In aqueous solution the ionization constants for carbonic acid are $$K_1 = 4.2 \times 10^{-7}$$ and $$K_2 = 4.8 \times 10^{-11}$$. Select the correct statement for a saturated $$0.034$$M solution of the carbonic acid.

For carbonic acid $$H_2CO_3$$ the two successive ionisations in water are

$$H_2CO_3 \;\rightleftharpoons\; H^+ + HCO_3^- \qquad K_1 = 4.2\times10^{-7}$$
$$HCO_3^- \;\rightleftharpoons\; H^+ + CO_3^{2-} \qquad K_2 = 4.8\times10^{-11}$$

The formal (analytical) concentration of the acid is $$C_0 = 0.034\;{\rm M}$$. Because both ionisation constants are very small, only a small fraction of $$H_2CO_3$$ dissociates, so we may take $$[H_2CO_3]\approx C_0$$ in the first approximation.

Step 1 : First-step dissociation
Let $$x$$ be the concentration produced by the first step:

$$H_2CO_3 \;\rightleftharpoons\; H^+ (x) + HCO_3^- (x)$$

Using $$K_1$$,

$$K_1 = \dfrac{[H^+][HCO_3^-]}{[H_2CO_3]} \approx \dfrac{x^2}{C_0}$$

so

$$x \approx \sqrt{K_1\,C_0} = \sqrt{(4.2\times10^{-7})(0.034)} \approx \sqrt{1.43\times10^{-8}} \approx 1.2\times10^{-4}\;{\rm M}$$

Step 2 : Second-step dissociation
Let $$y$$ be the amount of $$HCO_3^-$$ that dissociates further:

$$HCO_3^- \;\rightleftharpoons\; H^+ (+y) + CO_3^{2-} (+y)$$

At equilibrium $$[H^+] = x + y,\qquad [HCO_3^-] = x - y,\qquad [CO_3^{2-}] = y$$

Using $$K_2$$,

$$K_2 = \dfrac{[H^+][CO_3^{2-}]}{[HCO_3^-]} = \dfrac{(x+y)\,y}{\,x-y\,}$$

Because the second dissociation is much weaker, $$y \ll x$$, giving

$$K_2 \approx y \;\;\Longrightarrow\;\; y \approx 4.8\times10^{-11}\;{\rm M}$$

Step 3 : Comparing species

Final concentrations (to the significant figures needed for comparison):
$$[H^+] \;=\; x+y \;\approx\; 1.2\times10^{-4}\;{\rm M}$$
$$[HCO_3^-] \;=\; x-y \;\approx\; 1.2\times10^{-4}\;{\rm M}$$
$$[CO_3^{2-}] = y \approx 5\times10^{-11}\;{\rm M}$$

The results show

• $$[H^+]$$ and $$[HCO_3^-]$$ are practically the same.
• $$[CO_3^{2-}]$$ is many orders of magnitude smaller than either of them.

Examining the options
A. $$[CO_3^{2-}] = 0.034\;{\rm M}$$ ― false (actual value ≈ $$5\times10^{-11}\;{\rm M}$$).
B. $$[CO_3^{2-}] \gt [HCO_3^-]$$ ― false (it is much smaller).
C. $$[H^+]$$ and $$[HCO_3^-]$$ are approximately equal ― true, both ≈ $$1.2\times10^{-4}\;{\rm M}$$.
D. $$[H^+] = 2[CO_3^{2-}]$$ ― false (ratio is about $$2.4\times10^{6}$$).

Hence, the correct statement is:
Option C which is: The concentration of $$H^+$$ and $$HCO_3^-$$ are approximately equal.