How many litres of water must be added to 2 litre aqueous solution of `HCI` with a `pH` of 1 to create an aqueous solution with `pH` of 2 ?

Options

A. `20L`
B. `0.9L`
C. `2L`
D. `18L`

Explanation

To solve the problem of how many liters of water must be added to a 2-liter aqueous solution of HCl with a pH of 1 to create an aqueous solution with a pH of 2, we can follow these steps:

### Step 1: Determine the concentration of H⁺ ions in the initial solution (pH = 1)
The pH is related to the concentration of hydrogen ions (H⁺) by the formula:
\[ \text{pH} = -\log[H^+] \]

For pH = 1:
\[ [H^+] = 10^{-1} \, \text{M} = 0.1 \, \text{M} \]

### Step 2: Calculate the number of moles of H⁺ in the initial solution
Using the concentration and the volume of the solution, we can find the number of moles of H⁺:
\[ \text{Moles of } H^+ = [H^+] \times \text{Volume} \]
\[ \text{Moles of } H^+ = 0.1 \, \text{M} \times 2 \, \text{L} = 0.2 \, \text{moles} \]

### Step 3: Determine the concentration of H⁺ ions in the final solution (pH = 2)
For pH = 2:
\[ [H^+] = 10^{-2} \, \text{M} = 0.01 \, \text{M} \]

### Step 4: Set up the equation using the dilution formula
We can use the dilution equation \( M_1V_1 = M_2V_2 \), where:
- \( M_1 \) = initial concentration of H⁺ = 0.1 M
- \( V_1 \) = initial volume of solution = 2 L
- \( M_2 \) = final concentration of H⁺ = 0.01 M
- \( V_2 \) = final volume of solution (which we need to find)

Substituting the known values into the equation:
\[ 0.1 \, \text{M} \times 2 \, \text{L} = 0.01 \, \text{M} \times V_2 \]

### Step 5: Solve for \( V_2 \)
\[ 0.2 = 0.01 \times V_2 \]
\[ V_2 = \frac{0.2}{0.01} = 20 \, \text{L} \]

### Step 6: Calculate the volume of water to be added
The final volume of the solution is 20 L, and the initial volume is 2 L. Therefore, the volume of water to be added is:
\[ \text{Volume of water} = V_2 - V_1 = 20 \, \text{L} - 2 \, \text{L} = 18 \, \text{L} \]

### Final Answer
Thus, **18 liters of water must be added** to the 2-liter aqueous solution of HCl to create an aqueous solution with a pH of 2.

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