How many `mL` of a solution of concertration `100 mg Co^(2+)` per `mL` is needed to prepare `10 mL` of a solution of concentration `20 mg Co^(2+)` per `mL`.

Explanation

To solve the problem, we will use the dilution equation which relates the concentrations and volumes of two solutions:

\[ M_1 \times V_1 = M_2 \times V_2 \]

Where:
- \( M_1 \) = initial concentration of the solution (100 mg/mL)
- \( V_1 \) = volume of the initial solution we need to find
- \( M_2 \) = final concentration of the solution (20 mg/mL)
- \( V_2 \) = final volume of the solution (10 mL)

### Step-by-step Solution:

1. **Identify the given values:**
- \( M_1 = 100 \, \text{mg/mL} \)
- \( M_2 = 20 \, \text{mg/mL} \)
- \( V_2 = 10 \, \text{mL} \)

2. **Set up the equation using the dilution formula:**
\[ M_1 \times V_1 = M_2 \times V_2 \]
Plugging in the known values:
\[ 100 \, \text{mg/mL} \times V_1 = 20 \, \text{mg/mL} \times 10 \, \text{mL} \]

3. **Calculate the right side of the equation:**
\[ 20 \, \text{mg/mL} \times 10 \, \text{mL} = 200 \, \text{mg} \]

4. **Now, substitute this back into the equation:**
\[ 100 \, \text{mg/mL} \times V_1 = 200 \, \text{mg} \]

5. **Solve for \( V_1 \):**
\[ V_1 = \frac{200 \, \text{mg}}{100 \, \text{mg/mL}} = 2 \, \text{mL} \]

### Final Answer:
You will need **2 mL** of the solution with a concentration of 100 mg Co²⁺ per mL to prepare 10 mL of a solution with a concentration of 20 mg Co²⁺ per mL.