Question

Lead is a heavy metal that remains in the bloodstream, causing mental retardation in children. It is believed that \(3 \times 10^{-7} \mathrm{~g}\) of \(\mathrm{Pb}\) in \(1.00 \mathrm{~mL}\) of blood is a health hazard. For this amount of lead how many atoms of lead are there in one \(\mathrm{mL}\) of a child's blood?

Short answer

Answer: Approximately \(8.72 \times 10^{14}\) atoms of lead.

Step-by-step solution

Write down the given information

We are given the mass of lead (Pb) present in a 1.00 mL blood sample: \(3 \times 10^{-7} \mathrm{~g}\). We also know the molar mass of lead (Pb) which is approximately 207.2 g/mol and Avogadro's number: \(6.022 \times 10^{23} \mathrm{~atoms/mol}\).

Convert the mass of lead to moles

To convert the mass of lead to moles, we will use the formula: moles = mass / molar mass. Given mass of lead = \(3 \times 10^{-7} \mathrm{~g}\) Molar mass = 207.2 g/mol Moles of lead = \(\frac{3 \times 10^{-7} \mathrm{~g}}{207.2 \mathrm{~g/mol}}\)

Calculate the number of atoms of lead

Now we will use Avogadro's number to convert this amount of lead moles to atoms. Number of atoms of lead = Number of moles of lead x Avogadro's number Number of atoms of lead = \(\frac{3 \times 10^{-7} \mathrm{~g}}{207.2 \mathrm{~g/mol}} \times 6.022\times10^{23} \mathrm{~atoms/mol}\)

Calculate the final result

Now, we need to perform the calculations and get the final result. Number of atoms of lead = \(\frac{3 \times 10^{-7} \mathrm{~g}}{207.2 \mathrm{~g/mol}} \times 6.022\times10^{23} \mathrm{~atoms/mol} \approx 8.72\times10^{14} \mathrm{~atoms}\) Therefore, there are approximately \(8.72 \times 10^{14}\) atoms of lead in 1.00 mL of a child's blood when the lead concentration is at the health hazard level.

Key concepts

Molar Mass

Understanding the molar mass is crucial for many chemistry calculations. Molar mass is defined as the mass of one mole of a substance, and it has units of grams per mole (g/mol). Each element on the periodic table has a unique molar mass that corresponds to the mass of an Avogadro's number of atoms.

For example, the molar mass of lead (\textrm{Pb}) is approximately 207.2 g/mol. This means that one mole of lead atoms, or approximately 6.022\( \times \)10^{23} lead atoms, weighs about 207.2 grams. Grasping the concept of molar mass allows us to convert between mass and moles, which is a necessary step in many stoichiometric calculations.

Avogadro's Number

Avogadro's number, a cornerstone of chemistry, represents the number of particles in one mole of a substance. Precisely, it's 6.022\( \times \)10^{23} particles/mol. This constant allows chemists to relate macroscopic amounts of substances to the number of atoms or molecules they contain.

Whether the particles are atoms, ions, or molecules, Avogadro's number remains consistent across all substances. It serves as a bridge between the atomic world and the world we experience, making it easier to understand chemical reactions in quantitative terms and to perform calculations involving molecules or atoms.

Lead Health Hazard

Recognizing Lead Toxicity

Lead is a toxic heavy metal that poses significant health risks, especially to children. It can impair brain development, resulting in cognitive deficits and learning disabilities.

The health hazard threshold for lead in blood is thought to be 3\( \times \)10^{-7} grams per mL of blood. At such a concentration, even the smallest exposure can harm a child's nervous system. Understanding the impact of lead on health is crucial to enforce regulations that limit exposure and protect vulnerable populations such as children.

Mole to Atom Conversion

The conversion from moles to atoms involves using Avogadro's number and is a common practice in chemistry. Once you have the number of moles, you simply multiply it by Avogadro's number to get the total number of atoms.

In a step-by-step process, you first calculate the number of moles by dividing the mass of the substance by its molar mass. Then, multiply the result by Avogadro's number to obtain the atoms count. For example, converting 3\( \times \)10^{-7} g of lead to atoms requires dividing by lead's molar mass and then multiplying by Avogadro's number, enabling us to quantify the particles in a tangible measurement of lead.