A precipitate of AgCl is formed when equal volumes of the following are mixed `[K_(sp) "for" AgCl=10^(-10)]`

Options

A. `10^(-4) M AgNO_(3)` "and" `10^(-7)M HCl`
B. `10^(-5) M AgNO_(3)` "and" `10^(-6)M HCl`
C. `10^(-4) M AgNO_(3)` "and" `10^(-4)M HCl`
D. `10^(-4) M AgNO_(3)` "and" `10^(-4)M HCl`

Explanation

To solve the problem of precipitate formation of AgCl when mixing equal volumes of AgNO3 and HCl, we will follow these steps:

### Step 1: Understand the Condition for Precipitation
A precipitate forms when the reaction quotient (Qsp) exceeds the solubility product constant (Ksp). For AgCl, the Ksp is given as \(10^{-10}\).

### Step 2: Write the Relevant Equations
The dissolution of AgCl can be represented as:
\[ \text{AgCl (s)} \rightleftharpoons \text{Ag}^+ (aq) + \text{Cl}^- (aq) \]

The solubility product expression is:
\[ K_{sp} = [\text{Ag}^+][\text{Cl}^-] \]

### Step 3: Calculate the Initial Concentrations
We are given:
- Concentration of AgNO3 = \(10^{-5} \, \text{M}\)
- Concentration of HCl = \(10^{-4} \, \text{M}\)

When equal volumes of these solutions are mixed, the concentrations will change due to dilution. If we assume we mix 1 L of each solution, the new concentrations will be:
- Concentration of AgNO3 after mixing:
\[ [\text{Ag}^+] = \frac{10^{-5}}{2} = 5 \times 10^{-6} \, \text{M} \]
- Concentration of HCl after mixing:
\[ [\text{Cl}^-] = \frac{10^{-4}}{2} = 5 \times 10^{-5} \, \text{M} \]

### Step 4: Calculate Qsp
Now we can calculate Qsp using the concentrations found:
\[ Q_{sp} = [\text{Ag}^+][\text{Cl}^-] = (5 \times 10^{-6})(5 \times 10^{-5}) \]
\[ Q_{sp} = 25 \times 10^{-11} = 2.5 \times 10^{-10} \]

### Step 5: Compare Qsp with Ksp
Now we compare Qsp with Ksp:
- \( Q_{sp} = 2.5 \times 10^{-10} \)
- \( K_{sp} = 10^{-10} \)

Since \( Q_{sp} > K_{sp} \), a precipitate of AgCl will form.

### Conclusion
The formation of the precipitate indicates that the solution is supersaturated with respect to AgCl, confirming that AgCl will precipitate out of the solution.