Question

Iron has three major isotopes: \({ }^{54} \mathrm{Fe}\) (atomic mass = \(53.9396 \mathrm{u}\); abundance \(5.85 \%),{ }^{56} \mathrm{Fe}\) (atomic mass = 55.9349u; abundance 91.75\%), and \({ }^{57} \mathrm{Fe}\) (atomic mass \(=56.9354 \mathrm{u} ;\) abundance \(2.12 \%) .\) Calculate the atomic weight (average atomic mass) of iron.

Short answer

The atomic weight of iron is approximately 55.693 u.

Step-by-step solution

Understand the Components

The atomic weight (average atomic mass) of an element is calculated by adding the products of the atomic mass of each isotope and its respective percentage abundance.

Convert Percentage to Decimal

Convert each isotopic abundance percentage into a decimal format by dividing by 100. This gives you: \(\text{54Fe} = \frac{5.85}{100} = 0.0585\), \(\text{56Fe} = \frac{91.75}{100} = 0.9175\), and \(\text{57Fe} = \frac{2.12}{100} = 0.0212\).

Multiply Atomic Mass by Decimal Abundance

For each isotope, multiply the atomic mass by its decimal abundance: 1. \(53.9396 \times 0.0585 = 3.1545\) 2. \(55.9349 \times 0.9175 = 51.3319\) 3. \(56.9354 \times 0.0212 = 1.2066\)

Sum the Weighted Atomic Mass Values

Add all the products from the previous step to get the average atomic mass of iron: \(3.1545 + 51.3319 + 1.2066 = 55.693\).

Key concepts

Atomic Mass

Atomic mass is a key concept in understanding elements and their isotopes. It refers to the mass of a single atom of a chemical element. Normally, this mass is measured in atomic mass units (u). Each element on the periodic table is defined by the number of protons in its nucleus, called the atomic number, but it can have different forms, called isotopes, which vary depending on the number of neutrons they have. These differences in neutron numbers result in variations of atomic mass for the isotopes of the same element.
When discussing isotopes, the atomic mass tells us how much a single atom of that particular isotope weighs. The atomic mass of isotopes is crucial for calculating other important values, like the average atomic mass of an element, as isotopes contribute differently based on their mass and abundance.

Percentage Abundance

Percentage abundance is a term used to describe the proportion of each isotope present in a sample of an element. It is expressed as a percentage. Knowing the percentage abundance of each isotope is essential when trying to determine the average atomic mass of an element, because it shows how much each isotope contributes to the overall mass of the element.
To make calculations easier, we typically convert percentage abundances to decimals by dividing by 100. For example, an isotope with a percentage abundance of 5.85% becomes 0.0585 in decimal form. These decimal values then help in calculations involving the atomic masses of different isotopes to find the weighted average, which is crucial for predicting the properties of the actual element.

Average Atomic Mass

Average atomic mass is an important calculation that provides a weighted average of the masses of all isotopes of an element. This value gives us insight into the typical atomic mass of an element as it occurs naturally, taking into account the presence and abundance of all its isotopes.
To find the average atomic mass, we follow a methodical approach:

• First, convert each isotope's percentage abundance to a decimal.
• Then, multiply the atomic mass of each isotope by its decimal abundance. This gives the weighted mass for each isotope.
• Finally, add all these weighted masses together to get the average atomic mass of the element.

For instance, in the case of iron, with three isotopes having varying atomic masses and abundances, the resulting average atomic mass reflects the natural mixture you would typically find. This calculated average is what you would often see listed on the periodic table.