Problem 7 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

\(100 \mathrm{~mL}\) of a water sample contains \(0.81 \mathrm{~g}\) of calcium bicarbonate and \(0.73 \mathrm{~g}\) of magnesium bicarbonate. The hardness of this water sample expressed in terms of equivalents of \(\mathrm{CaCO}_{3}\) is: (molar mass of calcium bicarbonate is \(162 \mathrm{~g} \mathrm{~mol}^{-1}\) and magnesium bicarboante is \(146 \mathrm{~g} \mathrm{~mol}^{-1}\) ) [Main April 8, 2019 (I)] (a) \(5,000 \mathrm{ppm}\) (b) \(1,000 \mathrm{ppm}\) (c) \(100 \mathrm{ppm}\) (d) \(10,000 \mathrm{ppm}\)

Step-by-Step Solution

Answer

The hardness of the water is 1,000 ppm.

Step 1: Calculate Moles of Calcium Bicarbonate

To find the number of moles, we use the formula:\[ \text{moles} = \frac{\text{mass}}{\text{molar mass}} \]For calcium bicarbonate: \[ \text{moles of } (\text{Ca(HCO}_3)_2) = \frac{0.81 \text{ g}}{162 \text{ g/mol}} = 0.005 \text{ moles} \]

Step 2: Calculate Equivalents of Calcium Carbonate from Calcium Bicarbonate

Each mole of calcium bicarbonate is equivalent to one mole of \( \text{CaCO}_3 \). Therefore, the equivalents of \( \text{CaCO}_3 \) are equal to the moles calculated:\[ 0.005 \text{ equivalents of } \text{CaCO}_3 \]

Step 3: Calculate Moles of Magnesium Bicarbonate

Similarly, for magnesium bicarbonate, calculate the moles using the formula for moles:\[ \text{moles of } (\text{Mg(HCO}_3)_2) = \frac{0.73 \text{ g}}{146 \text{ g/mol}} = 0.005 \text{ moles} \]

Step 4: Calculate Equivalents of Calcium Carbonate from Magnesium Bicarbonate

Each mole of magnesium bicarbonate is equivalent to one mole of \( \text{CaCO}_3 \). Therefore, the equivalents of \( \text{CaCO}_3 \) are equal to the moles calculated:\[ 0.005 \text{ equivalents of } \text{CaCO}_3 \]

Step 5: Calculate Total Equivalents of Calcium Carbonate

Add the equivalents from calcium bicarbonate and magnesium bicarbonate to find the total equivalents of \( \text{CaCO}_3 \):\[ 0.005 + 0.005 = 0.01 \text{ equivalents of } \text{CaCO}_3 \]

Step 6: Express Hardness in PPM

To express the hardness in ppm, convert the equivalents of \( \text{CaCO}_3 \) to grams using its molar mass (100 g/mol) and then to ppm (mg/L). Each equivalent corresponds to 100 g/mol:\[ 0.01 \times 100 = 1 \text{ g/L} \]This is equivalent to 1000 ppm (since 1 g/L = 1000 mg/L).

Key Concepts

Calcium Bicarbonate

Calcium bicarbonate is a chemical compound with the formula \(\text{Ca(HCO}_3)_2\). It exists in the form of salts that dissolve in water, making the water "hard."
Calcium bicarbonate is not usually considered a dry compound because it only exists in aqueous solution. This substance forms when calcium, carbonate, carbon dioxide, and water interact in a system.
It plays a major role in the temporary hardness of water, which refers to the water's ability to deposit scale and reduce the effectiveness of soap and detergent.
When calcium bicarbonate breaks down, it releases carbon dioxide, water, and calcium carbonate (\(\text{CaCO}_3\)).

• This decomposition is key during the calculation of water hardness as calcium carbonate.
• One mole of calcium bicarbonate yields one mole of \(\text{CaCO}_3\), making conversion straightforward.

Understanding the behavior of calcium bicarbonate in solution is crucial in environmental chemistry and water treatment.

Magnesium Bicarbonate

Similar to calcium bicarbonate, magnesium bicarbonate \((\text{Mg(HCO}_3)_2)\) is another compound that contributes to the hardness of water.
Its presence in water results from the interaction of carbon dioxide, water, and magnesium carbonate.
Unlike permanent hardness, which is due to sulfates or chlorides of calcium and magnesium, temporary hardness is caused by the presence of bicarbonates.
When heated, magnesium bicarbonate also decomposes, forming magnesium carbonate, carbon dioxide, and water. This reaction is similar to calcium bicarbonate.

• Each mole of magnesium bicarbonate contributes the equivalent of one mole of calcium carbonate (\(\text{CaCO}_3\)).
• It's essential to account for both magnesium and calcium bicarbonates when calculating water hardness.

This is because both compounds convert equivalently to \(\text{CaCO}_3\) in the calculation of water hardness, indicating equal participatory role in making water hard.

Equivalent Weight

The concept of equivalent weight is pivotal in understanding chemical reactions, particularly those involving water hardness.
Equivalent weight refers to the mass of a substance that would react with or replace one mole of hydrogen ions.
For calcium bicarbonate and magnesium bicarbonate, each compound's equivalent weight is equal to its molar mass since they each react with one mole of \(\text{CaCO}_3\).

• Equivalent weight allows chemists to calculate how many grams of a compound produce a mole of a reaction, such as one mole of \(\text{CaCO}_3\).
• For instance, the equivalent weight of \((\text{Ca(HCO}_3)_2)\) and \((\text{Mg(HCO}_3)_2)\) equals their respective molar masses: 162 g/mol and 146 g/mol.

When considering water hardness and reactions in general, equivalent weight simplifies calculations by providing a direct basis for mole conversion when considering a standard reaction.

Parts Per Million (PPM)

Parts per million (PPM) is a unit of measurement used to express the concentration of one substance in a million parts of another.
In the context of water hardness, PPM assesses how much \(\text{CaCO}_3\) is present in a given volume of water.
One PPM is equivalent to 1 mg/L in aqueous solutions, making it a convenient measure for dissolved substances like calcium carbonate.
Expressing water hardness in PPM provides a quantitative figure to communicate water quality.

• This is essential for environments like homes and industries to determine the softness or hardness of their water supply.
• In the exercise, a calculation of PPM involves converting the total equivalents of \(\text{CaCO}_3\) into milligrams per liter, ultimately providing an easily understandable concentration measure.

Understanding PPM and its calculations is vital in pollution control, water treatment, and ensuring safety in the consumption and use of water.