Step 1: Understanding the Concept:
This problem requires calculating the pH of a strong base (NaOH) solution. The process involves finding the molar concentration of the base, determining the hydroxide ion concentration [OH$^-$], calculating the pOH, and finally finding the pH using the relationship between pH and pOH.
Step 2: Key Formula or Approach:
1. Calculate the number of moles of solute: moles = mass / molar mass.
2. Calculate the molarity of the solution: Molarity = moles / volume (in L).
3. For a strong base like NaOH, $[OH^-]$ = Molarity of the base.
4. Calculate pOH: pOH = $-\log[OH^-]$.
5. Calculate pH: pH + pOH = 14 (at 25$^\circ$C).
Step 3: Detailed Explanation:
Given:
- Mass of NaOH = 4 g.
- Volume of solution = 1.0 L.
- Molar mass of NaOH = 23 (Na) + 16 (O) + 1 (H) = 40 g/mol.
1. Calculate moles of NaOH:
\[ \text{moles} = \frac{4 \text{ g}}{40 \text{ g/mol}} = 0.1 \text{ mol} \]
2. Calculate Molarity of NaOH solution:
\[ \text{Molarity} = \frac{0.1 \text{ mol}}{1.0 \text{ L}} = 0.1 \text{ M} \]
3. Determine [OH$^-$]:
NaOH is a strong base, so it dissociates completely in water: NaOH $\rightarrow$ Na$^+$ + OH$^-$.
Therefore, the concentration of hydroxide ions is equal to the concentration of the NaOH solution.
\[ [OH^-] = 0.1 \text{ M} = 10^{-1} \text{ M} \]
4. Calculate pOH:
\[ \text{pOH} = -\log[OH^-] = -\log(10^{-1}) = -(-1) = 1 \]
5. Calculate pH:
\[ \text{pH} = 14 - \text{pOH} = 14 - 1 = 13 \]
Step 4: Final Answer:
The pH of the solution is 13. Therefore, option (A) is correct.