Problem 76 - Chemistry The Central Science In Si Units

Question

The CDC (Centers for Disease Control and Prevention) published a reference blood lead level (BLL), which is based on the BLL distribution among children. It is currently \(5 \mu \mathrm{g} / \mathrm{dL}\). (a) What is the molarity of an aqueous solution with this concentration? (b) Express this concentration in ppb.

Step-by-Step Solution

Answer

(a) Molarity: \(2.415 \times 10^{-4} \, M\). (b) Concentration: 50 ppb.

Step 1: Calculate Molality

Milli-molality (symbol: mM) is defined as the number of millimoles of solute per kilogram of solvent. To calculate the molality of lead in the solution, convert the concentration in µg/dL to mg/L by multiplying by 10 (since there are 10 deciliters in a liter). The concentration becomes 50 mg/L. Convert to mM by dividing 50 by the molar mass of lead, 207.2 g/mol, and converting mg to g. This gives \(50 \times 10^{-3} / 207.2 \approx 0.2415 \, mM\).

Step 2: Calculate Molarity

Since molarity is expressed in moles per liter, and the previously calculated value is essentially a molarity calculation (as density of water at low lead concentrations is approximately 1 g/mL), the molarity is the same as the calculated millimolarity: \( 2.415 \times 10^{-4} \, M \).

Step 3: Convert to ppb

Parts per billion (ppb) is calculated as the mass of solute per billion parts of the solution. First, use the concentration in micrograms per liter to find the concentration in ppb. Since 1 g/L is equivalent to 1,000,000 ppb, and the concentration of lead is 50 µg/L, it is equal to \(50 \times 1000 = 50 \times 10^{3} \, ppb\), which simplifies to 50 ppb.

Key Concepts

Molarity

Molarity is a measure of the concentration of a solute in a solution, expressed in moles of solute per liter of solution (mol/L). This is a fundamental concept in chemistry that helps us understand how much of a substance is dissolved in a given volume of liquid. To calculate the molarity, you need to take the number of moles of the solute and divide it by the volume of the solution in liters. For example, if you have a solution with a known mass of lead dissolved in it, like 50 mg/L from the exercise, you can convert this mass into moles using the molar mass of lead, which is 207.2 g/mol. By doing this conversion, you get the molarity of the lead in the solution. Understanding molarity is crucial because it allows chemists to accurately prepare solutions and perform reactions that depend on specific concentrations.

Molality

Molality is another way to express the concentration of a solution. However, unlike molarity, molality is defined as the number of moles of solute per kilogram of solvent (mol/kg). Molality is particularly useful when dealing with calculations involving changes in the temperature because it does not depend on volume, which can change with temperature. To determine molality, you first need to know the mass of the solvent and the number of moles of the solute. In our step-by-step solution, when we calculated milli-molality (or millimolality), the focus was on converting the concentration from micrograms per deciliter to milligrams per liter. This conversion helps in understanding how many millimoles of lead are present per kilogram of the solvent, allowing for precise characterization of the solution's concentration.

Parts Per Billion (ppb)

Parts per billion (ppb) is a unit used to describe the concentration of a component in a solution, particularly when dealing with very low concentrations. It is defined as the mass of the solute per billion mass units of the solution. In many contexts, ppb is a straightforward measure because it denotes thousands of micrograms per liter when dealing with aqueous solutions. From the original solution, we took the concentration of 50 micrograms per liter and recognized that this can be directly converted to ppb: 50 µg/L equals 50 ppb. This conversion is possible because of the definition where 1 g/L equals 1,000,000 ppb. This is especially useful in cases like monitoring contaminants, such as lead, where even trace amounts could be significant.

Lead Concentration

Lead concentration in a solution can have significant implications, especially when it comes to health standards, like those set by the CDC. The reference blood lead level (BLL) is determined based on adverse health effects and distribution data among populations. In our problem, the concentration of lead at 5 µg/dL is used for various calculations. This amount is compared to the permissible exposure limits and helps identify potential risks of exposure. Understanding the concentration in different units, such as molarity or ppb, aids in ensuring that measures are in place to mitigate lead exposure effectively. Ensuring accurate measurements is critical, as lead is a toxic element and poses serious health risks when exposure levels are high.

Unit Conversion

Unit conversion is a vital skill in chemistry and involves changing a measurement from one unit to another while keeping the quantity consistent. In our exercise, we converted the lead concentration from micrograms per deciliter (µg/dL) to milligrams per liter (mg/L) and eventually to micromoles and parts per billion. Such conversions are necessary to compare concentrations and perform calculations reliably. For instance, converting concentration units allows us to use the values in different computational frameworks, like converting from mass to molarity for reactions or from concentration to ppb for safety compliance. Understanding unit conversions empowers students to apply concepts across various scenarios in chemistry with confidence.