Problem 43 - General Chemistry: Principles and Modern Applications

Question

There are three naturally occurring isotopes of magnesium. Their masses and percent natural abundances are \(23.985042 \mathrm{u}, 78.99 \% ; 24.985837 \mathrm{u}, 10.00 \% ;\) and \(25.982593 \mathrm{u}, 11.01 \% .\) Calculate the weighted-average atomic mass of magnesium.

Step-by-Step Solution

Answer

The weighted-average atomic mass of magnesium is the sum of the weighted masses for all isotopes, calculated by multiplying each isotope's atomic mass by its relative abundance.

Step 1: Determine the weighted mass of each isotope

For each isotope, multiply the given atomic mass by its relative natural abundance (expressed as a decimal). For example, for the first isotope with atomic mass 23.985042 u and abundance 78.99%, the weighted mass is \(23.985042 \times \frac{78.99}{100}\). Repeat this for the other two isotopes.

Step 2: Calculate the sum of all weighted masses

Sum the weighted masses calculated in Step 1. This gives the total weighted mass for all isotopes of magnesium.

Step 3: Find the Weighted Average

Since the percentages of all isotopes add up to 100%, the sum of the weighted masses is the weighted-average atomic mass of magnesium.

Key Concepts

Isotopes

To understand atomic mass calculations, it's important first to grasp the concept of isotopes. Isotopes are different forms of the same chemical element, each having the same number of protons but a different number of neutrons. This means that while all isotopes of an element have the same atomic number, they have different atomic masses.
A great example is magnesium, which naturally occurs as a mix of three isotopes. Each isotope has its unique atomic mass:

• Isotope 1: 23.985042 u
• Isotope 2: 24.985837 u
• Isotope 3: 25.982593 u

Understanding isotopes is essential because they influence the element's atomic mass that we typically see on the periodic table. Isotopes of an element are identical in their chemical behavior but vary in physical properties due to their mass differences.

Natural Abundance

Natural abundance refers to the relative proportion of each isotope of an element found in nature. It is expressed as a percentage that reflects how common or rare a particular isotope is in a given sample of an element. For instance, in magnesium, the natural abundances are as follows:

• First Isotope: 78.99%
• Second Isotope: 10.00%
• Third Isotope: 11.01%

These values indicate the percentage of atoms of each isotope present in a natural sample of magnesium. Knowing the natural abundance helps us calculate the weighted average, which we'll discuss shortly. This percentage is converted into a decimal form for calculation purposes, which means dividing the percentage by 100.

Weighted-Average Atomic Mass

The weighted-average atomic mass of an element is a calculation that gives the average mass of an atom, considering the different isotopes and their relative abundances. This value is what we often see listed on the periodic table.

The process involves two major steps:

• Calculate the weighted mass of each isotope: Multiply the atomic mass of each isotope by its natural abundance (as a decimal). For example, for the first isotope of magnesium with a mass of 23.985042 u, the calculation would be 23.985042 times 0.7899.
• Add the weighted masses together: The sum of all these weighted masses gives the total weighted-average atomic mass of magnesium.

Since the sum of the natural abundances is 100%, or 1 in decimal form, the total weighted mass directly converts to the weighted-average atomic mass. This practical approach allows chemists and students alike to familiarize themselves with the average weight of an element's atom in natural conditions.