Question

Lead is a toxic metal that affects the central nervous system. A Pb- contaminated water sample contains \(0.0011 \% \mathrm{~Pb}\) by mass. How much of the water (in mL) contains \(150 \mathrm{mg}\) of \(\mathrm{Pb}\) ? (Assume a density of \(1.0 \mathrm{~g} / \mathrm{mL} .)\)

Short answer

To find the volume of water containing 150 mg of Pb, calculate the mass of water by dividing 150 mg of Pb by 0.000011, then convert the mass from mg to g, and use the density to find the volume in mL: Volume of water is 13636.36 mL.

Step-by-step solution

Convert Percentage to Mass Fraction

First, we need to convert the percentage of lead by mass into a decimal fraction. This is done by dividing the percentage value by 100. Therefore, \(0.0011\% \mathrm{~Pb}\) in decimal form is: \(0.0011 / 100 = 0.000011\).

Calculate the Mass of Water

Next, in order to find the mass of water containing 150 mg of Pb, we use the mass fraction of Pb in the water. Assuming that the mass of Pb is \(150 \mathrm{mg}\), we find the mass of the water by dividing the mass of Pb by the mass fraction: \(\frac{150 \mathrm{mg}}{0.000011}\).

Convert Mass to Volume

After finding the mass of water that contains 150 mg of Pb, we now need to convert this mass into volume. Since the density of water is given as \(1.0 \mathrm{~g} / \mathrm{mL}\), we can directly convert the mass of water in grams to the volume in milliliters. As 1 g is equivalent to 1 mL, we keep the numerical value the same and express it in mL.

Key concepts

Understanding Mass Fraction Calculation

Grasping the concept of mass fraction calculation is crucial when dealing with the concentration of substances in a mixture. Mass fraction is a way of expressing how much of one substance (the solute) is present in a total mixture. Essentially, it tells you the portion of the total mass represented by a particular component.

In chemistry, the mass fraction is often used to describe the concentration of an impurity or a solute within a solution. To calculate the mass fraction, one would simply divide the mass of the solute by the total mass of the solution. For example, if a solution is made by dissolving a certain mass of solute into a mass of solvent, the mass fraction of the solute would be the mass of the solute divided by the sum of the mass of solute plus the mass of solvent.

When applied to lead contamination problems, understanding the mass fraction of lead in water can help determine how much water is tainted by a specific mass of lead. This calculation is vital to ensure safety and compliance with health regulations.

Converting Percentages to Decimals

Often in science and mathematics, we work with percentages. However, most calculations require these values to be in decimal form. Converting percentages to decimals is a straightforward process that involves dividing the percentage by 100.

For instance, if you have a percentage, say 0.0011%, divide it by 100 to get the decimal equivalent, which would be 0.000011. This decimal form is much easier to work with in mathematical equations, especially when determining the concentration of a substance in a solution, such as the amount of lead in a water sample.

Why is this important?

To effectively use the mass fraction in calculations, the percentage form of the concentration must first be converted to a decimal. This simple conversion enables further calculations, such as determining how much water contains a certain amount of lead, thereby simplifying the process of analyzing mixtures and solutions.

Mass to Volume Conversion

Mass to volume conversion is a common practice in chemistry and is necessary when we want to relate the amount of a substance to its physical space, which is especially useful with fluids. Given a known density—mass per unit volume (typically grams per milliliter or kilograms per liter)—you can convert between mass and volume easily.

In our example involving lead-contaminated water, we use this conversion process to find out how much water, by volume, is contaminated with a specified mass of lead. Since the density of water is approximately 1 gram per milliliter, the mass of water (in grams) will be numerically equivalent to the volume of water (in milliliters).

Knowing how to perform this conversion is vital for a variety of applications, like dosing medication, engineering, and in environmental studies to evaluate pollution levels and execute remedial measures accordingly. It's a fundamental skill for students and professionals in the sciences.