In a saturated aqueous solution of [tex]$Ag_2SO_4$[/tex], the silver ion concentration is [tex]$3.11 \times 10^{-2} M$[/tex] and the sulfate ion concentration is [tex]$1.55 \times 10^{-2} M$[/tex]. Calculate the solubility product, [tex]$K_{\text{sp}}$[/tex], for [tex]$Ag_2SO_4$[/tex].

[tex]K_{\text{sp}} = [/tex] [tex]\square[/tex]

To calculate the solubility product, [tex]$ K_{\text{sp}} $[/tex], for [tex]$ \text{Ag}_2\text{SO}_4 $[/tex], we'll follow these steps:

1. Understand the Dissolution Process:
- When [tex]$ \text{Ag}_2\text{SO}_4 $[/tex] dissolves in water, it dissociates according to the following equation:
[tex]\[
\text{Ag}_2\text{SO}_4(s) \rightleftharpoons 2\text{Ag}^+(aq) + \text{SO}_4^{2-}(aq)
\][/tex]

2. Identify the Ion Concentrations:
- According to the problem, the concentration of silver ions, [tex]$[ \text{Ag}^+ ]$[/tex], is [tex]$3.11 \times 10^{-2} \, M$[/tex].
- The concentration of sulfate ions, [tex]$[ \text{SO}_4^{2-} ]$[/tex], is [tex]$1.55 \times 10^{-2} \, M$[/tex].

3. Write the Expression for [tex]$ K_{\text{sp}} $[/tex]:
- The solubility product, [tex]$ K_{\text{sp}} $[/tex], is given by the formula:
[tex]\[
K_{\text{sp}} = [\text{Ag}^+]^2 \times [\text{SO}_4^{2-}]
\][/tex]

4. Substitute the Known Values:
- Substitute the given concentrations into the [tex]$ K_{\text{sp}} $[/tex] expression:
[tex]\[
K_{\text{sp}} = (3.11 \times 10^{-2})^2 \times (1.55 \times 10^{-2})
\][/tex]

5. Calculate [tex]$ K_{\text{sp}} $[/tex]:
- First, calculate [tex]$(3.11 \times 10^{-2})^2$[/tex]:
[tex]\[
(3.11 \times 10^{-2})^2 = 9.6721 \times 10^{-4}
\][/tex]
- Now, multiply by [tex]$(1.55 \times 10^{-2})$[/tex] to find [tex]$ K_{\text{sp}} $[/tex]:
[tex]\[
K_{\text{sp}} = 9.6721 \times 10^{-4} \times 1.55 \times 10^{-2} = 1.4991755 \times 10^{-5}
\][/tex]

So, the solubility product, [tex]$ K_{\text{sp}} $[/tex], for [tex]$ \text{Ag}_2\text{SO}_4 $[/tex] is [tex]$ 1.4991755 \times 10^{-5} $[/tex].