Question

The maximum allowable concentration of lead in drinking water is 9.0 ppb. (a) Calculate the molarity of lead in a 9.0ppb solution. (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

The molarity of lead in a 9.0 ppb solution is \(9.0 \times 10^{-9} M\). The mass of lead in a swimming pool containing 9.0 ppb lead in 60 m³ of water is approximately 0.011 g.

Step-by-step solution

Convert ppb to molarity

First, we need to convert the given concentration in ppb (parts per billion) to molarity. The value of 9.0 ppb can be expressed as: 9.0 ppb = 9.0 x 10^{-9} Now we need to convert this fraction to molarity. The formula for molarity (M) is: \[ M = \frac{moles}{volume~in~liters} \] we can rearrange the formula as: \[ moles = M \times volume~in~liters \] In this problem, we are considering 1 liter of solution and m_n, as the number of moles in 1 liter of solution: \[ m_n = (9.0 \times 10^{-9}) \times 1 \]

Calculate molarity

Now that we have the number of moles per liter, we can find the molarity. Since we calculated the moles present in 1 liter of solution, the molarity is just the moles itself: Molarity of lead = \( m_n = 9.0 \times 10^{-9} \: M \) So, the molarity of lead in a 9.0 ppb solution is \(9.0 \times 10^{-9} M\).

Convert m³ to liters

Now, we want to find the mass of lead in a swimming pool containing 60 m³ of water. First, we need to convert the volume of water from m³ to liters. 1 m³ = 1000 liters So, 60 m³ = 60 × 1000 = 60,000 liters

Calculate moles and mass of lead

We will use the molarity (M) we calculated earlier to find the moles of lead in the swimming pool: Moles of lead = Molarity × Volume in liters Moles of lead = \( (9.0 \times 10^{-9} M) \times 60,000 L \) To find the mass of lead, we can use the molar mass of lead (Pb), which is 207.2 g/mol: Mass of lead = Moles of lead × Molar mass of lead

Final result

Now we have all the information needed to find the mass of lead in the swimming pool: Mass of lead = [\( (9.0 \times 10^{-9} M) \times 60,000 L \)] × 207.2 g/mol So, the mass of lead in the swimming pool containing 9.0 ppb lead in 60 m³ of water is approximately 0.011 g.

Key concepts

Molarity

Molarity is a measure of the concentration of a solute in a solution, expressed as moles of solute per liter of solution. This unit comes in handy when dealing with chemical solutions because it provides a straightforward way to represent the number of particles in a given volume.
To calculate molarity (\(M\)), use the formula:

• \[ M = \frac{\text{moles of solute}}{\text{volume in liters}} \]

Molarity is essential for understanding how concentrated a solution is and plays a pivotal role in stoichiometric calculations in chemistry. In the context of the original exercise, converting the concentration from ppb to molarity involves quantifying lead particles in terms of their moles per liter, which allows for comparisons across different types of measurements.

Concentration Conversion

Concentration conversion is the process of transforming measurement units for the concentration of a substance to different units. This conversion is vital when the measurement format provided is not what is required for specific calculations or comparisons.
One common conversion is between parts per billion (ppb) and molarity. To perform this conversion, it's necessary to consider the density of the solution, but often, the assumption is made that 1 liter of water is approximately equal to 1 kilogram, allowing conversion straightforwardly for diluted solutions.
Converting 9.0 ppb to molarity in the exercise involves expressing the lead concentration as a fraction of the liter instead of its presence per billion parts.

Molar Mass

Molar mass is fundamental in chemistry for relating the mass of a substance to the number of moles present. Molar mass is expressed in grams per mole (g/mol) and is used to convert between the amount of substance and its mass.
For lead, the molar mass is 207.2 g/mol. This value is crucial when determining the mass of lead from its molar quantity, as shown in the example of the swimming pool problem.
To find the mass from moles:

• Multiply the number of moles by the molar mass.
• \[ \text{Mass (g)} = \text{Moles} \times \text{Molar Mass (g/mol)} \]

For instance, using the molar mass of lead helps calculate its mass in the 60 m³ of pool water after determining how many moles are present at the given molarity.

Parts Per Billion (ppb)

Parts per billion (ppb) is a unit of concentration that denotes one part of solute per billion parts of the total mixture. This unit is often used in environmental chemistry when dealing with pollutants or trace elements because it effectively represents very low concentrations.
In the context of the original exercise, 9.0 ppb indicates that for every billion parts of water, there are 9 parts of lead.
To work with ppb in calculations involving moles and liters, converting this concentration to more standardized units like molarity can help clarify the amount of substance involved in chemical reactions or exposure analyses.

Volume Conversion

Volume conversion is crucial in chemical calculations when dealing with solutions of different total volumes. Common conversions include changing cubic meters into liters, which is more user-friendly in lab settings and for molarity-related calculations.
In the water context, where 1 m³ equals 1000 liters, converting the swimming pool's volume from 60 m³ to 60,000 liters allows for straightforward use in molarity computations.
Implementing correct volume conversions ensures calculations align properly with the units of other variables, maintaining the accuracy of chemical measurements and results.