Question

In a water sample, the concentrations of Ca\(^{2+}\), Mg\(^{2+}\) and HCO\(_3^-\) are 100 mg/L, 36 mg/L and 122 mg/L, respectively. The atomic masses of various elements are: Ca = 40, Mg = 24, H = 1, C = 12, O = 16.
The total hardness and the temporary hardness in the water sample (in mg/L as CaCO\(_3\)) will

• A. 400 and 100, respectively.
• B. 400 and 300, respectively.
• C. 500 and 100, respectively.
• D. 800 and 200, respectively.

Hint:

Total hardness is the sum of the contributions from calcium and magnesium, while temporary hardness is caused by bicarbonates.

Answer 1

The Correct Option is A

To calculate the total hardness and temporary hardness, we use the following formulas: 1. Temporary hardness is caused by the presence of bicarbonates (HCO\(_3^-\)). It can be calculated from the concentration of HCO\(_3^-\) using the formula: \[ \text{Temporary hardness} = \frac{[\text{HCO}_3^-] \times 50}{\text{Molar mass of HCO}_3^-} \] Where the molar mass of HCO\(_3^-\) is \(1 + 12 + 3 \times 16 = 61 \, \text{g/mol}\). Given that the concentration of HCO\(_3^-\) is 122 mg/L, the temporary hardness is: \[ \text{Temporary hardness} = \frac{122 \times 50}{61} = 100 \, \text{mg/L as CaCO}_3. \] 2. Total hardness is the sum of the hardness contributions from both calcium (Ca\(^{2+}\)) and magnesium (Mg\(^{2+}\)). We calculate the hardness from these ions as follows: For Ca\(^{2+}\): \[ \text{Hardness from Ca}^{2+} = \frac{[\text{Ca}^{2+}] \times 100}{\text{Molar mass of Ca}} \] Where the molar mass of Ca is 40 g/mol. Given that the concentration of Ca\(^{2+}\) is 100 mg/L: \[ \text{Hardness from Ca}^{2+} = \frac{100 \times 100}{40} = 250 \, \text{mg/L as CaCO}_3. \] For Mg\(^{2+}\): \[ \text{Hardness from Mg}^{2+} = \frac{[\text{Mg}^{2+}] \times 100}{\text{Molar mass of Mg}} \] Where the molar mass of Mg is 24 g/mol. Given that the concentration of Mg\(^{2+}\) is 36 mg/L: \[ \text{Hardness from Mg}^{2+} = \frac{36 \times 100}{24} = 150 \, \text{mg/L as CaCO}_3. \] Thus, the total hardness is: \[ \text{Total hardness} = 250 + 150 = 400 \, \text{mg/L as CaCO}_3. \] Thus, the total hardness is 400 mg/L and the temporary hardness is 100 mg/L, corresponding to option (A).