Problem 61 - General Chemistry: Principles and Modern Applications

Question

How many \(^{204} \mathrm{Pb}\) atoms are present in a piece of lead weighing \(215 \mathrm{mg} ?\) The percent natural abundance of \(204 \mathrm{Pb}\) is \(1.4 \%\)

Step-by-Step Solution

Answer

The final answer after combining all the results would be the number of \(^{204}\)Pb atoms in 215 mg of lead.

Step 1: Calculate the number of Pb atoms in 215 mg of lead

Firstly, we need to convert 215 mg to grams by dividing it by 1000. So, we get 0.215 g of lead. Next, we use Avogadro’s number (\(6.022 \times 10^{23}\) atoms/mol) and the atomic mass of lead (207.2 g/mol) in order to figure out the number of atoms. The formula used here is (mass of lead / atomic mass of lead) * Avogadro’s number = number of Pb atoms.

Step 2: Calculate \(^{204}Pb\) atoms

We know that the percent natural abundance of \(^{204}Pb\) is 1.4%. Hence, given so, the count of \(^{204}Pb\) atoms would simply be 1.4/100 multiplied by the total number of Pb atoms.

Step 3: Final answer

Combine the results from step 1 and step 2 to find the number of \(^{204}\)Pb atoms in 215 mg of lead.

Key Concepts

Atomic Mass

Atomic mass is the average mass of an element's atoms, usually measured in atomic mass units (amu). It's calculated based on the isotopes of the element and their distribution in nature. For example, the atomic mass of lead is roughly 207.2 g/mol. This value considers that lead has several isotopes with varying masses, but isotope distribution varies.
When calculating how many atoms are in a sample, the atomic mass tells you how many grams of an element contain one mole (or Avogadro's number) of atoms. By knowing the atomic mass, you can convert a given mass of a substance into moles, which further helps in quantifying the number of atoms using Avogadro's number.
This is crucial while performing isotope abundance calculations, like finding how many atoms of a certain isotope are in a sample based on its atomic mass and natural abundance.

Avogadro's Number

Avogadro's number is a fundamental constant with a value of approximately \(6.022 \times 10^{23}\). It defines how many atoms or molecules are in one mole of a substance. This huge number allows chemists to work with the macroscopic amounts of material in the laboratory.

In the context of isotope abundance calculations, Avogadro's number is instrumental in converting between moles and number of atoms. By taking the mass of lead and dividing it by its atomic mass, you obtain the number of moles. This number of moles, when multiplied by Avogadro's number, gives you the total number of atoms in the sample.

• Converting grams to moles: Divide by atomic mass (e.g., lead's atomic mass is 207.2 g/mol).
• Moles to atoms: Multiply by Avogadro's number.

Understanding these calculations is essential for determining the precise quantity of isotopes.

Percent Natural Abundance

Percent natural abundance refers to the relative amount of each isotope present in a naturally occurring sample of an element. It shows how much of each isotope exists compared to the total amount of the element. In this problem, the percent natural abundance of \(^{204}Pb\) is 1.4%, meaning this isotope makes up 1.4% of naturally occurring lead.
To find the number of atoms of a specific isotope, you first calculate the total number of atoms (using the mass, atomic mass, and Avogadro's number). Then, you multiply the total number of atoms by the percent natural abundance divided by 100.

• Example Calculation: If you have a complete count of total lead atoms, 1.4% of these are \(^{204}Pb\).
• Formula: \(\text{Number of }^{204}Pb = \left(\frac{1.4}{100}\right) \times \text{Total Number of Atoms}\)

This method helps in breaking down the total content to isolate specific isotopes for analysis in chemistry.