Question

The maximum allowable concentration of lead in drinking water is \(9.0\) 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?

Short answer

(a) The molarity of the lead in a 9.0 ppb solution is \(\frac{9 \times 10^{-9}}{207.2} \, \text{moles per liter}\). The assumption made in this calculation is that 1 liter of the solution weighs approximately 1 kg or 1000 grams. (b) The mass of lead in a swimming pool containing 9.0 ppb lead in 60 m³ of water is \( \frac{9 \times 10^{-9}}{207.2} \, \text{moles per liter} \times 60,000 \, \text{liters} \times 207.2 \, \text{grams/mole}\).

Step-by-step solution

Understanding Parts per Billion (ppb)

Parts per billion (ppb) is a unit for expressing concentrations. 1 ppb represents 1 part of a substance in 1,000,000,000 parts of the solution. To calculate molarity, we need to convert this concentration into moles per liter.

Calculate the Molarity of the 9.0 ppb Lead Solution

To calculate the molarity, we will convert the concentration given in ppb to moles per liter (M). In this case, we make the assumption that 1 liter of the solution weighs approximately 1 kg or 1000 grams. This assumption holds true for dilute aqueous solutions. 1. Convert the 9.0 ppb concentration to grams per liter: \[ \frac{9.0 \, \text{grams}}{1,000,000,000 \, \text{liters}} \] 2. Multiply by the molar mass of lead (Pb) to convert grams per liter to moles per liter: \[ \frac{9 \times 10^{-9} \, \text{grams}}{1 \, \text{liter}} \times \frac{1 \, \text{mole}}{207.2 \, \text{grams}} \] Calculate the molarity of the lead solution: \[ M =\frac{9 \times 10^{-9}}{207.2} \, \text{moles per liter} \]

Calculate the Mass of Lead in the Swimming Pool

Now that we know the molarity of the lead solution, we can find the mass of the lead present in a swimming pool with 60 m³ of water. First, convert the volume of the pool from 60 m³ to liters: \[ 60 \, \text{m}^3 \times \frac{1000 \, \text{liters}}{1 \, \text{m}^3} = 60,000 \, \text{liters} \] Now, using the molarity and the volume of the pool, we can calculate the moles of lead present in the swimming pool: \[ \text{moles} = \text{molarity} \times \text{volume} = \frac{9 \times 10^{-9}}{207.2} \, \text{moles per liter} \times 60,000 \, \text{liters} \] Then, multiply the moles of lead by the molar mass of lead (Pb, 207.2 grams/mole) to find the mass of lead in grams: \[ \text{mass of lead} = \text{moles} \times \text{molar mass} \] Calculate the mass of lead in the swimming pool. (a) Report the molarity of lead in the 9.0 ppb solution and the assumption made in this calculation. (b) Report the mass of lead in the swimming pool containing 9.0 ppb lead in 60 m³ of water.

Key concepts

Parts per Billion (ppb)

Understanding dilution levels is crucial when assessing the purity of substances or the severity of pollution. Parts per billion (ppb) is one such measure used to express extremely small concentrations of a substance. For example, if you're told the concentration of lead in water is 9.0 ppb, it means that for every billion parts of water, there are 9 parts of lead. It's like finding 9 seconds in over 31 years!

When dealing with ppb, it's essential to connect it to more commonly used scientific measurements, such as molarity, which expresses the number of moles of a substance in a liter of solution. We often need to convert from ppb to molarity for consistency and to apply the mole concept and conduct various calculations in chemistry.

Conversion of Concentration

Converting concentrations from one unit to another is a fundamental skill in chemistry. To convert from ppb to molarity, you first need to change the ppb value into a mass per volume ratio (like grams per liter), and then use the molar mass of the substance to find the number of moles per liter.

Step-by-Step Conversion:

• Identify the concentration in ppb and convert to grams per liter.
• Use the molar mass of the substance to change grams into moles.
• Since molarity is moles per liter, you now can express your concentration in the desired unit of molarity.

This process requires an understanding of the mole concept and the ability to perform mass-volume calculations accurately.

Mole Concept

In chemistry, the mole is a fundamental unit that measures the amount of a substance. One mole has exactly 6.022 x 10^23 (Avogadro's number) of particles (atoms, molecules, ions, etc.).

The mole concept connects the macroscopic world we can measure (like grams of a substance) to the microscopic world of atoms and molecules. Utilizing molar mass, which is the mass of one mole of a substance, you can convert between the mass of a substance and the number of moles it represents. It's a bit like using a conversion factor to switch between different units of measurement, say kilometers to miles, but instead, you're converting grams to moles for chemical computations.

Molar Mass of Lead

Every element on the periodic table has a unique molar mass that corresponds to the mass of one mole of its atoms. For lead (Pb), the molar mass is 207.2 grams per mole. This means that one mole of lead, which contains Avogadro's number of atoms, weighs 207.2 grams.

This information is essential when calculating molarity from ppb: once you figure out how many grams of lead are present in a solution, you can divide by the molar mass of lead to find out how many moles are in that mass. This step is critical in the conversion from a mass-based concentration (like ppb) to a molarity-based one.

Mass-Volume Calculations

In chemistry, mass-volume calculations are often used to determine the concentration of a substance in a solution. To perform these calculations, you need to know two things: the mass of the substance dissolved and the volume of the solution it's dissolved in.

In the context of our problem, to find the mass of lead in a swimming pool with 60 cubic meters of water, you need the concentration in molarity and the volume in liters. With these values, you can calculate the number of moles of lead and then multiply by the molar mass of lead to get the mass in grams. The key to success with mass-volume calculations lies in unit consistency and careful execution of each step in the conversion process.