Question

If hardness of water sample is \(200 \mathrm{ppm}\), then select the incorrect statement: (a) Mass ratio of \(\mathrm{CaCO}_{3}\) to \(\mathrm{H}_{2} \mathrm{O}\) is \(\frac{0.02}{100}\) (b) Mol ratio of \(\mathrm{CaCO}_{3}\) to \(\mathrm{H}_{2} \mathrm{O}\) is \(3.6 \times 10^{-5}\) (c) Mass of \(\mathrm{CaCO}_{3}\) present in hard water is \(0.2 \mathrm{~g} / \mathrm{L}\) (d) 1 miliequivalent of \(\mathrm{CaCO}_{3}\) present in \(1 \mathrm{~kg}\) of hard water

Short answer

Statement (d) is incorrect because 1 miliequivalent of CaCO3 is 0.05 g and with the given hardness, there are 0.2 g of CaCO3 per L, which equals 4 miliequivalents, not 1 miliequivalent per kg of hard water.

Step-by-step solution

Understanding Water Hardness

Water hardness is often given in terms of parts per million (ppm) of calcium carbonate (CaCO3). A hardness of 200 ppm means that there are 200 mg of CaCO3 per liter of water.

Analyzing Statement (a)

The mass ratio of CaCO3 to H2O given is 0.02 to 100. This means there are 0.02 g of CaCO3 in 100 g of water. Given that 1 L of water roughly equals 1000 g, the statement (a) suggests there would be 0.2 g of CaCO3 per 1000 g of water, which is consistent with the 200 ppm hardness (200 mg/L = 0.2 g/L). Therefore, statement (a) is correct.

Analyzing Statement (b)

To find the mol ratio of CaCO3 to H2O, we first need to convert the mass of CaCO3 to moles. The molar mass of CaCO3 is approximately 100 g/mol. 200 mg (0.2 g) of CaCO3 is 0.002 moles. Since 1 liter of water is roughly 1 kg or 1000 g, the number of moles of H2O is approximately 55.56 (1000 g divided by the molar mass of water which is 18 g/mol). The mol ratio is then 0.002 moles of CaCO3 to 55.56 moles of H2O, which is approximately 3.6 x 10^-5. Therefore, statement (b) is correct.

Analyzing Statement (c)

Statement (c) directly states that the mass of CaCO3 present in hard water is 0.2 g/L, which is exactly what a 200 ppm hardness means. Therefore, statement (c) is correct.

Analyzing Statement (d)

One equivalent of a substance is its gram equivalent weight per liter, which for CaCO3 (molar mass approximately 100 g/mol, and since Ca has a valence of 2, its equivalent weight is 50 g/equiv) is 50 g/equiv. Therefore, 1 miliequivalent would be 0.05 g. Given that statement (c) is correct, and there are 0.2 g of CaCO3 per L, this is 4 miliequivalents of CaCO3 per L, not 1 miliequivalent per kg. Therefore, statement (d) is incorrect.

Key concepts

Parts Per Million (ppm)

When measuring water hardness and other such concentrations, scientists often use a unit called parts per million (ppm). It's a way of expressing very dilute concentrations of substances. One ppm is equivalent to 1 milligram of a substance dissolved in one liter of water. This is because one liter of water has a mass of about one kilogram or one million milligrams, so the ratio is literally one part per million parts by weight.

Understanding ppm is essential in fields like environmental science, chemistry, and water treatment because it allows for a standard way of comparing concentrations irrespective of the volume of water or substance in question. In the context of the exercise we're discussing, a water hardness of 200 ppm implies that there are 200 milligrams of calcium carbonate (CaCO3) per liter of water. This concept helps tie in complex quantitative analysis into an easily understandable format.

Calcium Carbonate (CaCO3)

Calcium carbonate (CaCO3) is a common substance found in rocks as minerals like calcite and aragonite, most notably found in limestone. It is the primary cause of water hardness. As water percolates through limestone deposits, CaCO3 dissolves and introduces calcium (Ca2+) and carbonate (CO32-) ions into the water, increasing its hardness.

The concentration of CaCO3 in water is a crucial factor not just in the understanding of water chemistry but also in various industrial and domestic processes. Excessive hardness can lead to scaling in boilers and heaters, while insufficient CaCO3 levels can cause water to be too soft, leading to its own set of issues. This underlines why measuring the concentration of CaCO3, often in ppm, is essential in managing water systems.

Mole Concept

The mole concept is a fundamental principle in chemistry that serves as a bridge between the macroscopic world we observe and the microscopic world of atoms and molecules. A mole is simply a count of particles, usually atoms or molecules, and is roughly equivalent to Avogadro's number, which is approximately 6.022 x 10^23.

To apply the mole concept to our exercise, when we reference a 'mol ratio' we are discussing the ratio of the number of moles of one substance to the number of moles of another substance. The mole concept allows us to translate mass measurements into a count of molecules, which is critical for understanding and conducting chemical reactions. It effectively enables chemists to 'count' atoms and molecules by weighing them.

Equivalent Weight

Equivalent weight is a concept that was more common in older chemistry texts but remains relevant for understanding reactions, especially in titration and precipitation. It relates to the mass of a compound that combines with or replaces a fixed amount (usually one mole) of hydrogen atoms. In the case of ionic compounds, such as calcium carbonate, the equivalent weight is influenced by the valency of the ions present.

In the problem at hand, one miliequivalent of CaCO3 is actually 0.05 g, as the molar mass is about 100 g/mol and calcium has a valence of 2, hence 50 g/equiv. Understanding equivalent weights is important as it helps in calculations of chemical reactions, particularly when dealing with ions and their charges. It essentially helps in understanding how much of a substance will react with a certain amount of another substance.