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Calculate the solubility at [tex]$25^{\circ} C$[/tex] of AgCl in pure water and in a [tex]$0.0170 M \, \text{AgNO}_3$[/tex] solution. You'll find [tex][tex]$K_{sp}$[/tex][/tex] data in the ALEKS Data tab.

Round both of your answers to 2 significant digits.

Solubility in pure water:
[tex]\square \frac{ \text{g} }{ \text{L} }[/tex]

Solubility in [tex]$0.0170 M \, \text{AgNO}_3$[/tex] solution:
[tex]\square \frac{ \text{g} }{ \text{L} }[/tex]


Sagot :

Sure, let's go through the steps to calculate the solubility of AgCl in both pure water and a 0.0170 M AgNO_3 solution.

### Solubility in Pure Water

1. Identify the Solubility Product Constant (K_{sp}):
The solubility product constant for AgCl is given as [tex]\( K_{sp} = 1.77 \times 10^{-10} \)[/tex].

2. Set Up the Equilibrium Expression:
For the dissociation of AgCl in water:
[tex]\[ \text{AgCl}_{(s)} \leftrightarrow \text{Ag}^{+}_{(aq)} + \text{Cl}^{-}_{(aq)} \][/tex]
Let [tex]\( s \)[/tex] be the solubility of AgCl in [tex]\( \text{mol/L} \)[/tex].
At equilibrium, the concentrations of Ag⁺ and Cl⁻ ions will both be [tex]\( s \)[/tex].

3. Write the K_{sp} Expression:
[tex]\[ K_{sp} = [\text{Ag}^{+}] [\text{Cl}^{-}] = s \cdot s = s^2 \][/tex]

4. Solve for [tex]\( s \)[/tex]:
[tex]\[ s = \sqrt{K_{sp}} = \sqrt{1.77 \times 10^{-10}} = 1.33 \times 10^{-5} \, \text{mol/L} \][/tex]

5. Convert Solubility to g/L:
The molar mass of AgCl is 143.32 g/mol.
[tex]\[ \text{Solubility} = s \times \text{molar mass of AgCl} = 1.33 \times 10^{-5} \, \text{mol/L} \times 143.32 \, \text{g/mol} = 0.001905 \, \text{g/L} \][/tex]

6. Round to 2 Significant Digits:
[tex]\[ \text{Solubility in pure water} = 0.0 \, \text{g/L} \][/tex]

### Solubility in 0.0170 M AgNO_3 Solution

1. Set Up the Equilibrium Expression:
In the presence of AgNO_3, the concentration of [tex]\( \text{Ag}^+ \)[/tex] ions is already 0.0170 M.

2. Write the K_{sp} Expression:
[tex]\[ K_{sp} = [\text{Ag}^{+}] [\text{Cl}^{-}] \][/tex]
Given [Ag⁺] = 0.0170 M.
Let [tex]\( s' \)[/tex] be the solubility of AgCl in the AgNO_3 solution, representing the concentration of [tex]\( \text{Cl}^- \)[/tex].

3. Solve for [tex]\( [\text{Cl}^-] \)[/tex]:
[tex]\[ [\text{Cl}^-] = \frac{K_{sp}}{[\text{Ag}^{+}]} = \frac{1.77 \times 10^{-10}}{0.0170} = 1.04 \times 10^{-8} \, \text{mol/L} \][/tex]

4. Convert Solubility to g/L:
[tex]\[ \text{Solubility} = s' \times \text{molar mass of AgCl} = 1.04 \times 10^{-8} \, \text{mol/L} \times 143.32 \, \text{g/mol} = 0.000001492 \, \text{g/L} \][/tex]

5. Round to 2 Significant Digits:
[tex]\[ \text{Solubility in 0.0170 M AgNO_3 solution} = 0.0 \, \text{g/L} \][/tex]

To summarize:
- The solubility of AgCl in pure water is [tex]\( \boxed{0.0 \, \frac{g}{L}} \)[/tex].
- The solubility of AgCl in a 0.0170 M AgNO_3 solution is [tex]\( \boxed{0.0 \, \frac{g}{L}} \)[/tex].
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