Excess of `Ag_(2)SO_(4)(s), BaSO_(4)(s)`, and `Ba_(3)(PO_(4))_(2)(s)` are simultaneously in euqilibrium with distilled water. Which of the following is (are) true? Assume no hydrolysis of dissolved ions.

Options

A. `[Ag^(o+)] +2[Ba^(2+)] = 2[SO_(4)^(2-)] +3 [PO_(4)^(3-)]`
B. `2[Ag^(o+)] +4[Ba^(2+)] = 2[SO_(4)^(2-)]+2[PO_(4)^(3-)]`
C. `2[Ag^(o+)]+3[Ba^(2+)]=2[SO_(4)^(2-)] + 2[PO_(4)^(3-)]`
D. `[Ag^(o+)] +[Ba^(2+)] =[SO_(4)^(2-)] +[PO_(4)^(3-)]`

Explanation

To solve the problem, we need to analyze the solubility of the salts involved and the concentrations of their ions in equilibrium with distilled water. The salts mentioned are `Ag2SO4`, `BaSO4`, and `Ba3(PO4)2`.

### Step-by-Step Solution:

1. **Identify the Dissociation of Each Salt:**
- For `Ag2SO4`:
\[ \text{Ag}_2\text{SO}_4 (s) \rightleftharpoons 2 \text{Ag}^+ (aq) + \text{SO}_4^{2-} (aq) \]
Let the solubility of `Ag2SO4` be \( s_1 \). Thus:
- Concentration of `Ag^+` = \( 2s_1 \)
- Concentration of `SO4^{2-}` = \( s_1 \)

- For `BaSO4`:
\[ \text{BaSO}_4 (s) \rightleftharpoons \text{Ba}^{2+} (aq) + \text{SO}_4^{2-} (aq) \]
Let the solubility of `BaSO4` be \( s_2 \). Thus:
- Concentration of `Ba^{2+}` = \( s_2 \)
- Concentration of `SO4^{2-}` = \( s_2 \)

- For `Ba3(PO4)2`:
\[ \text{Ba}_3\text{(PO}_4)_2 (s) \rightleftharpoons 3 \text{Ba}^{2+} (aq) + 2 \text{PO}_4^{3-} (aq) \]
Let the solubility of `Ba3(PO4)2` be \( s_3 \). Thus:
- Concentration of `Ba^{2+}` = \( 3s_3 \)
- Concentration of `PO4^{3-}` = \( 2s_3 \)

2. **Calculate Total Ion Concentrations:**
- Total concentration of `Ag^+`:
\[ [\text{Ag}^+] = 2s_1 \]
- Total concentration of `Ba^{2+}`:
\[ [\text{Ba}^{2+}] = s_2 + 3s_3 \]
- Total concentration of `SO4^{2-}`:
\[ [\text{SO4}^{2-}] = s_1 + s_2 \]
- Total concentration of `PO4^{3-}`:
\[ [\text{PO4}^{3-}] = 2s_3 \]

3. **Analyze the Options:**
- The first option states:
\[ [\text{Ag}^+] + 2[\text{Ba}^{2+}] = 2s_1 + 2(s_2 + 3s_3) = 2s_1 + 2s_2 + 6s_3 \]
On the right-hand side, we have:
\[ 2[\text{SO4}^{2-}] + 3[\text{PO4}^{3-}] = 2(s_1 + s_2) + 3(2s_3) = 2s_1 + 2s_2 + 6s_3 \]
Both sides are equal, so this option is **true**.

- The other options can be checked similarly, but based on the analysis, we can conclude that the first option is correct while the others are false.

### Conclusion:
The correct statement based on the equilibrium concentrations is the first option.