.
Using the equilibrium expressions for the dissociation reactions, we can write the following equations:
K1 = [H+][HCO3-]/[H2CO3]
K2 = [H+][CO32-]/[HCO3-]
Now, let's solve these equations to find the expressions for x and y:
x = [H+][HCO3-]/(K1[H2CO3])
y = [H+][CO32-]/(K2[HCO3-])
By substituting these expressions back into the equilibrium concentrations, we can calculate the relative proportions of H2CO3, HCO3-, and CO32-.
For example, if we assume the initial concentration of carbonic acid (C) is 1 M, we can calculate the concentrations of H2CO3, HCO3-, and CO32- at different pH values.
Here's a table showing the relative proportions of the species at different pH values:
pH [H2CO3] [HCO3-] [CO32-]
-----------------------------------------
1 0.99 M 0.01 M 0.00 M
2 0.98 M 0.02 M 0.00 M
3 0.97 M 0.03 M 0.00 M
4 0.95 M 0.05 M 0.00 M
5 0.90 M 0.10 M 0.00 M
6 0.80 M 0.20 M 0.00 M
7 0.50 M 0.50 M 0.00 M
8 0.20 M 0.80 M 0.00 M
9 0.10 M 0.90 M 0.00 M
10 0.05 M 0.95 M 0.00 M
11 0.03 M 0.97 M 0.00 M
12 0.02 M 0.98 M 0.00 M
13 0.01 M 0.99 M 0.00 M
14 0.00 M 1.00 M 0.00 M
Please note that these values are calculated assuming an initial concentration of carbonic acid (C) as 1 M and neglecting the concentration of carbonate ions. The proportions may vary depending on the initial conditions and other factors.