Problem 76 - Chemistry: The Central Science in SI Units, Global Edition

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 \(5 \mu \mathrm{p} / \mathrm{dL} .\) (a) What is the molarity of an aqueous solution with this concentration? (b) Express this concentration in ppb.

Step-by-Step Solution

Answer

(a) The molarity of an aqueous solution with a BLL of 5 μg/dL is \(2.42 \times 10^{-7}\) M. (b) The concentration expressed in ppb is 50 ppb.

Step 1: Convert BLL to grams per liter (g/L)

BLL is given as 5 μg/dL. We will convert this to g/L: BLL = 5 μg/dL = 5 * \(1 \times 10^{-6}\) g/dL = \(5 \times 10^{-6}\) g/dL Now, convert deciliter (dL) to liter (L): BLL = \(5 \times 10^{-6}\) g/dL * \(\dfrac{1}{0.1}\) L/dL = \(5 \times 10^{-5}\) g/L

Step 2: Calculate moles of Pb

Use the molar mass of lead (Pb) to calculate the moles of Pb: Moles of Pb = \(\dfrac{5 \times 10^{-5}\text{g}}{207.2 \text{g/mol}}\)

Step 3: Calculate the molarity

Now, divide the moles of Pb by the volume (in liters) to get the molarity: Molarity = \(\dfrac{5 \times 10^{-5}\text{g}}{207.2 \text{g/mol} \times 1\text{L}}\) Molarity = \(2.42 \times 10^{-7}\) M #a. Converting the concentration to ppb#

Step 4: Determine the molar ratio of Pb to the solution

We have already calculated the molarity of Pb in the solution, which is the ratio of moles of Pb to liters of the solution. Now, we need to find the concentration ratio: Concentration ratio of Pb = \(\dfrac{2.42 \times 10^{-7}\text{mol}}{1\text{L}}\)

Step 5: Calculate the mass concentration of Pb in the solution (g/L)

To obtain the mass concentration in g/L, we multiply the concentration ratio by the molar mass of Pb: Mass concentration of Pb = \(2.42 \times 10^{-7}\text{mol} \times 207.2\text{g/mol}\) / 1 L = \(5 \times 10^{-5}\) g/L

Step 6: Convert the mass concentration to ppb

Now, we will convert the mass concentration from g/L to ppb (parts per billion): 1 g/L = \(1 \times 10^{9}\) ppb So, BLL in ppb = \(5 \times 10^{-5}\) g/L * \(1 \times 10^{9}\) ppb/g = \(50\) ppb Therefore, (a) the molarity of an aqueous solution with a BLL of 5 μg/dL is \(2.42 \times 10^{-7}\) M, and (b) the concentration expressed in ppb is 50 ppb.

Key Concepts

Concentration Conversion

Converting concentrations often involves changing the units of measurement to suit different contexts. In chemistry, concentrations can be represented in various forms such as moles per liter (molarity), grams per liter (g/L), or percentage by volume, among others. Understanding how to switch between these forms is crucial.
One way to convert between different concentration units is by using dimensional analysis—a technique that involves multiplying the given value by a series of conversion factors until the desired unit is obtained.
In the original exercise, blood lead level (BLL) was given in micrograms per deciliter (µg/dL). The first step was converting this to grams per liter (g/L), using these relationships:

• 1 µg = 1 x 10^-6 g
• 1 dL = 0.1 L

After these conversions, the BLL was then easily expressible in g/L.
This is a vital skill in many scientific disciplines, helping professionals maintain consistency in their measurements.

ppm and ppb

Parts per million (ppm) and parts per billion (ppb) are units that describe the concentration of a substance in a solution. They are especially useful when dealing with very diluted solutions.
Imagine ppm as one part of a substance in a million parts of the solution, and ppb as one part of a substance in a billion parts. These units help to easily quantify extremely low concentrations, such as trace amounts of pollutants in air or water.
To convert a concentration from grams per liter to ppm or ppb, use these conversions:

• 1 g/L = 1,000 ppm
• 1 g/L = 1,000,000 ppb

In the exercise, after finding the lead concentration in g/L, it was multiplied by 1,000,000 to convert it into ppb, resulting in 50 ppb. This process highlights the versatility and practicality of ppm and ppb in environmental sciences and health sectors.

Moles and Molar Mass

The concept of moles and molar mass is fundamental in chemistry, facilitating the quantification of substances.
A mole is a unit that measures the amount of substance, representing Avogadro's number (approximately 6.022 x 10^23) of particles, such as atoms or molecules.
Molar mass, on the other hand, is the mass of one mole of a substance, expressed in grams per mole (g/mol). This allows chemists to interconvert between the mass of a substance and the number of moles, using the formula:
\[ ext{Moles} = \frac{ ext{mass (g)}}{ ext{molar mass (g/mol)}}\] In our exercise, the molar mass of lead (Pb) enabled the calculation of the number of moles from the known mass. By dividing the mass concentration (g/L) by the molar mass of lead, we obtained the molarity, which is the moles of solute per liter of solution.
Understanding these conversions is crucial for experiments, where precise quantities of substances are necessary to obtain accurate results.