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^(-5) M AgNO_(3)` and `10^(-4) M HCl`
D. `10^(-6) M AgNO_(3)` and `10^(-6) M HCl`

Explanation

To determine when a precipitate of AgCl will form upon mixing equal volumes of two solutions, we need to consider the solubility product constant (Ksp) of AgCl, which is given as \( K_{sp} = 10^{-10} \). The condition for the formation of a precipitate is that the ionic product (IP) of the ions in solution must exceed the Ksp value.

### Step-by-Step Solution:

1. **Understand the Precipitation Condition**:
- A precipitate forms when the ionic product (IP) of the ions exceeds the Ksp. Mathematically, this can be expressed as:
\[ IP > K_{sp} \]

2. **Identify the Relevant Ions**:
- For AgCl, the ions involved are \( Ag^+ \) and \( Cl^- \). Therefore, we need to calculate the ionic product:
\[ IP = [Ag^+][Cl^-] \]

3. **Determine the Concentrations**:
- When equal volumes of two solutions are mixed, the concentrations of the ions will be halved. For example, if we have \( [Ag^+] \) from AgNO3 and \( [Cl^-] \) from HCl, we need to find their concentrations before and after mixing.

4. **Example Calculation**:
- Let's assume we are mixing \( 10^{-5} \, M \) AgNO3 (which provides \( Ag^+ \)) and \( 10^{-4} \, M \) HCl (which provides \( Cl^- \)).
- After mixing equal volumes, the concentrations will be:
\[ [Ag^+] = \frac{10^{-5}}{2} = 5 \times 10^{-6} \, M \]
\[ [Cl^-] = \frac{10^{-4}}{2} = 5 \times 10^{-5} \, M \]

5. **Calculate the Ionic Product**:
- Now, calculate the ionic product:
\[ IP = [Ag^+][Cl^-] = (5 \times 10^{-6})(5 \times 10^{-5}) = 25 \times 10^{-11} = 2.5 \times 10^{-10} \]

6. **Compare with Ksp**:
- Now compare the ionic product with the Ksp:
\[ 2.5 \times 10^{-10} > 10^{-10} \]
- Since the ionic product is greater than the Ksp, a precipitate of AgCl will form.

7. **Conclusion**:
- The correct option is the one that provides \( [Ag^+] = 10^{-5} \, M \) and \( [Cl^-] = 10^{-4} \, M \) after mixing, as this results in an ionic product that exceeds the Ksp.