`H_(2) SO_(4)` in aqueous medium ionises in two steps
`H_(2) SO_(4) (aq) + H_(2) O (i) to H_(3) O^(+) (aq) + HSO_(4)^(-) (aq) , K_(a_(1)) = x`
`HSO_(4)^(-) (aq) + H_(2) O(l) to H_(3) O^(+) (aq) + SO_(4)^(2-) , K_(a_(2)) = y`
What is relation between x and y ?

Options

A. ` x lt y`
B. ` x = y`
C. `x gt y`
D. `x gt gt y`

Answer

To determine the relationship between the dissociation constants \( K_{a1} \) (denoted as \( x \)) and \( K_{a2} \) (denoted as \( y \)) for the ionization of sulfuric acid (\( H_2SO_4 \)) in aqueous medium, we can analyze the two steps of ionization.

### Step 1: Ionization of Sulfuric Acid
The first step of ionization of sulfuric acid in water is represented as:
\[ H_2SO_4 (aq) + H_2O (l) \rightarrow H_3O^+ (aq) + HSO_4^- (aq) \]
This reaction involves the strong acid \( H_2SO_4 \) donating a proton (\( H^+ \)) to water, forming hydronium ions (\( H_3O^+ \)) and the bisulfate ion (\( HSO_4^- \)). The dissociation constant for this reaction is \( K_{a1} = x \).

### Step 2: Ionization of Bisulfate Ion
The second step involves the ionization of the bisulfate ion:
\[ HSO_4^- (aq) + H_2O (l) \rightarrow H_3O^+ (aq) + SO_4^{2-} (aq) \]
In this step, the bisulfate ion acts as a weak acid, donating a proton to water to form more hydronium ions and sulfate ions (\( SO_4^{2-} \)). The dissociation constant for this reaction is \( K_{a2} = y \).

### Comparison of \( K_{a1} \) and \( K_{a2} \)
1. **Strength of Acids**:
- \( H_2SO_4 \) is a strong acid, meaning it ionizes completely in the first step. Therefore, \( K_{a1} \) is very large (high value).
- \( HSO_4^- \) is a weak acid, meaning it does not ionize completely in the second step. Thus, \( K_{a2} \) is relatively smaller (lower value).

2. **Conclusion**:
- Since \( K_{a1} \) corresponds to the ionization of a strong acid and \( K_{a2} \) corresponds to the ionization of a weak acid, we can conclude that:
\[ K_{a1} > K_{a2} \quad \text{or} \quad x > y \]

### Final Relation
Thus, the relationship between \( x \) and \( y \) is:
\[ x > y \]