Question

Naturally occurring magnesium (one of the elements in milk of magnesia) is composed of \(78.99 \%\) of \({ }^{24} \mathrm{Mg}\) (atomic mass \(=23.9850 \mathrm{u}\) ) \(, 10.00 \%\) of \({ }^{25} \mathrm{Mg}\) (atomic mass \(=24.9858 \mathrm{u}\) ), and \(11.01 \%\) of \({ }^{26} \mathrm{Mg}\) (atomic mass \(=\) \(25.9826 \mathrm{u}\) ). Use these data to calculate the average atomic mass of magnesium.

Short answer

The average atomic mass of magnesium is calculated as follows: \(0.7899 \times 23.9850 \, \mathrm{u}\) + \(0.1000 \times 24.9858 \, \mathrm{u}\) + \(0.1101 \times 25.9826 \, \mathrm{u}\) = 24.3056 \, \mathrm{u}\).

Step-by-step solution

Identify the Contribution of Each Isotope

The average atomic mass of an element is the weighted average of the masses of its isotopes, weighted by their natural abundance. First, we need to identify the contribution of each isotope to the average atomic mass. This is done by multiplying the atomic mass of each isotope by its relative abundance (expressed as a decimal).

Convert Percent Abundances to Decimal Form

Convert the percent abundances of each magnesium isotopes to decimal form: 78.99% becomes 0.7899 for \( ^{24}Mg \), 10.00% becomes 0.1000 for \( ^{25}Mg \), and 11.01% becomes 0.1101 for \( ^{26}Mg \) .

Calculate the Contribution of Each Isotope

Calculate the contribution of each isotope to the average atomic mass by multiplying the atomic mass of each isotope by its abundance in decimal form.

Sum the Contributions

Add the contributions of all three isotopes to determine the average atomic mass of magnesium.

Report the Average Atomic Mass

The sum calculated in the previous step is the average atomic mass of magnesium, usually reported in unified atomic mass units (u).

Key concepts

Isotopic Abundance

Isotopic abundance refers to the percentage of a particular isotope naturally found in a sample of an element. It is important because isotopes of an element can have different numbers of neutrons, thus different masses, but their chemical behavior remains largely the same. Therefore, when calculating the average atomic mass of an element, which is what one usually encounters on the periodic table, you need to consider the isotopic abundance of each naturally occurring isotope.

Understanding isotopic abundance is key in various scientific fields, from archaeology (in radiocarbon dating) to physics (in nuclear reactions). In educational exercises, the concept is simplified to a set percentage, but in real-world situations, these abundances might be measured or determined through mass spectrometry or other techniques. The concept is pivotal to the correct calculation of the average atomic mass of elements, as seen in the calculation for magnesium.

Atomic Mass Unit (u)

The atomic mass unit (u), also known as the unified atomic mass unit, is a fundamental constant that expresses the mass of atoms and molecules. It is defined as one twelfth of the mass of a carbon-12 atom at rest and unbound. The value of 1 u is approximately equal to 1.660539040 x 10^-24 grams. This unit offers a convenient way to express atomic and molecular masses, since dealing with such tiny masses in grams would be cumbersome and impractical.

To understand calculations in chemistry, it is vital to recognize that while the concept of atomic mass unit can seem abstract, it forms the bridge between the mass of subatomic particles (like protons and neutrons) and the larger scales we can measure in the lab. It is central to the way we measure, communicate, and calculate chemical properties and reactions.

Magnesium Isotopes

Magnesium naturally consists of three stable isotopes: ²⁴Mg, ²⁵Mg, and ²⁶Mg. These isotopes have the same atomic number but differ in atomic mass because of their varying numbers of neutrons. ²⁴Mg is the most abundant, followed by ²⁶Mg and ²⁵Mg, respectively.

Each isotope plays an equal role chemically but contributes differently to the average atomic mass due to distinct isotopic abundances and masses. In exercises involving magnesium, one calculates its average atomic mass by considering these different contributions from each isotope – a literal balance of masses based on their natural occurrence. This fosters a deeper understanding of why magnesium's atomic mass on the periodic table is not a whole number, illustrating the mosaic of isotopes in nature.