Which of the following expression is/are true ?

Options

A. `[H^(+)]=[OH^(-)]=sqrt(K_(w))` for a netural solution
B. `[OH^(-)]ltsqrt(K_(w))`
C. pH+pOH=14 at all temperature
D. `[H^(-)]=10^(-7)` M for a netural solution at `25^(@)C`

Answer

To determine which of the given expressions are true regarding ionic equilibrium, we will analyze each expression step-by-step.

### Step 1: Analyze the first expression
**Expression:** \( [H^+] = [OH^-] = \sqrt{K_w} \) for a neutral solution.

- In a neutral solution, the concentration of hydrogen ions \([H^+]\) is equal to the concentration of hydroxide ions \([OH^-]\).
- The ion product of water \(K_w\) at 25°C is \(1.0 \times 10^{-14}\).
- Therefore, \([H^+] = [OH^-] = \sqrt{K_w} = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7}\) M.
- This expression is **true** for neutral solutions.

### Step 2: Analyze the second expression
**Expression:** \( [OH^-] < \sqrt{K_w} \)

- This expression is not universally true. In neutral solutions, \([OH^-]\) is equal to \(\sqrt{K_w}\).
- In acidic solutions, \([OH^-]\) will be less than \(\sqrt{K_w}\), but in neutral solutions, they are equal.
- Therefore, this expression is **false**.

### Step 3: Analyze the third expression
**Expression:** \( pH + pOH = 4 \) at 24°C.

- The relationship \(pH + pOH = 14\) holds true at 25°C, not at 24°C.
- At any temperature, the sum of \(pH\) and \(pOH\) is not necessarily equal to 4.
- Therefore, this expression is **false**.

### Step 4: Analyze the fourth expression
**Expression:** \( [H^+] = 10^{-7} \) for a neutral solution at 25°C.

- As established earlier, for a neutral solution at 25°C, \([H^+] = 10^{-7}\) M.
- This expression is **true**.

### Conclusion
Based on the analysis:
- The first expression is **true**.
- The second expression is **false**.
- The third expression is **false**.
- The fourth expression is **true**.

### Final Answer
The true expressions are:
1. \( [H^+] = [OH^-] = \sqrt{K_w} \) for a neutral solution.
2. \( [H^+] = 10^{-7} \) for a neutral solution at 25°C.