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\(\mathrm{ppb}\) solution. (b) How many grams of lead are in a swimming pool containing \(9.0 \mathrm{ppb}\) lead in \(60 \mathrm{~m}^{3}\) of water?

Step-by-Step Solution

Answer

(a) The molarity of lead in a 9.0 ppb solution is \(9\times10^{-9}\) M. (b) The amount of lead in a swimming pool containing 9.0 ppb lead in 60 \(m^{3}\) of water is approximately 0.0112 grams.

Step 1: 1. Convert ppb to molarity (M)

To convert ppb (parts per billion) to molarity, we need to use the conversion factor: 1 ppb = \(1\times10^{-9}\) M. Therefore, we have: Molarity (M) = 9.0 ppb × \(1\times10^{-9}\)

Step 2: 2. Calculate the molarity of lead in a 9.0 ppb solution

Now that we have the conversion factor, we can find the molarity of lead in the 9.0 ppb solution as follows: Molarity (M) = 9.0 ppb × \(1\times10^{-9}\) = \(9\times10^{-9}\) M So, the molarity of lead in a 9.0 ppb solution is \(9\times10^{-9}\) M.

Step 3: 3. Calculate the amount of lead in moles

In order to determine the amount of lead in grams, we need to know the volume of the swimming pool, which is given as 60 m³. We will first convert this volume to liters by using the conversion factor: 1 m³ = 1000 L. So, 60 m³ = 60000 L. Now, we can calculate the amount of lead in moles using the formula: Amount of lead (mol) = Molarity (M) × Volume (L) Amount of lead (mol) = \(9\times10^{-9}\) M × 60000 L

Step 4: 4. Calculate the amount of lead in grams

Now that we have the amount of lead in moles, we can convert it to grams using the molar mass of lead (Pb), which is 207.2 g/mol. We will use the formula: Amount of lead (g) = Amount of lead (mol) × Molar mass (g/mol) Amount of lead (g) = \(9\times10^{-9}\) M × 60000 L × 207.2 g/mol After calculating this expression, we will obtain the amount of lead in grams.

Step 5: 5. Results

(a) The molarity of lead in a 9.0 ppb solution is \(9\times10^{-9}\) M. (b) The amount of lead in the swimming pool containing 9.0 ppb lead in 60 \(m^{3}\) of water is \(9\times10^{-9}\) M × 60000 L × 207.2 g/mol ≈ 0.0112 g.

Key Concepts

Parts Per Billion (ppb)

Parts per billion (ppb) is a unit of measurement used to describe very small concentrations of substances. Imagine dividing a space or item into a billion tiny pieces and then focusing on just one of those pieces. That’s what ppb represents! This measurement is particularly useful when dealing with contaminants or trace elements in water or air, such as lead.

• 1 ppb is equivalent to 1 part of the substance per billion parts of the mixture.
• In terms of volume, you can think of it like 1 milligram of a substance in 1,000 liters of water.

Converting ppb to other measurements, like molarity, helps understand how much of the chemical substance is present on a deeper, molecular level.

Molar Mass

Molar mass is the weight of one mole of a given substance and is expressed in grams per mole (g/mol). For an element like lead (Pb), the molar mass is found by looking at the periodic table and reading off the atomic mass.
Lead has a molar mass of 207.2 g/mol.

• This value tells us how much one mole of lead atoms weighs.
• Molar mass is critical in the calculation of converting moles into grams.

When performing chemistry problems, this value serves as a bridge to convert between the number of atoms or molecules and the mass of a sample.

Conversion Factor

In chemistry, conversion factors are used to switch between different units of measurement, such as changing ppb to molarity. They are essential because units must be consistent to perform accurate calculations.
For this specific exercise, you'll want to know:

• 1 ppb = \(1\times10^{-9}\) molarity (M); this is useful for small concentrations.
• Also, when switching volumes, note that 1 m³ = 1000 liters.

Utilizing conversion factors correctly can simplify complex problems by allowing you to focus on just the numbers and relationships between units, making it easier to track what you are measuring.

Lead Concentration

Lead concentration in solutions like water is an important safety parameter, as high lead levels can harm health. In our exercise, the lead concentration in water is given as 9.0 ppb, a very low concentration indicating a trace amount of lead.
To convert this to a molar concentration:

• Multiply the ppb value by the conversion factor: 9.0 ppb multiplied by \(1\times10^{-9}\) gives a molarity of \(9\times10^{-9}\) M.

This low concentration means that, even over a large volume of water, there is a relatively tiny amount of lead, specifically around 0.0112 grams in 60 m³ of water. Understanding this concentration and being able to calculate it helps in ensuring compliance with safety regulations and assessing potential health risks.