Problem 97 - Chemistry The Central Science

Question

The maximum allowable concentration of lead in drinking water is \(9.0 \mathrm{ppb}\). (a) Calculate the molarity of lead in a 9.0-ppb solution. What assumption did you have to make in your calculation? (b) How many grams of lead are in a swimming pool containing 9.0 ppb lead in \(60 \mathrm{~m}^{3}\) of water?

Step-by-Step Solution

Answer

The molarity of lead in a 9.0 ppb solution is \(4.35 * 10^{-11} \mathrm{M}\). The assumption made was that the density of the solution is approximately the same as that of pure water. There are 0.00054 grams of lead in a swimming pool containing 9.0 ppb lead in 60 m^3 of water.

Step 1: (a) Calculate the molarity of lead in 9.0 ppb water

First, let's convert ppb (parts per billion) into moles per liter (molarity): ppb = parts per billion = mass of lead (g) / volume of water (L) * 1,000,000,000 g Now, we will solve for the mass of lead in grams and then convert this to moles. Finally, we will divide the moles of lead by the volume of water to get the molarity. 1. Convert ppb to mass of lead (g): \(9.0 \mathrm{ppb} = \frac{mass\: of\: lead}{volume\: of\: water} * 1,000,000,000\) Assuming 1 L of water: \(mass\: of\: lead = 9.0 * 10^{-9} \:g\) 2. Convert mass of lead to moles using the molar mass of lead (Pb): The molar mass of lead (Pb) is 207.2 g/mol. \(moles\: of\: lead = \frac{mass\: of\: lead}{molar\: mass\: of\: Pb}\) \(moles\: of\: lead = \frac{9.0 * 10^{-9}\: g}{207.2\: g/mol} = 4.35 * 10^{-11} \: mol\) 3. Find the molarity of lead in water: Since we assumed 1 L of water, molarity of lead = moles of lead / volume of water molarity of lead = \(4.35 * 10^{-11} \: mol / 1\: L\) The molarity of lead in 9.0 ppb solution is \(4.35 * 10^{-11} \mathrm{M}\). The assumption made was that the density of the solution is approximately the same as that of pure water, because the amount of lead is extremely small compared to the amount of water.

Step 2: (b) Calculate the grams of lead in a swimming pool

We are given the volume of the pool as 60 m^3 and the concentration of lead as 9.0 ppb. First, we will convert the volume of the pool from cubic meters to liters, and then use the given concentration to find the mass of lead present in the pool. 1. Convert the volume of the pool to liters: To convert volume from cubic meters to liters, multiply by 1000. \(volume\: of\: water = 60\: m^3 * 1,000\: L/m^3 = 60,000\: L\) 2. Calculate mass of lead in grams in the swimming pool: To find the mass of lead, we will multiply the volume of water with the concentration of lead in ppb, and then divide by one billion. \(mass\: of\: lead = \frac{volume\: of\: water * 9.0\: ppb}{1,000,000,000}\) \(mass\: of\: lead = \frac{60,000\: L * 9.0\: ppb}{1,000,000,000} = 0.00054\: g\) There are 0.00054 grams of lead in a swimming pool containing 9.0 ppb lead in 60 m^3 of water.

Key Concepts

Understanding Parts Per Billion (ppb)

When it comes to measuring the concentration of substances within a solution, parts per billion (ppb) can often be a go-to unit for expressing extremely low concentrations, such as pollutants in the environment, like lead in drinking water. To visualize what ppb means, imagine dividing a substance evenly into a billion parts; one part per billion represents one part of the substance in a billion parts of the total solution.

Calculating ppb involves knowing both the mass of the substance in question and the volume of the solution it's in. The formula for ppb is \[\begin{equation} ppb = \frac{\text{mass of substance (g)}}{\text{volume of solution (L)}} \times 1,000,000,000 \end{equation}\]To provide clarity, when tasked with converting ppb to a more commonly-used concentration metric like molarity, we first convert the ppb value to mass, and then to moles which can then be related to volume to get molarity.

Molarity: A Measure of Concentration in Chemistry

Molarity is one of the most fundamental units in chemistry for quantifying the concentration of a substance in a solution. Expressed in moles per liter (M), molarity is calculated by the number of moles of solute divided by the volume of solution in liters. The basic formula for molarity (\( M \)) is:\[\begin{equation} M = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \end{equation}\]For a student trying to find the molarity of a dilute substance like lead in water, the key steps would include converting the given ppb concentration to mass in grams, then using the molar mass of the substance to find the number of moles, and finally dividing by the volume of the solution to get the molarity. This process shows how interconnected these concepts are and demonstrates the importance of understanding each step in determining solution concentration.

The Role of Molar Mass in Chemical Calculations

The molar mass of a substance is one of the pillars of stoichiometry in chemistry. It expresses the mass of one mole of a substance (usually in grams per mole, g/mol) and is calculated by summing the atomic masses of all the atoms in a molecule of that substance. For example, the molar mass of lead (Pb) is 207.2 g/mol, meaning one mole of lead weighs 207.2 grams.

The molar mass is crucial when converting between grams and moles, a common task in chemistry. Here’s the formula:\[\begin{equation} \text{moles of substance} = \frac{\text{mass of substance}}{\text{molar mass}} \end{equation}\]In exercises calculating the concentration of lead in water, the molar mass allows us to convert a measured mass of lead (determined from the ppb concentration) into the number of moles, which is then used to calculate molarity. Understanding molar mass is essential for interpreting and performing these types of calculations accurately.