To determine which of the given solutions will have a pH of 9 at 298 K, we need to understand the relationship between pH, pOH, and the concentrations of hydrogen ions (H⁺) and hydroxide ions (OH⁻) in the solution.
### Step-by-Step Solution:
1. **Understanding pH and pOH**:
- The pH scale ranges from 0 to 14. A pH of 7 is neutral, below 7 is acidic, and above 7 is basic.
- The relationship between pH and pOH is given by the equation:
\[
pH + pOH = 14
\]
- Therefore, if we want a pH of 9, we can calculate pOH:
\[
pOH = 14 - pH = 14 - 9 = 5
\]
2. **Finding the concentration of OH⁻ ions**:
- The pOH is related to the concentration of hydroxide ions (OH⁻) by the formula:
\[
pOH = -\log[OH⁻]
\]
- Rearranging this gives:
\[
[OH⁻] = 10^{-pOH} = 10^{-5} \, \text{M}
\]
3. **Analyzing the given solutions**:
- **Option 1: HCl solution (10⁻⁹ M)**:
- HCl is a strong acid and will dissociate completely in solution, contributing H⁺ ions.
- The concentration of H⁺ ions from HCl is 10⁻⁹ M, and the concentration of H⁺ ions in pure water is approximately 10⁻⁷ M.
- The effective concentration of H⁺ will be dominated by water, leading to a pH less than 7. Thus, this option is incorrect.
- **Option 2: NaOH solution (10⁻⁵ M)**:
- NaOH is a strong base and will dissociate completely, contributing OH⁻ ions.
- The concentration of OH⁻ ions is 10⁻⁵ M, which is sufficient to yield a pOH of 5.
- Therefore, the pH will be 9, making this option correct.
- **Option 3: KOH solution (10⁻⁹ M)**:
- KOH is also a strong base, but with a concentration of 10⁻⁹ M, the contribution of OH⁻ ions is negligible compared to the concentration of OH⁻ in water (10⁻⁷ M).
- Thus, the effective concentration of OH⁻ will be less than needed to achieve a pH of 9, making this option incorrect.
- **Option 4: Both A and B**:
- Since option A is incorrect and option B is correct, this option is also incorrect.
4. **Conclusion**:
- The only solution that will have a pH of 9 at 298 K is the NaOH solution (option 2).
### Final Answer:
The solution that will have a pH of 9 at 298 K is **NaOH solution (10⁻⁵ M)**.
