Problem 116 - Chemistry Matter and Change

Question

Magnesium Isotopes Compute the mass number, \(X,\) of the third isotope of magnesium given that the respective abundances of the naturally occurring isotopes are: \(79.0 \%, 10 \%,\) and 11\(\%\) for \(_{12}^{24} \mathrm{Mg}_{12}^{25} \mathrm{Mg}_{12}^{X} \mathrm{Mg}.\) The relative atomic mass of magnesium is 24.305 amu.

Step-by-Step Solution

Answer

The mass number \( X \) of the third isotope of magnesium is 26.

Step 1: Understand the Given Information

We have isotopes of magnesium with masses of 24 amu, 25 amu, and an unknown mass \( X \). The natural abundances of these isotopes are given as 79%, 10%, and 11% respectively. The average atomic mass of magnesium is provided as 24.305 amu.

Step 2: Set Up the Weighted Average Equation

The relative atomic mass is the average of the isotopes' masses, weighted by their abundances. Set up the equation: \[ 0.79 \times 24 + 0.10 \times 25 + 0.11 \times X = 24.305 \].

Step 3: Simplify and Solve for X

First, calculate the contributions from the known isotopes: \( 0.79 \times 24 = 18.96 \) and \( 0.10 \times 25 = 2.5 \). Substitute these into the equation to get: \[ 18.96 + 2.5 + 0.11X = 24.305 \]. Simplify to find \( 0.11X = 24.305 - 18.96 - 2.5 \), which gives \( 0.11X = 2.845 \). Divide by 0.11: \( X = \frac{2.845}{0.11} \approx 25.86 \).

Step 4: Round to the Nearest Whole Number

Since mass numbers are whole numbers, round \( 25.86 \) to the nearest whole number to get \( X = 26 \). This is the mass number of the third magnesium isotope.

Key Concepts

Magnesium Isotopes

Magnesium is a chemical element represented by the symbol Mg. It is a shiny gray solid and is the eighth most abundant element in the Earth's crust. Isotopes are different "versions" of the same element that differ in the number of neutrons within their nuclei but share the same number of protons. For magnesium, the three naturally occurring isotopes are:

• \(_{12}^{24} \mathrm{Mg}\)
• \(_{12}^{25} \mathrm{Mg}\)
• \(_{12}^{26} \mathrm{Mg}\)

In this context, each isotope of magnesium has a different mass number, which accounts for the slight variations in their individual atomic masses.

Mass Number

The mass number of an isotope is the total count of protons and neutrons in the nucleus of that isotope. For magnesium isotopes, the atomic number is always twelve because each has twelve protons. The difference comes in the number of neutrons. The mass number gives us essential information about the isotope's identity when combined with the atomic number.For example:

• \(_{12}^{24} \mathrm{Mg}\): Contains 12 protons and 12 neutrons.
• \(_{12}^{25} \mathrm{Mg}\): Contains 12 protons and 13 neutrons.
• \(_{12}^{26} \mathrm{Mg}\): Contains 12 protons and 14 neutrons.

Understanding the mass number is crucial as it helps define the isotope and distinguishes it from other isotopes of the same element.

Atomic Mass

The atomic mass, or atomic weight, of an element is typically the weighted average of the masses of the isotopes of an element. These are expressed in atomic mass units (amu), where one amu is defined as one twelfth the mass of a carbon-12 atom. Magnesium has an average atomic mass of 24.305 amu. This value is calculated not by simply averaging the mass numbers but instead by using a weighted average of all isotopes, taking into account their natural abundances. Atomic masses are essential because they are used to quantify the amount of an element in a given sample and calculate how the element behaves in reactions and combinations.

Weighted Average Calculation

A weighted average is a type of average where each value in a data set is multiplied by a "weight" before summing. The weights reflect the importance or frequency of each value. This method is especially useful when calculating the atomic mass of elements with multiple isotopes.For magnesium:

• \(_{12}^{24} \mathrm{Mg}\): 79% abundance
• \(_{12}^{25} \mathrm{Mg}\): 10% abundance
• \(_{12}^{26} \mathrm{Mg}\): 11% abundance

In the case of magnesium isotopes, the weighted average is calculated using this formula:\[0.79 \times 24 + 0.10 \times 25 + 0.11 \times 26 = 24.305 \, \text{amu}\]This calculation shows how each isotope's mass contributes to the average atomic mass based on its relative abundance, ensuring precision in understanding the element's characteristics in nature.