Question

Wastewater from a cement factory contains \(0.25 \mathrm{~g}\) of \(\mathrm{Ca}^{2+}\) ions and \(0.056 \mathrm{~g}\) of \(\mathrm{Mg}^{2+}\) ions per \(100.0 \mathrm{~L}\) of solution. The solution density is \(1.001 \mathrm{~g} / \mathrm{mL}\). Calculate the \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) concentrations in ppm (by mass).

Short answer

\(\text{Ca}^{2+}\) concentration: 2.498 ppm, \(\text{Mg}^{2+}\) concentration: 0.559 ppm

Step-by-step solution

Convert Mass of Wastewater to Kilograms

Since the density of the solution is given as 1.001 g/mL, first convert the volume of wastewater to mass. The volume of wastewater is 100.0 L, which is equivalent to 100,000 mL. Therefore, mass in grams is:\[ \text{mass} = 100,000 \text{ mL} \times 1.001 \text{ g/mL} = 100,100 \text{ g} \]Convert grams to kilograms because ppm is typically expressed on a mass-per-mass basis, and 100,100 g is equal to 100.1 kg.

Calculate \(\text{Ca}^{2+}\) Concentration in ppm

To find the concentration of \(\text{Ca}^{2+}\) ions in ppm:\[ \text{ppm}(\text{Ca}^{2+}) = \frac{\text{mass of } \text{Ca}^{2+} \text{ ions}}{\text{mass of solution}} \times 10^6 \]Substitute the values:\[ \text{ppm}(\text{Ca}^{2+}) = \frac{0.25 \text{ g}}{100,100 \text{ g}} \times 10^6 = 2.498 \text{ ppm} \]

Calculate \(\text{Mg}^{2+}\ Concentration in ppm

To find the concentration of \(\text{Mg}^{2+}\) ions in ppm:\[ \text{ppm}(\text{Mg}^{2+}) = \frac{\text{mass of } \text{Mg}^{2+} \text{ ions}}{\text{mass of solution}} \times 10^6 \]Substitute the values:\[ \text{ppm}(\text{Mg}^{2+}) = \frac{0.056 \text{ g}}{100,100 \text{ g}} \times 10^6 = 0.559 \text{ ppm} \]

Key concepts

Calcium Ion Concentration

Calcium ion concentration refers to the amount of calcium ions \(\text{Ca}^{2+}\) present in a given volume of solution. It's often measured in parts per million (ppm), especially in environmental chemistry contexts like wastewater analysis. When calculating ppm, you consider the mass of the ions in relation to the total mass of the solution.
To find the \(\text{Ca}^{2+}\) concentration in ppm, you need to follow these steps:

• Determine the mass of \(\text{Ca}^{2+}\) ions.
• Calculate the total mass of the solution, considering both solute and solvent.
• Use the formula \[ \text{ppm}(\text{Ca}^{2+}) = \frac{\text{mass of } \text{Ca}^{2+} \text{ ions}}{\text{mass of solution}} \times 10^6 \]

For example, if a solution contains 0.25 grams of calcium ions in 100 liters of water, the mass of the solution (considering a density of 1.001 g/mL) is 100,100 grams. Therefore, the ppm calculation would look like this: \[ \text{ppm}(\text{Ca}^{2+}) = \frac{0.25 \text{ g}}{100,100 \text{ g}} \times 10^6 = 2.498 \text{ ppm} \]Understanding this calculation helps in gauging the level of calcium ion contamination in water, crucial for environmental and public health.

Magnesium Ion Concentration

Magnesium ion concentration measures the presence of magnesium ions \(\text{Mg}^{2+}\) in a solution. Just like \(\text{Ca}^{2+}\), magnesium ion concentration is also often expressed in ppm. This unit makes it easier to understand the amount of ions in large volumes of water.
To calculate \(\text{Mg}^{2+}\) concentration in ppm:

• First, know the mass of the \(\text{Mg}^{2+}\) ions.
• Find the total mass of the solution in which these ions are dissolved.
• Use the formula \[ \text{ppm}(\text{Mg}^{2+}) = \frac{\text{mass of } \text{Mg}^{2+} \text{ ions}}{\text{mass of solution}} \times 10^6 \]

For instance, if a solution contains 0.056 grams of magnesium ions in 100 liters of water, the total mass of the solution is again 100,100 grams (with a density of 1.001 g/mL). Utilizing the ppm formula: \[ \text{ppm}(\text{Mg}^{2+}) = \frac{0.056 \text{ g}}{100,100 \text{ g}} \times 10^6 = 0.559 \text{ ppm} \]This result portrays the low but significant presence of magnesium ions, important for water quality assessments.

Solution Density

Solution density is a crucial concept in chemistry, especially when working with concentrations. Density is the mass per unit volume of a solution and can vary depending on the amount and types of solutes dissolved.
For example, the given problem states that the density of the solution is 1.001 g/mL. This means that every milliliter of the solution weighs 1.001 grams. To convert larger volumes (like the 100 liters of wastewater) into mass, you multiply the volume by the density: \[ \text{mass} = \text{volume} \times \text{density} \]\