Question

Consider the three isotopes of magnesium discussed in Problem 17. Given that the relative natural abundances of these isotopes are \(79 \%, 10 \%,\) and \(11 \%\) respectively, without looking at the inside cover of this book, what is the approximate atomic molar mass of magnesium? Explain how you made your prediction.

Short answer

The approximate atomic molar mass of magnesium is 24.31 amu, calculated by finding the weighted average of the atomic masses of its three main isotopes (Mg-24, Mg-25, and Mg-26) using their given relative natural abundances: \(Atomic \thinspace molar \thinspace mass \thinspace of \thinspace Magnesium \approx (0.79 \times 23.98) + (0.10 \times 24.98) + (0.11 \times 25.98)\).

Step-by-step solution

Identify the isotopes of magnesium

First, we need to identify the isotopes of magnesium discussed in Problem 17. Magnesium has three main isotopes: Mg-24, Mg-25, and Mg-26. Their respective atomic masses are approximately 23.98 amu (Mg-24), 24.98 amu (Mg-25), and 25.98 amu (Mg-26).

Calculate the atomic molar mass of each isotope

Given the relative natural abundances of these isotopes, we can calculate their contribution to the overall atomic molar mass of magnesium using percentage values. 1. Mg-24: \(79 \% \times 23.98 \thinspace amu\) 2. Mg-25: \(10 \% \times 24.98 \thinspace amu\) 3. Mg-26: \(11 \% \times 25.98 \thinspace amu\)

Compute the overall atomic molar mass of magnesium

Now, sum the contributions of the isotopes to find the overall atomic molar mass of magnesium: \(Atomic \thinspace molar \thinspace mass \thinspace of \thinspace Magnesium = (79 \% \times 23.98 \thinspace amu) + (10 \% \times 24.98 \thinspace amu) + (11 \% \times 25.98 \thinspace amu)\)

Evaluate and approximate the result

Plug in the percentage values and evaluate the expression: \(Atomic \thinspace molar \thinspace mass \thinspace of \thinspace Magnesium \approx (0.79 \times 23.98) + (0.10 \times 24.98) + (0.11 \times 25.98)\) \(Atomic \thinspace molar \thinspace mass \thinspace of \thinspace Magnesium \approx 18.95 \thinspace amu + 2.50 \thinspace amu + 2.86 \thinspace amu,\) \(Atomic \thinspace molar \thinspace mass \thinspace of \thinspace Magnesium \approx 24.31 \thinspace amu\) Hence, the approximate atomic molar mass of magnesium is 24.31 amu.

Key concepts

magnesium isotopes

Magnesium has three naturally occurring isotopes: Mg-24, Mg-25, and Mg-26. Isotopes are variants of elements that have the same number of protons but different numbers of neutrons. Because magnesium is an essential element in both biology and industry, understanding its isotopes helps in various applications.
Mg-24 is the most abundant isotope, accounting for approximately 79% of magnesium found in nature. The less abundant Mg-25 and Mg-26 account for roughly 10% and 11% respectively. Each of these isotopes has a slightly different atomic mass, measured in atomic mass units (amu), which we will explore further. This difference in mass allows scientists to calculate the atomic molar mass of magnesium by considering the natural abundance of each isotope.

isotope natural abundance

The term 'isotope natural abundance' refers to the percentage of a particular isotope that occurs naturally out of all isotopes for an element. This percentage helps us to understand how common or rare each isotope is.
For magnesium, the natural abundances are crucial in determining its average atomic mass. When we say that Mg-24 is 79% abundant, we mean that in any sample of magnesium, 79% of the atoms are Mg-24. The other isotopes, Mg-25 and Mg-26, appear in smaller quantities, at 10% and 11% respectively. These percentages are vital for accurately calculating the element's atomic molar mass because they indicate how each isotope contributes to the overall mass of a typical sample of magnesium.

atomic mass unit (amu)

An atomic mass unit (amu) is a standard unit of mass used to express atomic and molecular weights. It is based on the mass of the carbon-12 isotope, which is defined as exactly 12 amu. Therefore, 1 amu is equal to one twelfth of the mass of a carbon-12 atom.
In the context of magnesium isotopes, each isotope has a specific atomic mass measured in amu. Mg-24 has a mass of approximately 23.98 amu, Mg-25 approximately 24.98 amu, and Mg-26 approximately 25.98 amu. These values help chemists and students alike to calculate the weighted average atomic mass of magnesium by using the isotopes' specific masses in amu coupled with their natural abundances.

weighted average

The concept of a weighted average is essential in determining the average atomic molar mass of magnesium. A weighted average accounts for the relative importance, or weight, of each component in a data set. Here, the components are the isotopes, and their natural abundances act as the weights.
To calculate the weighted average atomic molar mass of magnesium, you multiply the atomic mass of each isotope by its respective natural abundance (expressed as a decimal), sum the results, and then observe the overall mean. For example, the formula is: \[ \text{Atomic Molar Mass} = (0.79 \times 23.98) + (0.10 \times 24.98) + (0.11 \times 25.98) \]This gives you the average mass of magnesium as approximately 24.31 amu, considering all isotopes and their natural prevalences.