The hardness of water sample containing `0*002` mole of magnesium sulphate dissolved in a litre of water is expressed as

Options

A. 20 ppm
B. 200 ppm
C. 2000 ppm
D. 120 ppm.

Explanation

To solve the problem of determining the hardness of a water sample containing 0.002 moles of magnesium sulfate (MgSO4) dissolved in a liter of water, we will follow these steps:

### Step 1: Calculate the mass of MgSO4 in grams
We know that the number of moles of MgSO4 is given as 0.002 moles. The molar mass of MgSO4 can be calculated as follows:
- Magnesium (Mg) = 24 g/mol
- Sulfur (S) = 32 g/mol
- Oxygen (O) = 16 g/mol × 4 = 64 g/mol

So, the molar mass of MgSO4 = 24 + 32 + 64 = 120 g/mol.

Now, we can calculate the mass of MgSO4:
\[ \text{Mass of MgSO4} = \text{Number of moles} \times \text{Molar mass} = 0.002 \, \text{moles} \times 120 \, \text{g/mol} = 0.24 \, \text{grams} = 240 \, \text{milligrams} \]

### Step 2: Convert the mass of MgSO4 to the equivalent mass of CaCO3
To express the hardness in terms of CaCO3, we need to know the relationship between MgSO4 and CaCO3. The equivalence is as follows:
- 1 mole of MgSO4 is equivalent to 1 mole of CaCO3.

The molar mass of CaCO3 is:
- Calcium (Ca) = 40 g/mol
- Carbon (C) = 12 g/mol
- Oxygen (O) = 16 g/mol × 3 = 48 g/mol

So, the molar mass of CaCO3 = 40 + 12 + 48 = 100 g/mol.

Now we can set up a proportion to find the equivalent mass of CaCO3 for the mass of MgSO4 we calculated:
\[ \text{240 mg of MgSO4} \rightarrow \text{X mg of CaCO3} \]
Using the molar masses:
\[ \frac{240 \, \text{mg}}{120 \, \text{g/mol}} = \frac{X \, \text{mg}}{100 \, \text{g/mol}} \]

Cross-multiplying gives:
\[ 240 \times 100 = 120 \times X \]
\[ 24000 = 120X \implies X = \frac{24000}{120} = 200 \, \text{mg} \]

### Step 3: Express the hardness in ppm
Since 1 mg/L is equivalent to 1 ppm, the hardness of the water sample can be expressed as:
\[ \text{Hardness} = 200 \, \text{mg/L} = 200 \, \text{ppm} \]

### Conclusion
The hardness of the water sample containing 0.002 moles of magnesium sulfate dissolved in a liter of water is **200 ppm**.

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