If `0.561 g` of `(KOH)` is dissolved in water to give. `200 mL` of solution at `298 K`. Calculate the concentration of potassium, hydrogen and hydroxyl ions. What is its `pH`?

Explanation

To solve the problem step by step, we will follow the outlined approach:

### Step 1: Calculate the Molecular Weight of KOH
The molecular weight of potassium hydroxide (KOH) can be calculated as follows:
- Potassium (K) = 39 g/mol
- Oxygen (O) = 16 g/mol
- Hydrogen (H) = 1 g/mol

**Molecular Weight of KOH = 39 + 16 + 1 = 56 g/mol**

### Step 2: Calculate the Number of Moles of KOH
To find the number of moles of KOH, we use the formula:
\[ \text{Number of moles} = \frac{\text{Given weight}}{\text{Molecular weight}} \]

Given weight of KOH = 0.561 g

\[ \text{Number of moles of KOH} = \frac{0.561 \text{ g}}{56 \text{ g/mol}} = 0.01002 \text{ mol} \]

### Step 3: Calculate the Volume of the Solution in Liters
The volume of the solution is given as 200 mL. We need to convert this to liters:
\[ \text{Volume in liters} = \frac{200 \text{ mL}}{1000} = 0.2 \text{ L} \]

### Step 4: Calculate the Molarity of KOH
Molarity (M) is defined as the number of moles of solute per liter of solution:
\[ \text{Molarity (M)} = \frac{\text{Number of moles}}{\text{Volume in liters}} \]

\[ \text{Molarity of KOH} = \frac{0.01002 \text{ mol}}{0.2 \text{ L}} = 0.0501 \text{ M} \]

### Step 5: Determine the Concentration of Ions
When KOH dissolves in water, it dissociates completely into potassium ions (K⁺) and hydroxide ions (OH⁻):
\[ \text{KOH} \rightarrow \text{K}^+ + \text{OH}^- \]

Thus, the concentration of K⁺ ions and OH⁻ ions will be equal to the molarity of KOH:
- Concentration of K⁺ = 0.0501 M
- Concentration of OH⁻ = 0.0501 M

### Step 6: Calculate the Concentration of H⁺ Ions
We can use the ion product of water (Kw) to find the concentration of H⁺ ions:
\[ K_w = [H^+][OH^-] \]
At 298 K, \( K_w = 1.0 \times 10^{-14} \).

Rearranging the equation gives:
\[ [H^+] = \frac{K_w}{[OH^-]} \]

Substituting the values:
\[ [H^+] = \frac{1.0 \times 10^{-14}}{0.0501} = 1.99 \times 10^{-13} \text{ M} \]

### Step 7: Calculate the pH of the Solution
The pH is calculated using the formula:
\[ \text{pH} = -\log[H^+] \]

Substituting the value of [H⁺]:
\[ \text{pH} = -\log(1.99 \times 10^{-13}) \]

Using logarithmic properties:
\[ \text{pH} = -\log(1.99) - \log(10^{-13}) \]
\[ \text{pH} = -0.301 - (-13) \]
\[ \text{pH} = 12.699 \]

### Final Results
- Concentration of K⁺ = 0.0501 M
- Concentration of OH⁻ = 0.0501 M
- Concentration of H⁺ = 1.99 × 10⁻¹³ M
- pH = 12.7