A 1 litre solution of pH=1 diluted upto 10 times.What volume of a solutions with pH =2 is to be added in diluted solution so that final pH remains '2'.

Options

A. 1 litre
B. 10 litre
C. 100 litre
D. 25 litre

Explanation

To solve the problem step by step, we will analyze the dilution and the addition of the solution with pH = 2.

### Step 1: Understand the initial solution
The initial solution has a pH of 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 concentration after dilution
The solution is diluted 10 times. Therefore, the new concentration of hydrogen ions after dilution is:
\[ [H^+]_{\text{diluted}} = \frac{0.1 \text{ M}}{10} = 0.01 \text{ M} \]
This corresponds to a pH of:
\[ \text{pH} = -\log(0.01) = 2 \]

### Step 3: Set up the equation for the final solution
We need to add a volume \(V\) of a solution with pH = 2 to this diluted solution. The concentration of hydrogen ions in this solution is:
\[ [H^+]_{\text{added}} = 10^{-2} \text{ M} = 0.01 \text{ M} \]

### Step 4: Use the dilution equation
Let \(V_1\) be the volume of the diluted solution (1 L) and \(V_2\) be the volume of the solution with pH = 2 that we need to add. The equation for the final concentration of hydrogen ions after mixing is given by:
\[ \frac{[H^+]_{\text{diluted}} \cdot V_1 + [H^+]_{\text{added}} \cdot V_2}{V_1 + V_2} = [H^+]_{\text{final}} \]
Substituting the known values:
\[ \frac{(0.01 \text{ M}) \cdot (1 \text{ L}) + (0.01 \text{ M}) \cdot V}{1 + V} = 0.01 \text{ M} \]

### Step 5: Solve the equation
Substituting the values into the equation:
\[ \frac{0.01 + 0.01V}{1 + V} = 0.01 \]
Cross-multiplying gives:
\[ 0.01 + 0.01V = 0.01(1 + V) \]
Expanding the right side:
\[ 0.01 + 0.01V = 0.01 + 0.01V \]
This equation holds true for any value of \(V\), meaning that any volume of the solution with pH = 2 can be added, and the final pH will still be 2.

### Conclusion
Thus, the volume of the solution with pH = 2 that can be added to the diluted solution while maintaining a final pH of 2 can be any volume.