Question

Magnesium has three naturally occurring isotopes, \({ }^{24} \mathrm{Mg}\) (isotopic mass \(=23.9850\) amu, abundance \(=78.99 \%),{ }^{25} \mathrm{Mg}\) (isotopic mass \(=24.9858\) amu, abundance \(=10.00 \%\) ), and \({ }^{26} \mathrm{Mg}\) (isotopic mass \(=25.9826\) amu, abundance \(=11.01 \%\) ). Calculate the atomic mass of magnesium.

Short answer

The atomic mass of magnesium is approximately 24.3116 amu.

Step-by-step solution

Identify the given data

List the isotopic masses and their respective abundances: - \({ }^{24} \mathrm{Mg}\) with mass \(23.9850\) amu and abundance \(78.99 \%\)- \({ }^{25} \mathrm{Mg}\) with mass \(24.9858\) amu and abundance \(10.00 \%\)- \({ }^{26} \mathrm{Mg}\) with mass \(25.9826\) amu and abundance \(11.01 \%\)

Convert the percentages to decimal form

Divide each percentage by \100\ to convert to decimal form:- \({ }^{24} \mathrm{Mg}\) abundance: \(0.7899\)- \({ }^{25} \mathrm{Mg}\) abundance: \(0.1000\)- \({ }^{26} \mathrm{Mg}\) abundance: \(0.1101\)

Calculate the weighted average

Use the formula for the atomic mass \[ \text{Atomic mass} = (\text{mass of isotope 1} \times \text{abundance of isotope 1}) + (\text{mass of isotope 2} \times \text{abundance of isotope 2}) + (\text{mass of isotope 3} \times \text{abundance of isotope 3}) \]Substitute the values into the formula:\[ \text{Atomic mass} = (23.9850 \times 0.7899) + (24.9858 \times 0.1000) + (25.9826 \times 0.1101) \]

Compute each term individually

Calculate each term of the expression:\[ 23.9850 \times 0.7899 = 18.9512 \]\[ 24.9858 \times 0.1000 = 2.4986 \]\[ 25.9826 \times 0.1101 = 2.8618 \]

Sum the individual contributions

Add the results from the previous step to find the atomic mass:\[ 18.9512 + 2.4986 + 2.8618 = 24.3116 \]

Key concepts

Isotopes

Isotopes are different forms of the same element that contain the same number of protons but different numbers of neutrons. This results in different atomic masses for each isotope of the element. For example, magnesium has three naturally occurring isotopes: \( \text{^{24}Mg} \), \( \text{^{25}Mg} \), and \( \text{^{26}Mg} \).

Each isotope of magnesium has 12 protons, but they differ in the number of neutrons:

• \( \text{^{24}Mg} \) has 12 neutrons (12 protons + 12 neutrons = 24)
• \( \text{^{25}Mg} \) has 13 neutrons (12 protons + 13 neutrons = 25)
• \( \text{^{26}Mg} \) has 14 neutrons (12 protons + 14 neutrons = 26)

Isotopes have nearly identical chemical properties because the chemistry of an element is primarily determined by the number of protons and electrons, not neutrons.

Weighted Average

The atomic mass of an element that has multiple isotopes is calculated as a weighted average of the isotopic masses. This weighted average considers both the mass and the relative abundance of each isotope.

The formula used to calculate the atomic mass is:
\ \text{Atomic mass} = (\text{mass of isotope 1} \times \text{abundance of isotope 1}) + (\text{mass of isotope 2} \times \text{abundance of isotope 2}) + ... \

In the case of magnesium:
- \( \text{^{24}Mg} \) with mass 23.9850 amu and abundance 78.99%
- \( \text{^{25}Mg} \) with mass 24.9858 amu and abundance 10.00%
- \( \text{^{26}Mg} \) with mass 25.9826 amu and abundance 11.01%
To calculate the weighted average, convert the percentages to decimal form:

• \( 78.99\textrightarrow 0.7899 \)
• \( 10.00\textrightarrow 0.1000 \)
• \( 11.01\textrightarrow 0.1101 \)

Now, use the formula to find the atomic mass: \[ \text{Atomic mass} = (23.9850 \times 0.7899) + (24.9858 \times 0.1000) + (25.9826 \times 0.1101) \] This results in an atomic mass of approximately 24.3116 amu.

Abundance

When we talk about the abundance of isotopes, we are referring to how common each isotope is in a sample of the element. The abundance is expressed as a percentage or a decimal fraction of the whole.

For magnesium, the isotopic abundances are:

• \( \text{^{24}Mg} \) has an abundance of 78.99%
• \( \text{^{25}Mg} \) has an abundance of 10.00%
• \( \text{^{26}Mg} \) has an abundance of 11.01%

These percentages tell us how much of each isotope is present in a naturally occurring sample of magnesium.
To use these abundances in calculations, they need to be converted from percentages to decimals. This is done by dividing each percentage by 100.

Understanding the concept of abundance is crucial for determining the weighted average of atomic masses, as it influences the final value significantly.