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Answer :
To find the solubility product constant ([tex]\(K_{\text{sp}}\)[/tex]) for [tex]\( \text{Ag}_2\text{SO}_3 \)[/tex] using its molar solubility in pure water, follow these steps:
1. Understand the Dissociation Reaction:
[tex]\(\text{Ag}_2\text{SO}_3\)[/tex] dissociates in water according to the following equation:
[tex]\[
\text{Ag}_2\text{SO}_3(s) \rightleftharpoons 2\text{Ag}^+(aq) + \text{SO}_3^{2-}(aq)
\][/tex]
2. Use the Molar Solubility:
Given that the molar solubility of [tex]\(\text{Ag}_2\text{SO}_3\)[/tex] is [tex]\(1.55 \times 10^{-5} \, \text{M}\)[/tex]. Let's denote this molar solubility as [tex]\( s \)[/tex].
3. Determine Ion Concentrations at Equilibrium:
- Concentration of [tex]\(\text{Ag}^+\)[/tex] ions will be [tex]\(2s\)[/tex] because there are 2 moles of [tex]\(\text{Ag}^+\)[/tex] produced for each mole of [tex]\(\text{Ag}_2\text{SO}_3\)[/tex].
- Concentration of [tex]\(\text{SO}_3^{2-}\)[/tex] ions will be [tex]\(s\)[/tex].
4. Write the Expression for [tex]\(K_{\text{sp}}\)[/tex]:
[tex]\[
K_{\text{sp}} = [\text{Ag}^+]^2 \times [\text{SO}_3^{2-}]
\][/tex]
5. Substitute the Equilibrium Concentrations:
[tex]\[
K_{\text{sp}} = (2s)^2 \times (s) = 4s^3
\][/tex]
6. Calculate [tex]\(K_{\text{sp}}\)[/tex]:
[tex]\[
K_{\text{sp}} = 4 \times (1.55 \times 10^{-5})^3
\][/tex]
7. Result:
After performing the calculation, the solubility product constant [tex]\(K_{\text{sp}}\)[/tex] is found to be:
[tex]\[
K_{\text{sp}} = 1.49 \times 10^{-14}
\][/tex]
This is the [tex]\(K_{\text{sp}}\)[/tex] value for [tex]\( \text{Ag}_2\text{SO}_3 \)[/tex] to three significant figures.
1. Understand the Dissociation Reaction:
[tex]\(\text{Ag}_2\text{SO}_3\)[/tex] dissociates in water according to the following equation:
[tex]\[
\text{Ag}_2\text{SO}_3(s) \rightleftharpoons 2\text{Ag}^+(aq) + \text{SO}_3^{2-}(aq)
\][/tex]
2. Use the Molar Solubility:
Given that the molar solubility of [tex]\(\text{Ag}_2\text{SO}_3\)[/tex] is [tex]\(1.55 \times 10^{-5} \, \text{M}\)[/tex]. Let's denote this molar solubility as [tex]\( s \)[/tex].
3. Determine Ion Concentrations at Equilibrium:
- Concentration of [tex]\(\text{Ag}^+\)[/tex] ions will be [tex]\(2s\)[/tex] because there are 2 moles of [tex]\(\text{Ag}^+\)[/tex] produced for each mole of [tex]\(\text{Ag}_2\text{SO}_3\)[/tex].
- Concentration of [tex]\(\text{SO}_3^{2-}\)[/tex] ions will be [tex]\(s\)[/tex].
4. Write the Expression for [tex]\(K_{\text{sp}}\)[/tex]:
[tex]\[
K_{\text{sp}} = [\text{Ag}^+]^2 \times [\text{SO}_3^{2-}]
\][/tex]
5. Substitute the Equilibrium Concentrations:
[tex]\[
K_{\text{sp}} = (2s)^2 \times (s) = 4s^3
\][/tex]
6. Calculate [tex]\(K_{\text{sp}}\)[/tex]:
[tex]\[
K_{\text{sp}} = 4 \times (1.55 \times 10^{-5})^3
\][/tex]
7. Result:
After performing the calculation, the solubility product constant [tex]\(K_{\text{sp}}\)[/tex] is found to be:
[tex]\[
K_{\text{sp}} = 1.49 \times 10^{-14}
\][/tex]
This is the [tex]\(K_{\text{sp}}\)[/tex] value for [tex]\( \text{Ag}_2\text{SO}_3 \)[/tex] to three significant figures.
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