500 gms of water contains `6 times 10^(-3)` gms of dissolved `MgSO_(4)` in it. Calculate the hardness of water in ppm of `CaCO_(3)`?

Options

A. 20 ppm
B. 30 ppm
C. 40 ppm
D. 10 ppm

Explanation

To calculate the hardness of water in ppm of CaCO₃ based on the given amount of dissolved MgSO₄, follow these steps:

### Step 1: Identify the given data
- Mass of water = 500 g
- Mass of dissolved MgSO₄ = 6 x 10^(-3) g

### Step 2: Calculate the molecular weight of MgSO₄
- The molecular weight of MgSO₄ can be calculated as:
- Magnesium (Mg) = 24 g/mol
- Sulfur (S) = 32 g/mol
- Oxygen (O) = 16 g/mol (there are 4 oxygen atoms in MgSO₄)

Thus, the molecular weight of MgSO₄ = 24 + 32 + (16 x 4) = 120 g/mol.

### Step 3: Calculate the equivalent mass of CaCO₃ for the given mass of MgSO₄
- The molecular weight of CaCO₃ is:
- Calcium (Ca) = 40 g/mol
- Carbon (C) = 12 g/mol
- Oxygen (O) = 16 g/mol (there are 3 oxygen atoms in CaCO₃)

Thus, the molecular weight of CaCO₃ = 40 + 12 + (16 x 3) = 100 g/mol.

### Step 4: Set up the proportion to find the equivalent mass of CaCO₃
- Using the ratio of the molecular weights, we can set up the equation:

\[ \frac{6 \times 10^{-3} \text{ g MgSO₄}}{120 \text{ g MgSO₄}} = \frac{x \text{ g CaCO₃}}{100 \text{ g CaCO₃}} \]

### Step 5: Solve for x (mass of CaCO₃)
- Cross-multiplying gives:

\[ x = \frac{6 \times 10^{-3} \times 100}{120} \]

- Simplifying this:

\[ x = \frac{6 \times 10^{-1}}{120} = 5 \times 10^{-3} \text{ g CaCO₃} \]

### Step 6: Calculate the hardness in ppm
- The formula for hardness in ppm is given by:

\[ \text{Hardness (ppm)} = \left( \frac{\text{mass of CaCO₃}}{\text{mass of water sample}} \right) \times 10^6 \]

- Substituting the values:

\[ \text{Hardness (ppm)} = \left( \frac{5 \times 10^{-3} \text{ g}}{500 \text{ g}} \right) \times 10^6 \]

### Step 7: Calculate the final result
- Performing the calculation:

\[ \text{Hardness (ppm)} = \left( \frac{5 \times 10^{-3}}{500} \right) \times 10^6 = \left( 1 \times 10^{-5} \right) \times 10^6 = 10 \text{ ppm} \]

### Final Answer
The hardness of water in ppm of CaCO₃ is **10 ppm**.

---