Question

Naturally occurring magnesium has the following isotopic abundances: $$ \begin{array}{ccc} \hline \text { Isotope } & \text { Abundance } & \text { Atomic mass (amu) } \\\ \hline{ }^{24} \mathrm{Mg} & 78.99 \% & 23.98504 \\ { }^{25} \mathrm{Mg} & 10.00 \% & 24.98584 \\ { }^{26} \mathrm{Mg} & 11.01 \% & 25.98259 \\ \hline \end{array} $$ (a) What is the average atomic mass of \(\mathrm{Mg}\) ? (b) Sketch the mass spectrum of \(\mathrm{Mg}\).

Short answer

The average atomic mass of Mg is 24.30284 amu. To sketch the mass spectrum of Mg, draw a graph with "Intensity" or "Abundance" on the vertical axis and "m/z" or "Atomic Mass (amu)" on the horizontal axis. Mark the m/z values for \(^{24}\mathrm{Mg}\), \(^{25}\mathrm{Mg}\), and \(^{26}\mathrm{Mg}\) and draw vertical peaks with heights proportional to their isotopic abundances (78.99%, 10.00%, and 11.01%). Label the peaks with their isotopes and intensities.

Step-by-step solution

Calculate the weighted average

To calculate the average atomic mass of Mg, we need to find the weighted average of the atomic masses of its isotopes, considering their abundances. The formula for the weighted average is: \[ \text{Average Atomic Mass} = \sum_{i=1}^n (\text{Abundance}_i \times \text{Atomic Mass}_i) \] where n is the number of isotopes.

Find the average atomic mass of Mg

Using the given data, we have: Average Atomic Mass for Mg = (0.7899 × 23.98504 amu) + (0.1000 × 24.98584 amu) + (0.1101 × 25.98259 amu) Now, calculate the sum of these three values: Average Atomic Mass for Mg = 18.94279 + 2.49858 + 2.86147 = 24.30284 amu The average atomic mass of Mg is 24.30284 amu.

Sketch the mass spectrum of Mg

To sketch the mass spectrum of Mg, we will represent the isotopic abundances as the height of the peaks and the atomic masses as their locations (in terms of mass-to-charge ratio, m/z) on the x-axis. Follow these steps: 1. Draw a vertical axis labeled "Intensity" or "Abundance" on the left side of the graph, and a horizontal axis labeled "m/z" or "Atomic Mass (amu)" at the bottom. 2. Mark the m/z values (23.98504, 24.98584, and 25.98259 amu) on the horizontal axis, corresponding to the isotopes of Mg (\(^{24}\mathrm{Mg}\), \(^{25}\mathrm{Mg}\), and \(^{26}\mathrm{Mg}\)). 3. Draw a vertical line (peak) at each of these m/z values, with heights proportional to their respective isotopic abundances (78.99%, 10.00%, and 11.01%). 4. Label these peaks with their respective isotopes, and their corresponding intensities. Following the steps above, you will have a mass spectrum showing the three isotopes of Mg and their respective abundances.

Key concepts

Isotopic Abundances

Isotopic abundances refer to the relative quantities of the different isotopes of a chemical element found in nature. Each isotope of an element has the same number of protons but a different number of neutrons, causing variations in their atomic masses. Understanding isotopic abundances is crucial in several fields, including chemistry, geology, and physics.

For instance, naturally occurring magnesium consists of (^{24}Mg), (^{25}Mg), and (^{26}Mg) isotopes with abundances of 78.99%, 10.00%, and 11.01% respectively. These values represent the proportion of each isotope found in a sample of magnesium, and they play a critical role in determining the element's average atomic mass. Isotopic abundances are also essential for applications such as radiometric dating and understanding climatic changes through ice core analyses.

Weighted Average Calculation

When calculating the average atomic mass of an element, a simple arithmetical average does not suffice because it does not account for the relative abundances of the isotopes. Instead, a weighted average calculation is used, factoring in both the mass and the relative abundance of each isotope.

To calculate the weighted average, each isotope's atomic mass is multiplied by its abundance (expressed as a fraction), and then the products are added together. The formula looks like this:
\[\begin{equation}\text{Average Atomic Mass} = \sum_{i=1}^{n} (\text{Abundance}_i \times \text{Atomic Mass}_i)\end{equation}\]
For magnesium with three isotopes, the calculation incorporates the abundance and mass of each one, contributing to the final weighted average atomic mass of approximately 24.30284 amu. This approach ensures a more accurate representation of the element's mass as it appears on the periodic table.

Mass Spectrum

A mass spectrum is a graphical representation that displays the isotopic composition of a sample, primarily used in the technique of mass spectrometry. Through this method, researchers can determine the different isotopes present in a sample and their relative abundances.

To sketch a mass spectrum of magnesium (Mg), one would plot the intensity (abundance) of ions against the mass-to-charge ratio (m/z). On the x-axis, you'll indicate the m/z ratios corresponding to the atomic masses of Mg isotopes, and on the y-axis, the relative abundance of each isotope. The result is a series of peaks, each reflecting the presence of an isotope; the height of each peak correlates with its relative abundance. In the case of Mg, there would be prominent peaks at 23.98504 amu, 24.98584 amu, and 25.98259 amu, with their heights proportionate to their abundances of 78.99%, 10.00%, and 11.01%, respectively.

These peaks allow scientists to quickly identify the isotopes and understand their proportions within the sample, which is invaluable in fields ranging from pharmaceuticals to environmental analysis.