If `K_(a_1),K_(a_2)` and `K_a_3)` be the first, second and third dissociation constant of `H_(3)PO_(4)` and `K_(a_1)gtgt K_(2_a) gtgtK_(a_3)` whis is/are correct :

Options

A. `[H^(+)]~~sqrt(K_(a_1)[H_(3)PO_(4)]`
B. `[H^(+)]~~[HPO_(4)^(2-)]`
C. `K_(a_2)~~[HPO_(4)^(2-)]`
D. `[HPO_(4)^(-2)]=[PO_(4)^(3-)]`

Answer

To solve the problem regarding the dissociation constants of phosphoric acid \( H_3PO_4 \), we need to analyze the dissociation steps and the relationships between the constants \( K_{a1}, K_{a2}, \) and \( K_{a3} \).

### Step-by-Step Solution

1. **Understanding Dissociation Constants**:
- Phosphoric acid \( H_3PO_4 \) dissociates in three steps:
1. \( H_3PO_4 \rightleftharpoons H^+ + H_2PO_4^- \) with dissociation constant \( K_{a1} \)
2. \( H_2PO_4^- \rightleftharpoons H^+ + HPO_4^{2-} \) with dissociation constant \( K_{a2} \)
3. \( HPO_4^{2-} \rightleftharpoons H^+ + PO_4^{3-} \) with dissociation constant \( K_{a3} \)

2. **Given Information**:
- It is given that \( K_{a1} \gg K_{a2} \gg K_{a3} \). This indicates that the first dissociation is much stronger than the second, and the second is much stronger than the third.

3. **Analyzing Each Dissociation**:
- For the first dissociation:
\[ K_{a1} = \frac{[H^+][H_2PO_4^-]}{[H_3PO_4]} \]
- For the second dissociation:
\[ K_{a2} = \frac{[H^+][HPO_4^{2-}]}{[H_2PO_4^-]} \]
- For the third dissociation:
\[ K_{a3} = \frac{[H^+][PO_4^{3-}]}{[HPO_4^{2-}]} \]

4. **Assumptions**:
- Since \( K_{a1} \) is much larger than \( K_{a2} \) and \( K_{a3} \), we can assume that at equilibrium, the concentration of \( H_3PO_4 \) will be significantly less than the initial concentration, and thus \( [H_2PO_4^-] \) will be approximately equal to \( [H^+] \) generated from the first dissociation.
- For the second dissociation, since \( K_{a2} \) is smaller, we can assume that the change in concentration of \( H_2PO_4^- \) is negligible compared to its initial concentration.

5. **Conclusion**:
- From the analysis, we can conclude that:
- \( K_{a1} \) is the strongest dissociation and contributes most to the \( H^+ \) concentration.
- \( K_{a2} \) is significant but less than \( K_{a1} \).
- \( K_{a3} \) is the weakest and can be considered negligible in comparison to the first two.

6. **Correct Statements**:
- Based on the relationships and the analysis, the correct statements regarding the dissociation constants can be determined. The options that are correct are those that reflect the hierarchy of the dissociation constants.