Problem 17 - Chemistry: Principles and Reactions

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 (a) how many atoms of lead are there in one \(\mathrm{mL}\) of a child's blood? (b) how many moles of lead are in \(1.00 \mathrm{~L}\) of blood?

Step-by-Step Solution

Answer

Question: In 1 mL of a child's blood, there are \(3 \times 10^{-7}\mathrm{~g}\) of lead. Calculate the following: a) The number of lead atoms in this 1 mL sample. b) The number of moles of lead in 1 L (1000 mL) of blood. Answer: a) The number of lead atoms in 1 mL of blood is \((3 \times 10^{-7} \mathrm{~g})/(207.2 \mathrm{~g/mol}) \times 6.022 \times 10^{23}\mathrm{~atoms/mol}\). b) The number of moles of lead in 1 L of blood is \(((3 \times 10^{-7}\mathrm{g/mL})(1\ 000\mathrm{~mL}))/(207.2 \mathrm{~g/mol})\).

Step 1: Convert mass of lead to moles

To convert the mass of lead to moles, we can use the equation: moles = mass of substance / molecular weight In this case, the mass of lead in 1 mL of blood is \(3 \times 10^{-7} \mathrm{~g}\), and the molecular weight of lead is 207.2 g/mol. So, moles of lead in 1 mL of blood is: moles = \((3 \times 10^{-7} \mathrm{~g})/(207.2 \mathrm{~g/mol})\)

Step 2: Calculate the number of lead atoms

Once we have the moles of lead, we can calculate the number of lead atoms by using Avogadro's number: Number of lead atoms = moles x Avogadro's number Now, let's calculate the number of lead atoms in 1 mL of a child's blood. Number of lead atoms = \((3 \times 10^{-7} \mathrm{~g})/(207.2 \mathrm{~g/mol}) \times 6.022 \times 10^{23}\mathrm{~atoms/mol}\) #b) Finding the number of moles in 1 L of blood#

Step 3: Calculate the mass of lead in 1 L of blood

Since there are 1000mL in 1 L, the mass of lead in 1 L is 1000 times the mass in 1mL: mass of lead in 1 L = \((3 \times 10^{-7}\mathrm{g/mL})(1\ 000\mathrm{~mL})\)

Step 4: Convert mass of lead in 1 L to moles

Using the same method as in part (a), we can convert the mass of lead in 1 L to the number of moles using the equation: moles in 1 L = mass of lead in 1 L / molecular weight of lead moles in 1 L = \(((3 \times 10^{-7}\mathrm{g/mL})(1\ 000\mathrm{~mL}))/(207.2 \mathrm{~g/mol})\)

Key Concepts

Mole Calculation

Mole calculation is a fundamental concept in chemistry that helps us determine the number of moles in a given substance, based on its mass and molecular weight. This is crucial when dealing with substances like lead in the blood, where understanding the quantity of atoms present is important for health and safety assessments.
The mole is a counting unit used by chemists, just like a dozen is used to count items such as eggs. To calculate the number of moles, you divide the mass of the substance by its molecular weight.

• Molecular weight is the sum of the atomic weights of all atoms in a molecule.
• The mole relates the mass of a substance to the number of entities (atoms, molecules) it contains.

In our exercise, we have a mass of lead of \(3 \times 10^{-7} \text{g}\) and the molecular weight of lead as 207.2 g/mol. To find the moles of lead, use the formula:\[\text{moles} = \frac{\text{mass of substance}}{\text{molecular weight}}\]By substituting the given values, we can find the moles of lead in a specific volume of blood, facilitating further calculations related to the toxicity levels of lead.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, representing the number of atoms or molecules in one mole of a substance. This number is \(6.022 \times 10^{23}\) and is essential for converting moles into the actual number of atoms or molecules.

• This conversion factor allows chemists to understand and count the microscopic entities in macroscopic quantities of material.
• One mole of any substance contains exactly Avogadro's number of atoms or molecules.

In the context of our problem, once the moles of lead are known, Avogadro's number helps in calculating the number of lead atoms within the blood. By multiplying the number of moles of lead by Avogadro's constant, you obtain the total number of lead atoms present in the given volume:\[\text{Number of lead atoms} = \text{moles} \times 6.022 \times 10^{23} \text{ atoms/mol}\]Understanding and using Avogadro's number is crucial in both theoretical and practical chemistry applications, such as assessing contamination levels of hazardous substances like lead.

Molecular Weight of Lead

Molecular weight, also known as molar mass, is the mass of one mole of a given substance. For elements, this is equivalent to the atomic weight found on the periodic table, expressed in grams per mole (g/mol). For compounds, it is the sum of the atomic weights of its constituent elements.

• Lead (Pb) has a molecular weight of 207.2 g/mol.
• This value is used to convert the mass of lead to moles, which is necessary for calculating the number of atoms in a sample.

In our exercise, knowing the molecular weight of lead allows us to transition from mass to moles seamlessly. This conversion is done by dividing the total mass of lead by its molecular weight:\[\text{moles} = \frac{\text{mass of lead}}{207.2 \text{ g/mol}}\]Understanding molecular weight is fundamental for precise scientific calculations, particularly when evaluating the potential health impact of toxic substances like lead in the human bloodstream.