Step 1: Understanding the Concept:
This problem involves the dilution of a solution. During dilution, the amount (moles) of the solute remains constant; only the volume of the solvent is increased, which decreases the concentration.
Step 2: Key Formula or Approach:
The principle of dilution is expressed by the formula:
\[ M_1V_1 = M_2V_2 \]
where:
- $M_1$ and $V_1$ are the molarity and volume of the initial (concentrated) solution.
- $M_2$ and $V_2$ are the molarity and volume of the final (diluted) solution.
The number of moles of solute, $n = M \times V$, is constant before and after dilution.
Step 3: Detailed Explanation:
Given:
- Initial molarity, $M_1 = 0.1$ M.
- Initial volume, $V_1 = x$ ml.
- Final molarity, $M_2 = 0.01$ M.
- Final volume, $V_2 = 250$ ml.
We can plug these values into the dilution formula:
\[ M_1V_1 = M_2V_2 \]
\[ (0.1 \text{ M}) \times (x \text{ ml}) = (0.01 \text{ M}) \times (250 \text{ ml}) \]
Now, solve for x:
\[ 0.1 \times x = 0.01 \times 250 \]
\[ 0.1x = 2.5 \]
\[ x = \frac{2.5}{0.1} \]
\[ x = 25 \]
The value of x is 25 ml.
Step 4: Final Answer:
The initial volume of the NaOH solution required is 25 ml. Therefore, option (B) is correct.