Problem 22 - Chemistry and Chemical Reactivity

Question

Verify that the atomic weight of magnesium is 24.31, given the following information: \(^{24} \mathrm{Mg},\) mass \(=23.985042 \mathrm{u} ;\) percent abundance \(=78.99 \%\) \(^{25} \mathrm{Mg},\) mass \(=24.985837 \mathrm{u} ;\) percent abundance \(=10.00 \%\) \(^{26} \mathrm{Mg},\) mass \(=25.982593 \mathrm{u} ;\) percent abundance \(=11.01 \%\)

Step-by-Step Solution

Answer

Yes, the calculated atomic weight is approximately 24.30, confirming the given value of 24.31 for magnesium.

Step 1: Understanding Atomic Weight

The atomic weight of an element is the weighted average of the masses of its isotopes, based on their natural abundance. We calculate it by summing up the products of the isotopic masses and their percent abundances.

Step 2: Convert Percent Abundances to Fractions

To perform calculations using abundances, we convert the percent abundances into fractions:\[\text{Abundance of } ^{24}Mg = \frac{78.99}{100} = 0.7899\] \[\text{Abundance of } ^{25}Mg = \frac{10.00}{100} = 0.10\]\[\text{Abundance of } ^{26}Mg = \frac{11.01}{100} = 0.1101\]

Step 3: Calculate Contribution of Each Isotope

We multiply each isotope's mass by its fractional abundance to get its contribution to the atomic weight.\[^{24}Mg: 23.985042 \times 0.7899 = 18.9405 \] \[^{25}Mg: 24.985837 \times 0.10 = 2.4986 \] \[^{26}Mg: 25.982593 \times 0.1101 = 2.8622 \]

Step 4: Sum the Contributions

Adding up the contributions from all isotopes gives us the atomic weight of magnesium:\[18.9405 + 2.4986 + 2.8622 = 24.3013 \]

Step 5: Verify the Atomic Weight

The calculated atomic weight is approximately 24.30, which is very close to the given atomic weight of 24.31. Any small discrepancies are due to rounding during calculation.

Key Concepts

Isotopic Abundance

Isotopic abundance refers to the relative proportion of each isotope of an element found in nature. Isotopes are variations of elements that have the same number of protons but different numbers of neutrons. Isotopic abundance is usually expressed as a percentage; for instance, out of 100 atoms of magnesium, approximately 78.99 will be the isotope with a mass of 23.985042 u (u for unified atomic mass unit).
This percentage is crucial for calculating the average atomic mass of elements. The understanding of isotopic abundance allows for more accurate calculations of atomic weights, as it provides insight into which isotope is more prevalent in nature.
These percentages can be converted into decimals by dividing by 100, which simplifies calculations when determining the atomic weight. Thus, a 78.99% abundance becomes 0.7899 in calculations.

Mass Spectrometry

Mass spectrometry is a powerful analytical technique used to measure the mass-to-charge ratio of ions. It is particularly useful in determining the isotopic composition and quantifying the abundance of different isotopes in a sample.
This process involves ionizing chemical compounds to produce charged molecules or molecule fragments and measuring their mass-to-charge ratios. The data gathered helps scientists determine precise and accurate isotopic masses and their relative abundances.
Mass spectrometry outputs a mass spectrum, which is a plot of the ion signal as a function of the mass-to-charge ratio. This graph allows us to interpret which isotopologues are present and in what quantities, providing essential information needed for calculating isotopic abundances that lead to the determination of atomic weights.

Weighted Average

A weighted average is a mathematical concept used to determine the average of a set of values, where each value has a specified weight or importance. In the context of atomic weights, this means that the contribution of each isotope is proportional to its natural occurrence or abundance.
The process of calculating the atomic weight of magnesium, for instance, relies on a weighted average of the isotopic masses. Here, we multiply each isotope's mass by its abundance (converted to a fraction) and sum the results. This reflects the fact that isotopes present in higher abundance have a greater influence on the average.
To calculate the atomic weight using a weighted average, it’s crucial to correctly handle these weights (or abundances), as they determine the significance of each isotope's mass in the overall calculation. This method ensures that the calculated atomic weight is reflective of what is observed in nature.

Isotopes of Magnesium

Magnesium, a chemical element found in Group 2 of the periodic table, has three naturally occurring isotopes:

• ²⁴Mg: This is the most abundant isotope, making up about 78.99% of natural magnesium. It has an atomic mass of 23.985042 u.
• ²⁵Mg: This isotope comprises around 10.00% of magnesium found in nature with a slightly heavier atomic mass of 24.985837 u.
• ²⁶Mg: The least abundant of the three, it makes up 11.01% of magnesium and has the heaviest atomic mass of 25.982593 u.

Understanding the isotopes of magnesium is essential for calculating its atomic weight. Each isotope's unique mass and abundance contribute to the overall weighted average. By considering these differences, scientists can accurately determine the properties of magnesium as it occurs naturally.