In aqueous solution the ionization constants for carbonic acid are:
`K_(1)=4.2xx10^(-7)and K_(2)=4.8xx10^(-11)`
Select the correct statement for a saturated `0.034M` solution of the carbonic acid.

Options

A. The concentration of `H^(+)` is double that of `CO_(3)^(2-)`
B. The concentration of `CO_(3)^(2-)` is `0.034`
C. The concentration of `CO_(3)^(2-)` is greater then that of `HCO_(3)^(-)`
D. The concentration of `H^(+)` and `HCO_(3)^(-)` are approximately equal

Answer

To solve the problem regarding the ionization of carbonic acid in a saturated solution, we will follow these steps:

### Step 1: Write the Ionization Reactions
Carbonic acid (H₂CO₃) dissociates in two steps:
1. **First Ionization:**
\[ \text{H}_2\text{CO}_3 \rightleftharpoons \text{H}^+ + \text{HCO}_3^- \]
The ionization constant for this reaction is \( K_1 = 4.2 \times 10^{-7} \).

2. **Second Ionization:**
\[ \text{HCO}_3^- \rightleftharpoons \text{H}^+ + \text{CO}_3^{2-} \]
The ionization constant for this reaction is \( K_2 = 4.8 \times 10^{-11} \).

### Step 2: Analyze the Ionization Constants
Since \( K_1 \) is significantly larger than \( K_2 \), it indicates that the first ionization is much more favorable than the second. Therefore, we can expect that most of the dissociation will occur in the first step.

### Step 3: Set Up the Equilibrium Expressions
Let the initial concentration of carbonic acid be \( [\text{H}_2\text{CO}_3] = 0.034 \, M \).

At equilibrium, let:
- \( [\text{H}^+] = x \)
- \( [\text{HCO}_3^-] = x \)
- \( [\text{CO}_3^{2-}] = y \)

From the first ionization:
\[ K_1 = \frac{[\text{H}^+][\text{HCO}_3^-]}{[\text{H}_2\text{CO}_3]} \approx \frac{x^2}{0.034 - x} \]

Since \( K_1 \) is much larger than \( K_2 \), we can assume \( x \) is small compared to 0.034, simplifying to:
\[ K_1 \approx \frac{x^2}{0.034} \]

### Step 4: Solve for \( x \)
Substituting the value of \( K_1 \):
\[ 4.2 \times 10^{-7} \approx \frac{x^2}{0.034} \]
\[ x^2 \approx 4.2 \times 10^{-7} \times 0.034 \]
\[ x^2 \approx 1.428 \times 10^{-8} \]
\[ x \approx \sqrt{1.428 \times 10^{-8}} \approx 3.78 \times 10^{-5} \, M \]

### Step 5: Determine Concentrations
- \( [\text{H}^+] \approx 3.78 \times 10^{-5} \, M \)
- \( [\text{HCO}_3^-] \approx 3.78 \times 10^{-5} \, M \)

For the second ionization, since \( K_2 \) is very small, we can assume:
\[ K_2 = \frac{[\text{H}^+][\text{CO}_3^{2-}]}{[\text{HCO}_3^-]} \approx \frac{(3.78 \times 10^{-5})y}{3.78 \times 10^{-5}} \]
This implies \( y \) is negligible compared to \( x \).

### Step 6: Conclusion
Since \( K_2 \) is very small, the concentration of \( \text{CO}_3^{2-} \) is negligible, and the concentrations of \( \text{H}^+ \) and \( \text{HCO}_3^- \) are approximately equal.

### Final Answer
The correct statement for a saturated \( 0.034M \) solution of carbonic acid is: "The concentration of \( \text{H}^+ \) and \( \text{HCO}_3^- \) are approximately equal."

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