Question

Calculate Three magnesium isotopes have atomic masses and relative abundances of 23.985 amu \((79.99 \%), 24.986\) amu \((10.00 \%),\) and 25.982\((11.01 \%) .\) Calculate the atomic mass of magnesium.

Short answer

The atomic mass of magnesium is approximately 24.5501 amu, which is calculated using a weighted average of its isotopes: Atomic mass = (23.985 × 0.7999) + (24.986 × 0.1000) + (25.982 × 0.1101) ≈ 24.5501.

Step-by-step solution

Set up the equation

We'll use the weighted average formula mentioned in the analysis: Atomic mass = (isotope 1 mass × isotope 1 abundance) + (isotope 2 mass × isotope 2 abundance) + (isotope 3 mass × isotope 3 abundance) We are given the following information: - Isotope 1: mass = 23.985 amu, abundance = 79.99% - Isotope 2: mass = 24.986 amu, abundance = 10.00% - Isotope 3: mass = 25.982 amu, abundance = 11.01% We need to convert the abundance percentages into decimals (i.e., divide by 100) before plugging them into the formula.

Convert percentages to decimals

Divide each abundance percentage by 100 to get the decimal form: - Isotope 1 abundance: 79.99% / 100 = 0.7999 - Isotope 2 abundance: 10.00% / 100 = 0.1000 - Isotope 3 abundance: 11.01% / 100 = 0.1101

Plug the values into the formula

Now we can substitute the masses and abundances into the formula: Atomic mass = (23.985 × 0.7999) + (24.986 × 0.1000) + (25.982 × 0.1101)

Calculate the atomic mass

We can now compute the atomic mass of magnesium: Atomic mass = (23.985 × 0.7999) + (24.986 × 0.1000) + (25.982 × 0.1101) ≈ 19.1914 + 2.4986 + 2.8601 ≈ 24.5501 The atomic mass of magnesium is approximately 24.5501 amu.

Key concepts

Isotopes

In understanding how the atomic mass of an element is calculated, it's crucial to know what isotopes are. Isotopes are different forms of the same element that have the same number of protons but different numbers of neutrons. Due to this difference, while they are chemically alike, they have different atomic masses. For example, carbon has two stable isotopes, carbon-12 and carbon-13, where the numbers 12 and 13 refer to the atomic mass of each isotope.

Most elements consist of a mixture of isotopes, and this variety is key in calculating the average atomic mass found on the periodic table. To make this concept more accessible to students, imagining isotopes as different 'flavors' of the same ice cream can be helpful; they're fundamentally the same but with a slight variation that changes them slightly.

Relative Abundance

When calculating the atomic mass of an element, the concept of relative abundance plays an integral role. Relative abundance is the percentage of each isotope present in a natural sample of the element. To simplify this idea, consider a bag of mixed candies. If one were to calculate the average weight of a candy, it wouldn't be enough to just know the weights of different candy types; it's essential to know how many of each type are in the bag. Similarly, scientists calculate the atomic mass of an element by considering how much of each isotope (candy type) exists in nature.

It is important to understand that these percentages must be converted into decimal form before they are used in calculations, to correctly weigh each isotope's contribution to the atomic mass. Thus, the relative abundance is like telling us what fraction of the bag consists of each candy type, allowing us to calculate a more accurate average.

Weighted Average

The atomic mass of an element, as seen on the periodic table, is actually a weighted average of all its isotopes' masses based on their natural abundance. This means that more abundant isotopes have a greater influence on the atomic mass. Let's liken this to grading in class: if a project counts for 50% of your grade and the final exam counts for 50%, an average grade will reflect performance in both equally. But if the project is 70% of the grade and the final is 30%, then the project holds more weight in determining the final grade. In the case of isotopes, an isotope that is more abundant will 'count for more' in the calculation of the average atomic mass.

To calculate this weighted average, each isotope's mass is multiplied by its relative abundance in decimal form. Afterward, these products are summed to provide the average atomic mass of the element. It is the method we use to ensure that the mass accurately represents the mixture of isotopes in the natural world.