Problem 35 - Chemistry: The Central Science

Question

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

Step-by-Step Solution

Answer

The average atomic mass of Mg is approximately 24.30231 amu. The mass spectrum of magnesium consists of three peaks at m/z values of 24, 25, and 26, corresponding to the isotopes 24Mg, 25Mg, and 26Mg, with heights proportional to their abundances (78.99%, 10.00%, and 11.01%, respectively). The peak at m/z = 24 is the tallest.

Step 1: Calculate the weighted average atomic mass

To find the average atomic mass of magnesium, we need to consider the atomic masses of each isotope and their abundances. We can calculate the weighted average atomic mass using the following formula: Average atomic mass = (Isotope 1 Abundance × Isotope 1 Atomic mass) + (Isotope 2 Abundance × Isotope 2 Atomic mass) + (Isotope 3 Abundance × Isotope 3 Atomic mass) Plug in the given values: Average atomic mass = (0.7899 * 23.98504) + (0.1000 * 24.98584) + (0.1101 * 25.98259) Calculate: Average atomic mass = 18.94538 + 2.49858 + 2.85835 Average atomic mass = 24.30231 Therefore, the average atomic mass of magnesium is approximately 24.30231 amu.

Step 2: Sketch the mass spectrum

The mass spectrum of an element is a graphical representation of the isotopes and their abundances. It has mass-to-charge (m/z) ratio on the x-axis and abundance on the y-axis. For magnesium, we have three isotopes and their given abundances. To sketch the mass spectrum: 1. Mark the m/z values on the x-axis corresponding to the atomic masses of the isotopes (rounded to the nearest integer): 24, 25, and 26 for 24Mg, 25Mg, and 26Mg, respectively. 2. Draw vertical lines at these m/z values with heights proportional to the isotopic abundances: 78.99%, 10.00%, and 11.01% for 24Mg, 25Mg, and 26Mg, respectively. Your sketch should show three distinct peaks corresponding to the magnesium isotopes, with the peak at m/z = 24 being the tallest, followed by the peaks at m/z = 26 and m/z = 25.

Key Concepts

Isotopic Abundances

Isotopic abundances refer to the percentage of each isotope that occurs in a natural sample of an element. These abundances are crucial for calculating the average atomic mass of an element because each isotope has a slightly different mass. For elements like magnesium, which can have multiple naturally occurring isotopes, a weighted average is used to determine the atomic mass that is listed on the periodic table.

To calculate the average atomic mass, each isotope's atomic mass is multiplied by its abundance, and the products are summed. Taking the example of magnesium from your textbook exercise, this involves using the given isotopic abundances of Magnesium-24, Magnesium-25, and Magnesium-26. Properly understanding isotopic abundance can aid in grasping concepts like radioactivity and the formation of elements in stars.

To enhance comprehension of concepts such as these, it's beneficial to look at real-world examples or participate in hands-on laboratory experiments, where students can measure isotopic abundances and calculate average atomic masses for themselves.

Mass Spectrum

The mass spectrum is a graph that shows the mass-to-charge ratio (m/z) of ions on the horizontal axis against their relative abundance on the vertical axis. This technique is often used in mass spectrometry to identify the different isotopes of an element, as well as to determine the structure of molecules. In the case of magnesium as laid out in your exercise, a mass spectrum would display three peaks corresponding to its natural isotopes: Mg-24, Mg-25, and Mg-26.

To better understand mass spectra, one might compare them to a fingerprint for elements and molecules — they are unique and can be used for identification. Each peak in the spectrum corresponds to an isotope, and its height represents the isotope's abundance. As demonstrated in the step-by-step solution, to create a rudimentary sketch of the mass spectrum, you align the peaks with the isotopes' mass numbers and scale their height according to the isotopic abundances. For enhancing learning, students could use simulation software to visualize how changes in isotopic abundances affect the peaks in the mass spectrum.

Atomic Mass Unit (amu)

The atomic mass unit, abbreviated as amu, is a standard unit of mass that quantifies the mass of atoms and molecules. It is defined as one twelfth of the mass of a carbon-12 atom, which is approximately equal to 1.66053906660 × 10^-24 grams. This tiny unit is incredibly useful when dealing with the minuscule masses of individual atoms and isotopes that would be impractical to express in grams.

Understanding the concept of amu is essential when calculating average atomic masses and interpreting mass spectra. The textbook exercise illustrates the use of amu in expressing the masses of isotopes. Developing a firm grasp of the amu allows for a deeper comprehension of the scales involved in atomic physics and chemistry. To facilitate the learning of this concept, practical examples involving calculations with atomic masses, such as balancing chemical equations or stoichiometry, can be applied to reinforce the significance of the amu as a measurement unit.