Question

A sample of water is known to contain \(Mg(HCO_3)_2=7.3\ \text{mg/L\), \(Ca(HCO_3)_2=8.1\ \text{mg/L}\), and \(27.2\ \text{mg/L}\) of \(CaSO_4\). The total hardness associated with water sample in ppm in equivalents of \(CaCO_3\) is \[ (\text{At.wt. }H=1,\ C=12,\ O=16,\ Mg=24,\ Ca=40,\ S=32) \] }

• A. \(20\)
• B. \(25\)
• C. \(30\)
• D. \(40\)

Hint:

To convert hardness into \(CaCO_3\) equivalent, use \(\frac{\text{salt amount}\times 50}{\text{equivalent weight of salt}}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem requires calculating the total hardness of a water sample based on the concentrations of various salts.
Hardness is conventionally expressed in terms of parts per million (ppm) of Calcium Carbonate (\(CaCO_3\)) equivalents.
Step 2: Key Formula or Approach:
Hardness in terms of \(CaCO_3\) equivalent = \(\left( \frac{\text{Mass of hardness causing substance}}{\text{Molecular weight of substance}} \right) \times 100\)
Molecular weights:
\(Mg(HCO_3)_2 = 24 + 2(1 + 12 + 48) = 24 + 122 = 146\) g/mol.
\(Ca(HCO_3)_2 = 40 + 2(1 + 12 + 48) = 40 + 122 = 162\) g/mol.
\(CaSO_4 = 40 + 32 + 64 = 136\) g/mol.
\(CaCO_3 = 40 + 12 + 48 = 100\) g/mol.
Step 3: Detailed Explanation:

Calculating Individual Equivalents:
1. For \(Mg(HCO_3)_2\):
Concentration = 7.3 mg/L.
\(CaCO_3\) equivalent = \(\frac{7.3}{146} \times 100 = 0.05 \times 100 = 5\) ppm.
2. For \(Ca(HCO_3)_2\):
Concentration = 8.1 mg/L.
\(CaCO_3\) equivalent = \(\frac{8.1}{162} \times 100 = 0.05 \times 100 = 5\) ppm.
3. For \(CaSO_4\):
Concentration = 27.2 mg/L.
\(CaCO_3\) equivalent = \(\frac{27.2}{136} \times 100 = 0.2 \times 100 = 20\) ppm.

Calculating Total Hardness:
Total Hardness = Hardness due to \(Mg(HCO_3)_2\) + Hardness due to \(Ca(HCO_3)_2\) + Hardness due to \(CaSO_4\).
Total Hardness = \(5 + 5 + 20 = 30\) ppm.
Note: Both temporary hardness (bicarbonates) and permanent hardness (sulphates) contribute to total hardness.

Step 4: Final Answer:
The total hardness of the water sample expressed in \(CaCO_3\) equivalents is 30 ppm.