Question

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

Short answer

The average atomic mass of the element is approximately 207.212 u, and the element is Lead (Pb).

Step-by-step solution

Set up the problem

Write down the given percentage and mass of each isotope. - Isotope 1: 1.40% and 203.973 u - Isotope 2: 24.10% and 205.9745 u - Isotope 3: 22.10% and 206.9759 u - Isotope 4: 52.40% and 207.9766 u

Convert percentages to decimals

Divide each percentage by 100 to convert them to decimals. - Isotope 1: 0.0140 and 203.973 u - Isotope 2: 0.2410 and 205.9745 u - Isotope 3: 0.2210 and 206.9759 u - Isotope 4: 0.5240 and 207.9766 u

Calculate the weighted average

Multiply the fraction of each isotope by its respective atomic mass and then add up these products to find the average atomic mass. The average atomic mass = (Fraction of Isotope 1 × Mass of Isotope 1) + (Fraction of Isotope 2 × Mass of Isotope 2) + (Fraction of Isotope 3 × Mass of Isotope 3) + (Fraction of Isotope 4 × Mass of Isotope 4) The average atomic mass = (0.0140 × 203.973 u) + (0.2410 × 205.9745 u) + (0.2210 × 206.9759 u) + (0.5240 × 207.9766 u)

Calculate the result

Plug the values into the equation to calculate the average atomic mass. The average atomic mass = (0.0140 × 203.973 u) + (0.2410 × 205.9745 u) + (0.2210 × 206.9759 u) + (0.5240 × 207.9766 u) = 2.85562 u + 49.639545 u + 45.741709 u + 108.975424 u The average atomic mass = 207.212291 u

Identify the element

Consult the periodic table to identify the element with an average atomic mass of approximately 207.212 u. The element is found to be Lead (Pb), which has an atomic mass of approximately 207.2 u. In conclusion, the average atomic mass of the element is approximately 207.212 u, and the element is Lead (Pb).

Key concepts

Isotopes

Isotopes are different forms of the same element that contain equal numbers of protons, but different numbers of neutrons. This means isotopes of an element have the same atomic number but different atomic masses. Isotopes play a crucial role in determining the atomic mass of an element because they contribute to the average atomic mass based on their abundance and individual masses.

Each isotope of an element has its own unique mass number due to the varying number of neutrons. For instance, in the exercise, the element had four isotopes, each with a distinct atomic mass. Knowing the mass and abundance of each isotope allows us to calculate the element's average atomic mass.

Weight Calculation

Weight calculation, in the context of average atomic mass, involves determining the mean mass of an element's isotopes based on their abundance. To find the average atomic mass, each isotope's mass is weighted by its abundance and then summed together.

The process involves a few simple steps:

• Convert the isotope abundances from percentages to decimal form.
• Multiply each isotope's mass by its relative abundance (in decimal form).
• Sum up all these products to find the average atomic mass.

This calculation gives us a weighted mean, which accurately reflects the average mass of an atom as found in nature.

Lead

Lead (symbol Pb, from the Latin $"plumbum"$) is a chemical element belonging to the carbon group in the periodic table. It is renowned for being dense, soft, and malleable. Lead's atomic weight is usually listed as approximately 207.2 u on the periodic table.

In the exercise, we solved for the average atomic mass based on isotopic mass contributions and found it to be around 207.212 u. This matched the atomic mass of lead, confirming its identity. Understanding isotopic calculation helps us precisely identify elements by matching calculated masses to standard ones on the periodic table.

Periodic Table

The periodic table is a vital tool in chemistry, organizing all known elements based on their atomic number. Each element is represented with its symbol, atomic number, and atomic mass. The periodic table helps identify elements and understand their properties and relationships with other elements.

When we calculated the average atomic mass of the isotopes in the exercise, we then used the periodic table to identify the element with an atomic mass close to our result. Finding that this mass corresponded to lead (Pb) helped accurately identify the element derived from the isotopic data. This highlights the periodic table's role in linking atomic properties to chemical identity.