`2.65gm` of a diacidic base was dissolved in 500 ml of water. Twenty millilitres of this solution required 12 ml of `(N)/(6)` HCl solution. Calculate the equivalent mass and molucular mass of the base.

Options

A. 35, 70
B. 24, 48
C. 18, 36
D. 53, 106

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given data
- Mass of the diacidic base = 2.65 g
- Volume of the solution = 500 mL
- Volume of the titrated solution = 20 mL
- Normality of HCl = \( \frac{N}{6} \)
- Volume of HCl used = 12 mL

### Step 2: Calculate the equivalents of HCl used
The equivalents of HCl can be calculated using the formula:
\[ \text{Equivalents of HCl} = \text{Normality} \times \text{Volume (in L)} \]
Substituting the values:
\[ \text{Equivalents of HCl} = \left(\frac{1}{6}\right) \times \left(\frac{12}{1000}\right) = \frac{12}{6000} = \frac{1}{500} \text{ equivalents} \]

### Step 3: Calculate the equivalents of the base in the 20 mL solution
Since the equivalents of the base will equal the equivalents of the acid (HCl) used for neutralization, we have:
\[ \text{Equivalents of Base} = \text{Equivalents of HCl} = \frac{1}{500} \]

### Step 4: Calculate the molarity of the base solution
The molarity of the base solution can be calculated using the formula:
\[ \text{Equivalents of Base} = \text{n-factor} \times \text{Molarity} \times \text{Volume (in L)} \]
For a diacidic base, the n-factor = 2. The volume of the base solution used in the titration is 20 mL, which is 0.020 L. Therefore:
\[ \frac{1}{500} = 2 \times \text{Molarity} \times 0.020 \]
Rearranging gives:
\[ \text{Molarity} = \frac{1}{500} \div (2 \times 0.020) = \frac{1}{500} \div 0.040 = \frac{1}{20} \text{ M} \]

### Step 5: Calculate the number of moles of the base in the 500 mL solution
The total volume of the solution is 500 mL, which is 0.500 L. Therefore, the number of moles of the base in the entire solution is:
\[ \text{Number of moles} = \text{Molarity} \times \text{Volume (in L)} = \frac{1}{20} \times 0.500 = \frac{0.500}{20} = 0.025 \text{ moles} \]

### Step 6: Calculate the molecular mass of the base
Using the formula:
\[ \text{Molecular Mass} = \frac{\text{Mass of Base}}{\text{Number of moles}} \]
Substituting the values:
\[ \text{Molecular Mass} = \frac{2.65}{0.025} = 106 \text{ g/mol} \]

### Step 7: Calculate the equivalent mass of the base
The equivalent mass can be calculated using the formula:
\[ \text{Equivalent Mass} = \frac{\text{Molecular Mass}}{\text{n-factor}} \]
Substituting the values:
\[ \text{Equivalent Mass} = \frac{106}{2} = 53 \text{ g/equiv} \]

### Final Results
- **Molecular Mass of the base** = 106 g/mol
- **Equivalent Mass of the base** = 53 g/equiv