Problem 97 - Chemistry The Central Science In Si Units

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. (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

(a) Molarity is \(4.34 \times 10^{-8} M\). (b) There are 0.54 grams of lead.

Step 1: Understanding Parts Per Billion (ppb)

Parts per billion (ppb) is a unit that quantifies the concentration of a substance in a solution. 1 ppb means 1 part of a substance per billion parts of the solution. For water, which is denser enough to assume 1 kg is approximately 1 L, 9.0 ppb of lead suggests there are 9.0 grams of lead in one billion grams of water.

Step 2: Convert ppb to Molarity for Part (a)

To find the molarity, we convert 9.0 grams of lead per billion grams of water to moles per liter. The molar mass of lead (Pb) is 207.2 g/mol. The density of water is assumed to be 1.0 kg/L, so:\[ \text{Moles of Pb} = \frac{9.0\,g}{207.2\,g/mol} = 0.0434\, mol \]Since 1 g of water corresponds to 1 mL, or 1000 g of water correspond to 1 L, then 1 billion grams of water is 1 million liters. Thus:\[ \text{Molarity} = \frac{0.0434\, mol}{1 \times 10^6 \ L} = 4.34 \times 10^{-8} \ M \]

Step 3: Calculate Volume of Water in Liters for Part (b)

The volume of the swimming pool is given as 60 m³. Since 1 m³ is equivalent to 1000 liters, the volume of the pool in liters is:\[ 60 \ m^3 \times 1000 \ \frac{liters}{m^3} = 60,000 \ liters \]

Step 4: Find Amount of Lead in Grams for Part (b)

Now, calculate the total grams of lead present in 60,000 liters of water at 9.0 ppb. The concentration is 9.0 grams per billion liters (which equates to 1,000,000 liters for 9.0 grams of lead). Therefore,:\[ \text{grams of lead} = \frac{9.0 \ g}{10^9 \ L} \times 60,000 \ L = 0.54 \ grams \]

Step 5: Conclusion of Calculations

For part (a), the molarity of lead in a 9.0 ppb solution is \(4.34 \times 10^{-8} M\). For part (b), there are 0.54 grams of lead in a swimming pool containing 9.0 ppb lead in 60 m³ of water.

Key Concepts

Parts Per Billion (ppb)

The term "parts per billion", abbreviated as ppb, is a measure of concentration commonly used in environmental science and chemistry. It denotes the presence of one part of a substance for every billion parts of the total solution.
For example, 9.0 ppb of lead in water implies there are 9.0 grams of lead dispersed in one billion grams of water. Since water can be approximated as having the density of 1 g/mL, this means 9.0 grams in one million liters of water, as 1,000 liters equals 1 cubic meter.

• This unit of measure is especially useful for detecting trace amounts of pollutants, as it easily expresses minute concentrations.
• For perspective, one in a billion is equivalent to one second in approximately 32 years.

To convert ppb to other concentration units like molarity, careful attention must be given to the molar mass and volume.

Molar Mass

Molar mass is a fundamental chemical concept that relates to the mass of one mole of a substance. It is expressed in g/mol and can be found by summing the atomic masses of the elements in a compound according to their proportions.
For instance, the molar mass of lead (Pb) is 207.2 g/mol. This means that one mole of lead contains 207.2 grams and Avogadro's number of atoms:

• To find the number of moles of a substance, divide the mass by the molar mass.
• Molar mass serves as a conversion factor, bridging mass and mole-based calculations.

Knowing the molar mass is crucial when converting concentrations like ppb into molarity, as it allows for calculation of the moles of solute present.

Solution Concentration

Solution concentration expresses the amount of a solute present in a certain volume of solvent, indicating how dilute or concentrated a solution is. Molarity ( M ), measured in moles of solute per liter of solution, is a common unit for expressing concentration:

• In molarity calculations, both the number of moles and the volume of the solution are key.
• For example, to convert ppb to molarity, calculate the moles of solute and divide by the volume in liters.

In the lead example, 9.0 grams per billion grams convert to a molarity of 4.34 × 10^{-8} M by considering that 1 billion grams is 1 million liters.

Unit Conversion

Unit conversion is the process of converting one unit of measure to another, ensuring that all units in a calculation are consistent. Proper unit conversion is essential in chemistry to allow calculations using different units to communicate accurately.
In converting from ppb to molarity, it's important to:

• Convert mass to moles using molar mass.
• Translate the volume from cubic meters to liters if necessary.

For instance, when calculating the amount of lead in a swimming pool, the pool's volume was given in cubic meters, which required conversion to liters (1 m³ = 1000 liters). This consistent use of units ensures that complex problems remain manageable and solutions accurate.