`100 mL` of a sample of hard water requires `25.1 mL` of `0.02 N H_(2)SO_(4)` for complete reaction, The hardness of water ( density `1g//mL)` is:

Options

A. 200 ppm
B. 250 ppm
C. 251 ppm
D. 258 ppm

Explanation

To calculate the hardness of water based on the given data, we will follow these steps:

### Step 1: Calculate the milliequivalents of H₂SO₄ used
We know that:
\[ \text{Milliequivalents} = \text{Volume (mL)} \times \text{Normality (N)} \]
Given:
- Volume of H₂SO₄ = 25.1 mL
- Normality of H₂SO₄ = 0.02 N

Calculating the milliequivalents:
\[ \text{Milliequivalents of H₂SO₄} = 25.1 \, \text{mL} \times 0.02 \, \text{N} = 0.502 \, \text{mEq} \]

### Step 2: Relate the milliequivalents of H₂SO₄ to the hardness of water
The milliequivalents of hardness in water can be considered equal to the milliequivalents of H₂SO₄ used:
\[ \text{Milliequivalents of hardness} = 0.502 \, \text{mEq} \]

### Step 3: Calculate the equivalent weight of calcium carbonate (CaCO₃)
The equivalent weight of CaCO₃ can be calculated using the formula:
\[ \text{Equivalent weight} = \frac{\text{Molar mass}}{n} \]
Where:
- Molar mass of CaCO₃ = 100 g/mol
- n (number of replaceable H⁺ ions) = 2

Thus,
\[ \text{Equivalent weight of CaCO₃} = \frac{100}{2} = 50 \, \text{g/equiv} \]

### Step 4: Calculate the weight of CaCO₃ corresponding to the hardness
Using the formula:
\[ \text{Weight} = \text{Milliequivalents} \times \text{Equivalent weight} \]
Convert milliequivalents to equivalents:
\[ 0.502 \, \text{mEq} = 0.502 \times 10^{-3} \, \text{Eq} \]
Now, calculate the weight of CaCO₃:
\[ \text{Weight of CaCO₃} = 0.502 \times 10^{-3} \, \text{Eq} \times 50 \, \text{g/equiv} = 0.0251 \, \text{g} \]

### Step 5: Calculate the hardness in ppm
Hardness in ppm can be calculated using the formula:
\[ \text{Hardness (ppm)} = \left( \frac{\text{Weight of CaCO₃ (g)}}{\text{Volume of water (L)}} \right) \times 10^6 \]
Given that the volume of water is 100 mL (or 0.1 L):
\[ \text{Hardness (ppm)} = \left( \frac{0.0251 \, \text{g}}{0.1 \, \text{L}} \right) \times 10^6 = 251 \, \text{ppm} \]

### Final Answer
The hardness of the water sample is **251 ppm**.

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