Question

Lead is a poisonous metal that especially affects children because children retain a larger fraction of lead than adults do. To date, there is no "safe" concentration of lead in blood. Research shows that \(100 \mu \mathrm{g} / \mathrm{L}\) ( 2 significant figures) of lead in the blood of young children can cause delayed cognitive development. How many moles of lead per liter in a child's blood does \(100 \mu \mathrm{g} \mathrm{b} / \mathrm{L}\) represent? How many ppm? Assume the density of blood is \(1.00 \mathrm{~g} / \mathrm{mL}\)

Short answer

Answer: The concentration of lead in the child's blood is approximately \(4.83 \times 10^{-7}\) mol Pb/L and 100 ppm.

Step-by-step solution

Convert µg to g

To convert 100 µg of lead to grams, you need to know that 1 µg = 1e-6 g. Let's apply this conversion factor: $$ 100 \mu g \times \frac{1g}{10^6 \mu g} = 100 \times 10^{-6} g $$

Calculate the moles of Pb in 100 µg

Now, we have the mass in grams, we can calculate the moles of lead using the molar mass of lead (Pb). The molar mass of Pb is 207.2 g/mol. We can apply the formula: moles = mass (g) / molar mass (g/mol) $$ \text{moles of Pb} = \frac{100 \times 10^{-6} g}{207.2 \frac{g}{\text{mol}}} $$

Calculate moles per liter concentration

Since the given concentration is per liter, we can write the moles per liter directly as $$ \text{Moles of Pb per liter} = \frac{100 \times 10^{-6} g}{207.2 \frac{g}{\text{mol}}} \frac{\text{mol}}{\text{L}} $$

Calculate Lead concentration in ppm

We are given the density of blood, 1.00 g/mL, and we know that there are 1000 mL in 1 liter. To find the ppm concentration, we can use the formula: ppm = (mass of solute / mass of solution) × 1,000,000 $$ \text{ppm} = \frac{100 \times 10^{-6} g}{1.00\text{ g/mL} \times 1000 \text{ mL}} \times 1,000,000 $$

Simplify both expressions

Finally, simplify both expressions to get the moles per liter and ppm concentration of lead in the child's blood: $$ \text{Moles of Pb per liter} = \frac{100 \times 10^{-6} g}{207.2 \frac{g}{\text{mol}}} \frac{\text{mol}}{\text{L}} \approx 4.83 \times 10^{-7} \text{ mol/L} $$ $$ \text{ppm} = \frac{100 \times 10^{-6} g}{1.00\text{ g/mL} \times 1000 \text{ mL}} \times 1,000,000 = 100 \text{ ppm} $$ So, 100 µg the lead per liter represents approximately \(4.83 \times 10^{-7}\) mol Pb/L and 100 ppm.

Key concepts

Molarity Calculation

Understanding molarity is crucial for mastering chemistry problems involving solutions. Molarity, commonly represented by the capital letter 'M', defines the concentration of a solution by indicating the number of moles of a solute present per liter of solution. Calculating molarity helps in comparing the strength of various solutions and preparing desired solution concentrations for experiments.

To calculate molarity, we use the formula:
\( M = \frac{\text{moles of solute}}{\text{liters of solution}} \).
For instance, in the textbook exercise, we need to determine the molarity of lead in a child's blood when given a mass concentration in micrograms. The first step involves converting the mass of lead to moles, by using the atomic mass of lead, and then dividing by the volume of the solution, which is one liter in this case. This gives us the concentration in moles per liter (mol/L), allowing us to understand the degree of lead exposure quantitatively and its potential impact on health.

Unit Conversion

Unit conversion is vital for accurately interpreting and solving scientific problems. It's the process of expressing a measurement in a different unit, while maintaining its original value. To convert between units, we often use conversion factors, which are ratios that represent the relationship between two different units.

In our exercise, converting micrograms (µg) to grams (g) is necessary, as most chemical calculations are done in grams or moles. Knowing that \(1 \mu g = 10^{-6} g\), we apply this conversion factor to convert the mass of lead into grams before proceeding to molarity and ppm calculations. Being adept at unit conversion allows students to navigate through different measurement systems with ease, which is indispensable for various scientific disciplines.

Parts per Million (ppm)

Parts per million (ppm) is a dimensionless quantity used in science to express very dilute concentrations of substances. One part per million means one part of solute per one million parts of solvent or solution. It's like measuring one minute of time in 1.9 years.

To convert to ppm, you multiply the ratio of the mass of the solute to the total mass of the solution by one million (\(10^6\)). In the context of the exercise, we're given the density of blood, enabling us to determine the total mass of one liter of blood. We divide the mass of lead by this total mass and then multiply by one million to find the ppm concentration of lead.

Understanding ppm is important, particularly in fields like environmental science and toxicology, where it helps quantify the concentration of pollutants or toxins. For instance, knowing that 100 µg/L of lead in a child's blood is equivalent to 100 ppm helps grasp the severity of contamination in a meaningful way.