Problem 18 - Chemistry and Chemical Reactivity

Question

Verify that the atomic weight of magnesium is 24.31 given the following information: $$\begin{aligned} &^{24} \mathrm{Mg}, \text { mass }=23.985042 \mathrm{u} ; \text { percent abundance }=78.99 \%\\\ &^{25} \mathrm{Mg}, \text { mass }=24.985837 \mathrm{u} ; \text { percent abundance }=10.00 \%\\\ &^{26} \mathrm{Mg}, \text { mass }=25.982593 \text { u; percent abundance }=11.01 \% \end{aligned}$$

Step-by-Step Solution

Answer

The atomic weight of magnesium is verified as approximately 24.31 using isotopic masses and abundances.

Step 1: Understand the Concept

The atomic weight of an element is the weighted average of the masses of its isotopes based on their natural abundance. This is calculated by multiplying the mass of each isotope by its relative abundance (as a fraction) and summing these values.

Step 2: Convert Percentages to Decimals

Given the percent abundances of the isotopes, convert these percentages to decimals for calculation:- For \(^{24} \mathrm{Mg}\), abundance = \( \frac{78.99}{100} = 0.7899 \)- For \(^{25} \mathrm{Mg}\), abundance = \( \frac{10.00}{100} = 0.1000 \)- For \(^{26} \mathrm{Mg}\), abundance = \( \frac{11.01}{100} = 0.1101 \)

Step 3: Calculate Contribution of Each Isotope

Multiply the mass of each isotope by its corresponding decimal abundance:- Contribution of \(^{24} \mathrm{Mg} = 23.985042 \times 0.7899 = 18.945355 \)- Contribution of \(^{25} \mathrm{Mg} = 24.985837 \times 0.1000 = 2.498584 \)- Contribution of \(^{26} \mathrm{Mg} = 25.982593 \times 0.1101 = 2.860236 \)

Step 4: Sum the Contributions

Add the individual contributions of the isotopes to find the atomic weight of magnesium:\[ 18.945355 + 2.498584 + 2.860236 = 24.304175 \]

Step 5: Verify and Compare

The calculated weighted atomic weight is approximately \(24.30\). This value is very close to the given atomic weight of magnesium, which is \(24.31\), possibly differing due to rounding.

Key Concepts

Isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but different numbers of neutrons in their nuclei. This difference in neutron count results in isotopes of a given element having different mass numbers, even though they exhibit similar chemical behavior.
Magnesium, for example, possesses three naturally occurring isotopes: \(^{24} \text{Mg}\), \(^{25} \text{Mg}\), and \(^{26} \text{Mg}\). Each has a distinct mass due to their different numbers of neutrons.
Understanding isotopes is crucial for calculating the atomic weight of an element because these variations account for the different mass contributing to the element's overall atomic mass.

Weighted Average

The concept of a weighted average is integral to calculating the atomic weight of elements with multiple isotopes. In a weighted average, different values contribute unequally to the final average, which reflects the significance of each value.
For atomic weight, each isotope's mass is multiplied by its fractional abundance (as a decimal, derived from the percentage). These products are then summed to determine the atomic weight.

• Example: For \(^{24} \text{Mg}\) with a mass of 23.985042 u and 78.99% abundance, the contribution to the weighted average is \(23.985042 \times 0.7899\).
• All isotope contributions are added: Weighted Atomic Weight = Sum of all contributions.

This weighted sum gives a more accurate measure of the element's "average" atomic mass, taking into account how common each variant is.

Natural Abundance

Natural abundance refers to the relative proportion of each isotope of an element that occurs in nature. Expressed as percentages, these tell us how much of each isotope is found compared to the total amount of the element.
For magnesium, the natural abundance values are 78.99% for \(^{24} \text{Mg}\), 10.00% for \(^{25} \text{Mg}\), and 11.01% for \(^{26} \text{Mg}\). Knowing these values is essential for calculating the atomic weight, as they inform the weighted average calculations, indicating how each isotope influences the overall atomic mass.

Atomic Mass Unit

The atomic mass unit (u) is a standard unit of mass that quantifies the masses of atoms and their subatomic components. It is defined as one-twelfth the mass of a carbon-12 atom, approximately equivalent to \(1.66053906660 \times 10^{-27}\) kilograms.
When calculating atomic weights, such as that of magnesium, the mass of each isotope is expressed in atomic mass units. This allows scientists to express the incredibly small masses of atoms in a manageable and comparable form. Each isotope's mass in atomic mass units, paired with its natural abundance, provides the foundation for computing the element's overall atomic weight.

• Example: \(\text{Mass of } ^{24} \text{Mg} = 23.985042 \text{ u}\)
• Used for accuracy and consistency in atomic weight calculations

By understanding atomic mass units, you can grasp the scale and precision required in atomic evaluations.