Question

Calculate the atomic mass for magnesium given the following data for its natural isotopes: $$ \begin{array}{lll} { }^{24} \mathrm{Mg} & 23.985 \mathrm{amu} & 78.70 \% \\ { }^{25} \mathrm{Mg} & 24.986 \mathrm{amu} & 10.13 \% \\ { }^{26} \mathrm{Mg} & 25.983 \mathrm{amu} & 11.17 \% \end{array} $$

Short answer

The atomic mass of magnesium is approximately 24.305 amu.

Step-by-step solution

Understand the Problem

We need to calculate the average atomic mass of magnesium by using the given isotopic masses and their relative abundances. This is determined by summing the products of each isotope's mass and its abundance.

Convert Percentages to Decimals

Convert the percentage abundances into decimals by dividing each percentage by 100. For example, for \(^{24}\text{Mg}\), the abundance is \( \frac{78.70}{100} = 0.7870 \). Apply the same conversion for \(^{25}\text{Mg}\) and \(^{26}\text{Mg}\).

Calculate Contribution of Each Isotope

Multiply each isotope's atomic mass by its decimal abundance to find the contribution of that isotope to the average atomic mass. For example, contribution of \(^{24}\text{Mg}\) is \(23.985 \times 0.7870\). Compute similarly for the other isotopes.

Sum the Contributions

Add the contributions of each isotope to find the total average atomic mass of magnesium. Sum the contributions: \(23.985\times0.7870 + 24.986\times0.1013 + 25.983\times0.1117\).

Final Calculation and Verification

Perform the arithmetic calculations to find the total sum and verify each step to ensure accuracy. The calculated sum gives the average atomic mass of magnesium.

Key concepts

Isotopic Composition

Every element consists of atoms, and atoms of the same element can have different numbers of neutrons.
This leads to the presence of various forms of the element, known as isotopes. Isotopic composition refers to the proportion of different isotopes present in a sample of an element. For any given element, isotopes have the same number of protons but differing numbers of neutrons. This difference in neutron number means that each isotope has a different atomic mass.

When dealing with isotopes, it's important to understand:

• **Isotopes are chemically similar but vary in mass.** Since isotopes contain different numbers of neutrons, they have different masses but behave similarly chemically.
• **The isotopic composition affects the average atomic mass of the element.** The more abundant an isotope, the more it influences the atomic mass.

Understanding isotopic composition is crucial when calculating the atomic mass, as it provides necessary information on how each isotope contributes to the overall average atomic mass of an element.

Average Atomic Mass

The average atomic mass of an element is a weighted average of the masses of its isotopes, taking into account their abundances. This means that isotopes with higher natural abundances impact the average atomic mass more significantly.

Here is how the calculation usually works:

• **Convert percentage abundance to decimals.** This step involves dividing the percentage by 100, transforming percent figures into a usable decimal form for multiplication, e.g., 78.70% becomes 0.7870.
• **Calculate each isotope's contribution.** Multiply each isotope's mass by its corresponding decimal abundance. This yields the contribution of each isotope to the average mass.
• **Add up contributions.** The sum of all these contributions represents the element’s average atomic mass.

The process of calculating average atomic mass ensures that each isotope's effect on the weight of an element is accurately represented according to its presence. This average is what is typically listed on the periodic table next to the symbol of a given element.

Magnesium Isotopes

Magnesium is a commonly occurring element in nature, featuring three stable isotopes:

• **^{24}Mg** - This is the most abundant magnesium isotope, accounting for 78.70% of natural magnesium.
• **^{25}Mg** - This isotope makes up 10.13% of magnesium found in nature.
• **^{26}Mg** - Comprising 11.17% of naturally occurring magnesium, this is the least abundant stable isotope.

Each of these isotopes has a slightly different atomic mass due to the variation in neutron number, which affects the overall atomic weight of magnesium.

The average atomic weight of magnesium depends on the weighted contributions of all these isotopes.
Understanding the isotopes and their respective abundances is key to correctly calculating the atomic weight of magnesium, as it reflects the most accurate figure based on naturally occurring samples. This knowledge helps in numerous scientific applications ranging from geology to manufacturing and even in biological processes where magnesium plays a significant role.