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Chapter 5: Problem 34

Calculate the atomic mass for magnesium given the following data for its natural isotopes: $$ \begin{array}{lll} { }^{24} \mathrm{Mg} & 23.985 \mathrm{amu} & 78.70 \% \\ { }^{25} \mathrm{Mg} & 24.986 \mathrm{amu} & 10.13 \% \\ { }^{26} \mathrm{Mg} & 25.983 \mathrm{amu} & 11.17 \% \end{array} $$

Short Answer

Expert verified
The atomic mass of magnesium is approximately 24.31 amu.

Step by step solution

01

Understand the Problem

To solve this problem, we need to calculate the weighted average of the atomic masses of magnesium's isotopes, considering their relative abundances. The atomic mass of an element is determined by the sum of the products of each isotope's mass and its relative abundance.
02

Convert Percentages to Fractions

Convert the percentage of each isotope's relative abundance into a fraction by dividing it by 100. For \( ^{24}\mathrm{Mg} \), the fraction is \( 0.7870 \). For \( ^{25}\mathrm{Mg} \), the fraction is \( 0.1013 \). For \( ^{26}\mathrm{Mg} \), the fraction is \( 0.1117 \).
03

Calculate Each Contribution

Multiply the atomic mass of each isotope by its fractional abundance to find each isotope's contribution to the atomic mass of magnesium. The calculations are:- \( 23.985 \text{ amu} \times 0.7870 = 18.8768 \text{ amu} \)- \( 24.986 \text{ amu} \times 0.1013 = 2.5310 \text{ amu} \)- \( 25.983 \text{ amu} \times 0.1117 = 2.9010 \text{ amu} \)
04

Sum the Contributions

Add the contributions of each isotope to find the atomic mass of magnesium:\[ 18.8768 \text{ amu} + 2.5310 \text{ amu} + 2.9010 \text{ amu} = 24.3088 \text{ amu} \]
05

Verify and Conclude

Verify the calculations to ensure there are no mistakes. The resulting atomic mass of magnesium, considering its natural isotopic composition, is approximately \( 24.31 \text{ amu} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isotope
In chemistry, an **isotope** refers to different forms of the same element that differ in the number of neutrons. Isotopes of an element have the same number of protons but varying numbers of neutrons, which gives them different atomic masses. These differing forms do not change the chemical properties of the element, but they do influence the atomic mass, which is incredibly significant for calculations. For magnesium given in the exercise, the isotopes are:
  • \(^{24} \text{Mg}\) with a mass of 23.985 amu.
  • \(^{25} \text{Mg}\) with a mass of 24.986 amu.
  • \(^{26} \text{Mg}\) with a mass of 25.983 amu.
Understanding isotopes is crucial in understanding how atomic masses are calculated and why natural elements have fractional atomic masses rather than whole numbers.
Relative Abundance
**Relative abundance** refers to how often an isotope occurs in nature compared to the total amount of that element. It is usually expressed as a percentage. For the atomic mass calculation, knowing the relative abundance is important because it allows us to weigh the isotope鈥檚 contribution to the overall atomic mass. For magnesium, the isotopes' relative abundances are:
  • 78.70% for \(^{24}\text{Mg}\)
  • 10.13% for \(^{25}\text{Mg}\)
  • 11.17% for \(^{26}\text{Mg}\)
These values highlight the prevalence of each isotope and inform how much each isotope mass will influence the atomic mass when determining the average. The larger the relative abundance, the more significantly an isotope will impact the atomic mass.
Weighted Average
A **weighted average** is a calculation that takes into account the varying "weights" or significance of the individual numbers in a dataset. When calculating the atomic mass of an element based on its isotopes, we are determining a weighted average of the isotopic masses. This is done by multiplying the atomic mass of each isotope by its relative abundance (expressed as a fraction), to get the isotope's contribution to the total atomic mass:
  • For \(^{24}\text{Mg}\), the contribution is \(23.985 \text{ amu} \times 0.7870 = 18.8768 \text{ amu}\).
  • For \(^{25}\text{Mg}\), the contribution is \(24.986 \text{ amu} \times 0.1013 = 2.5310 \text{ amu}\).
  • For \(^{26}\text{Mg}\), the contribution is \(25.983 \text{ amu} \times 0.1117 = 2.9010 \text{ amu}\).
These weighted contributions are then added together to yield the average atomic mass, allowing us to reflect the real-world composition of the element.
Fraction Conversion
In the context of atomic mass calculation, **fraction conversion** is a crucial step. It involves converting percentages of relative abundance into decimal fractions, enabling us to perform precise calculations. Since the relative abundance values are usually provided in percentages (as seen with magnesium鈥檚 isotopes), dividing these values by 100 will convert them into fractions:
  • 78.70% converts to the fraction 0.7870.
  • 10.13% converts to the fraction 0.1013.
  • 11.17% converts to the fraction 0.1117.
Using decimals instead of percentages makes it easier to calculate contributions to the atomic mass by allowing straightforward multiplication with the isotope masses. This conversion is a foundation of ensuring precise and accurate results in chemistry calculations.

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