Calculation of Total Hardness of Water

To determine the total hardness of water, we need to consider the presence of magnesium sulfate (MgSO4) and calcium bicarbonate (Ca(HCO3)2) in the given sample.

Step 1: Convert mg to g
We start by converting the given quantities from milligrams (mg) to grams (g) to ensure consistent units.

Given:
- Mass of water = 5 kg = 5000 g
- Mass of MgSO4 = 24 mg = 0.024 g
- Mass of Ca(HCO3)2 = 32.4 mg = 0.0324 g

Step 2: Determine the molar masses
Next, we need to determine the molar masses of MgSO4 and Ca(HCO3)2 in order to calculate the number of moles present.

The molar masses are:
- MgSO4: 24.31 g/mol (Molar mass of Mg) + 32.06 g/mol (Molar mass of S) + 4 * 16.00 g/mol (Molar mass of O) = 120.37 g/mol
- Ca(HCO3)2: 40.08 g/mol (Molar mass of Ca) + 2 * (1.01 g/mol (Molar mass of H) + 12.01 g/mol (Molar mass of C) + 3 * 16.00 g/mol (Molar mass of O)) = 162.12 g/mol

Step 3: Calculate the number of moles
The number of moles of each compound can be calculated using the formula:

Number of moles = Mass of compound / Molar mass of compound

For MgSO4:
Number of moles of MgSO4 = 0.024 g / 120.37 g/mol = 0.0001995 mol

For Ca(HCO3)2:
Number of moles of Ca(HCO3)2 = 0.0324 g / 162.12 g/mol = 0.0001999 mol

Step 4: Determine the hardness
The hardness of water is typically expressed in terms of calcium carbonate (CaCO3) equivalents.

For every mole of MgSO4, there are two moles of positive charges, equivalent to two moles of CaCO3.
For every mole of Ca(HCO3)2, there is one mole of positive charges, equivalent to one mole of CaCO3.

Therefore, the total number of moles of CaCO3 equivalents in the given sample can be calculated as follows:

Total moles of CaCO3 equivalents = 2 * (Number of moles of MgSO4) + Number of moles of Ca(HCO3)2
Total moles of CaCO3 equivalents = 2 * 0.0001995 mol + 0.0001999 mol = 0.000799 mol

Step 5: Calculate the total hardness
Finally, we can calculate the total hardness of water by multiplying the total moles of CaCO3 equivalents by the molar mass of CaCO3.

Total hardness = Total moles of CaCO3 equivalents * Molar mass of CaCO3

The molar mass