Question

An element consists of \(1.40 \%\) of an isotope with mass \(203.973\) amu, \(24.10 \%\) of an isotope with mass \(205.9745\) amu, \(22.10 \%\) of an isotope with mass \(206.9759 \mathrm{amu}\), and \(52.40 \%\) of an isotope with mass \(207.9766\) amu. Calculate the average atomic mass and identify the element.

Short answer

The average atomic mass of the element with the given isotopic composition is approximately \(207.42 \mathrm{amu}\). The element is lead (Pb).

Step-by-step solution

Find the weighted mean of the atomic masses

To find the weighted mean, we will multiply the atomic mass of each isotope by its percentage abundance, and then add them together. In mathematical terms: Average atomic mass = (Percentage of isotope 1 × mass of isotope 1) + (Percentage of isotope 2 × mass of isotope 2) + (Percentage of isotope 3 × mass of isotope 3) + (Percentage of isotope 4 × mass of isotope 4) Substitute the given data: Average atomic mass = (1.40% × 203.973 amu) + (24.10% × 205.9745 amu) + (22.10% × 206.9759 amu) + (52.40% × 207.9766 amu) To convert percentages to decimals, divide by 100: Average atomic mass = (0.014 × 203.973 amu) + (0.241 × 205.9745 amu) + (0.221 × 206.9759 amu) + (0.524 × 207.9766 amu)

Calculate the average atomic mass

Now we will perform the multiplications and additions: Average atomic mass = (2.85562 amu) + (49.6404145 amu) + (45.7409479 amu) + (109.1832944 amu) Average atomic mass = 207.4202788 amu

Identify the element

The calculated average atomic mass is approximately 207.42 amu. To identify the element, we will refer to the periodic table and look for an element with an atomic mass close to 207.42 amu. In the periodic table, we can find that the element with an atomic mass close to 207.42 amu is lead (Pb) with an atomic mass of approximately 207.2 amu. The element with the given isotope composition and average atomic mass is lead (Pb).

Key concepts

Understanding Isotope Composition

Isotope composition refers to the different types of atoms (isotopes) of an element that have the same number of protons but a different number of neutrons in their nuclei. This means that while isotopes of an element have the same chemical properties, their masses differ. These varied mass numbers come from the addition or loss of neutrons. For example, lead has multiple isotopes with mass numbers like 203.973, 205.9745, 206.9759, and 207.9766 amu.

In the example, isotopes contribute differently to the whole element due to their relative amounts, expressed as percentages. By knowing these percentages, you can understand how each isotope influences the overall atomic mass.

Weighted Average Explained

A weighted average accounts for the varying percentage contribution of each part to the total. In the context of isotopes, the weighted average gives the average atomic mass. Each isotope does not contribute equally, but proportionately to its abundance. Think of it like this:

• The percentage (by abundance) represents the weight.
• The isotopic mass is the value being weighted.

To calculate the weighted average:

• Multiply each isotope's mass by its fractional abundance (percentage divided by 100).
• Add these results together to find the overall average atomic mass.

This ensures that isotopes with greater abundance have a bigger impact on the calculated average than those with lesser abundance.

Lead (Pb) Isotopic Details

Lead, with its chemical symbol Pb, is a well-known heavy metal. Its isotopes primarily include those with mass numbers 204, 206, 207, and 208. In natural occurrence, these isotopes mix in different proportions.

The isotopes of lead do not significantly differ in terms of chemical reactions, but their differing mass numbers affect the overall atomic mass when you calculate it as a weighted average. The ample variation in isotopic mass helps determine the weighted average atomic mass of lead close to 207.2 amu, reflecting the mixture of its isotopes naturally found.

Calculating Atomic Mass through Isotopes

Calculating the atomic mass of an element using its isotopes involves using the weighted average method. Here's how it works in practice:

• Start by identifying each isotope's mass and its relative abundance.
• Convert the abundance percentages into decimal form by dividing by 100.
• Multiply each isotope’s mass by its abundance in decimal form to find the contribution of each isotope.
• Add all contributions together to find the average atomic mass.

For the given isotopes of lead:

• Calculate: - (0.014 × 203.973) + (0.241 × 205.9745) + (0.221 × 206.9759) + (0.524 × 207.9766)
- Result: 207.42 amu approximately

This calculated average atomic mass aligns closely with lead's known atomic mass found in the periodic table, helping confirm the identity of the element.