Problem 35 - Chemistry: The Central Science in SI Units, Global Edition

Question

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

Step-by-Step Solution

Answer

The average atomic mass of iron can be calculated using the weighted average of its isotopes' atomic masses and their respective abundances. After converting the abundance percentages into fractions, we multiply each fraction by its corresponding isotope's atomic mass and sum them up to get the average atomic mass of iron, which is approximately 55.715 u.

Step 1: Convert the percentages to fractions

To convert a percentage to a fraction, divide the percentage by 100. \[\frac{5.85}{100} = 0.0585\] \[\frac{91.75}{100} = 0.9175\] \[\frac{2.12}{100} = 0.0212\] Now we have the fractions of the abundance of the isotopes: 54Fe: 0.0585 56Fe: 0.9175 57Fe: 0.0212

Step 2: Calculate the weighted average of atomic masses

Now that we have the fractions of the abundances of each isotope, we can multiply these fractions by the atomic masses of the isotopes, and sum them up to find the weighted average of atomic masses. Weighted average of atomic masses = (fraction of 54Fe × atomic mass of 54Fe) + (fraction of 56Fe × atomic mass of 56Fe) + (fraction of 57Fe × atomic mass of 57Fe) Weighted average = (0.0585 × 53.9396 u) + (0.9175 × 55.9349 u) + (0.0212 × 56.9354 u)

Step 3:Perform the calculations

Now, calculate the weighted average: Weighted average = (0.0585 × 53.9396) + (0.9175 × 55.9349) + (0.0212 × 56.9354) Weighted average ≈ 3.1552 + 51.3521 + 1.2076 Weighted average ≈ 55.715 The average atomic mass of iron is approximately 55.715 u.

Key Concepts

Isotopes

Isotopes are variants of elements that have the same number of protons but differ in the number of neutrons present in their nuclei. This means that while isotopes of an element share the same atomic number, their atomic masses vary because of the difference in neutron count. For instance, iron has several isotopes, such as

• \(^{54}\text{Fe}\)
• \(^{56}\text{Fe}\)
• \(^{57}\text{Fe}\)

These isotopes all have 26 protons, which is characteristic of iron, but a different number of neutrons, resulting in different overall atomic masses.

Weighted Average

A weighted average is a calculation that reflects both the value of different items and their significance or frequency. In the context of chemistry, it takes into account the abundance of each isotope of an element to calculate a more representative average atomic mass. This calculation is particularly important because different isotopes occur naturally in varying proportions. By using weighted averages, chemists can determine a more accurate representation of an element's atomic mass as it naturally occurs. The formula for a weighted average is \[\text{Average} = \sum (\text{fraction of isotope} \times \text{atomic mass of isotope})\] This ensures that each isotope's contribution to the atomic mass reflects not just its mass, but also how commonly it occurs in nature.

Atomic Mass Calculation

Calculating the atomic mass of an element with isotopes involves several steps. First, the natural abundances of the isotopes are converted from percentages to decimal form. For example, if the isotope

• \(^{54}\text{Fe}\) is 5.85% abundant, it converts to 0.0585 in decimal form.

Next, multiply each isotope's decimal abundance by its respective atomic mass. Finally, add these values together to get the element's average atomic mass. The use of decimal abundances ensures that each isotope's importance is accurately weighed in the final computation.This method gives a clearer indication of the element's true atomic mass as found in nature.

Iron Isotopes

Iron is a common element with isotopes that include \(^{54}\text{Fe}\), \(^{56}\text{Fe}\), and \(^{57}\text{Fe}\). These isotopes vary based on their atomic masses and natural abundances. When calculating the average atomic mass of iron, it's essential to consider these isotopes and their relative occurrence. The calculation uses the following data:

• \(^{54}\text{Fe}\) has an atomic mass of 53.9396 u and is 5.85% abundant.
• \(^{56}\text{Fe}\) has an atomic mass of 55.9349 u and is 91.75% abundant.
• \(^{57}\text{Fe}\) has an atomic mass of 56.9354 u and is 2.12% abundant.

By gathering this information and calculating the weighted average, scientists determine the most accurate atomic mass reflective of the naturally occurring iron isotopes. This calculated average is crucial for practical applications and theoretical studies involving iron.