Question

Find the degree of hardness of a sample of water containing \(12 \mathrm{mg}\) of magnesium sulphate (molar mass 120) per kg of water. (a) \(0.1 \mathrm{ppm}\) (b) \(1 \mathrm{ppm}\) (c) \(10 \mathrm{ppm}\) (d) \(100 \mathrm{ppm}\)

Short answer

The degree of hardness is 10 ppm (option c).

Step-by-step solution

Understanding the Concept of Hardness

Water hardness is measured in parts per million (ppm), which indicates the number of milligrams of calcium carbonate equivalent in one liter or one kilogram of water. The question provides magnesium sulphate in mg, and we must convert it to an equivalent calcium carbonate amount.

Convert Magnesium Sulphate to Calcium Carbonate Equivalent

To find the equivalent calcium carbonate (CaCO_3), we need to multiply the mass of magnesium sulphate by the ratio of the molar masses of calcium carbonate to magnesium sulphate. The molar mass of CaCO_3 is 100 g/mol and the molar mass of MgSO_4 is 120 g/mol. The formula is: \[\text{Equivalent mass} = \left( \frac{\text{mass of MgSO}_4}{\text{molar mass of MgSO}_4} \right) \times \text{molar mass of CaCO}_3\]Substitute the values: \[\text{Equivalent mass} = \left( \frac{12 \text{ mg}}{120 \text{ g/mol}} \right) \times 100 \text{ g/mol}\]Apply the calculation, remember the conversion from grams to milligrams (1 g = 1000 mg).

Calculate the Equivalent in ppm

First, calculate the equivalent mass: \[\text{Equivalent mass} = \left( \frac{12}{120} \right) \times 100 = 10 \text{ mg}\]Since ppm means mg per kg, and the mass of water is 1 kg, the hardness in ppm is directly equal to the equivalent mass in mg, which is 10 ppm.

Determine the Correct Answer

Based on the calculation, the degree of hardness is 10 ppm. Thus, the correct option from the given choices is (c) \(10\ ppm\).

Key concepts

Magnesium Sulphate

Magnesium sulphate, also known as Epsom salt, is a chemical compound with the formula MgSO₄. In the context of water hardness, magnesium sulphate represents the magnesium ion's contribution to the total hardness. Hardness in water is often caused by the presence of dissolved calcium and magnesium salts.
Understanding magnesium sulphate's role is essential when calculating water hardness because it affects how we determine the equivalent quantity of another compound, such as calcium carbonate, which allows us to express hardness in standardized terms.

Calcium Carbonate Equivalent

Calcium carbonate equivalent (CCE) is a standardized measure used to express hardness in terms of calcium carbonate (CaCO₃). This approach simplifies comparisons across different water samples regardless of the actual minerals in them.
To find the calcium carbonate equivalent for any compound that contributes to hardness, we use the formula: \( \text{Equivalent mass} = \left( \frac{\text{mass of compound}}{\text{molar mass of compound}} \right) \times \text{molar mass of CaCO}_3 \). This conversion ensures consistent measurement by translating other compounds' effects into the common language of calcium carbonate.

Parts Per Million (PPM)

Parts per million (PPM) is a unit used to describe the concentration of one substance relative to a million parts of another. In water hardness, PPM indicates the amount of calcium carbonate equivalent in water.
When measuring water hardness, PPM is crucial because it provides an easy way to express small concentrations. For instance, 10 PPM denotes 10 milligrams of calcium carbonate equivalent per kilogram of water. This unit is beneficial for understanding and communicating water hardness levels in a clear and concise manner.

Molar Mass Conversion

Molar mass conversion plays a pivotal role in the calculation of water hardness. Every compound has a molar mass, which is the mass of one mole of that substance. For magnesium sulphate, it's 120 g/mol, and for calcium carbonate, it's 100 g/mol.
To convert the mass of magnesium sulphate to its calcium carbonate equivalent, we use the molar masses. The conversion formula \( \left( \frac{\text{mass of MgSO}_4}{\text{molar mass of MgSO}_4} \right) \times \text{molar mass of CaCO}_3 \) allows us to adjust the mass to reflect its effect as if it were calcium carbonate, ensuring uniformity in reporting water hardness.