Problem 38 - Chemistry An Atoms First Approach

Question

An element "X" has five major isotopes, which are listed below along with their abundances. What is the element? $$\begin{aligned} &\begin{array}{|ccc|}\hline \text { Isotope } & \text { Percent Natural Abundance } & \text { Mass (u) } \\\\\hline 46 \mathrm{X} & 8.00 \% & 45.95232 \\\^{47} \mathrm{X} & 7.30 \% & 46.951764 \\\^{48} \mathrm{X} & 73.80 \% & 47.947947 \\\^{49} \mathrm{X} & 5.50 \% & 48.947841 \\\^{50} \mathrm{X} & 5.40 \% & 49.944792 \\\\\hline\end{array}\\\&\end{aligned}$$

Step-by-Step Solution

Answer

Average atomic mass of "X" = \(3.6761856 \, \text{u} + 3.42542868 \, \text{u} + 35.369306866 \, \text{u} + 2.69213365 \, \text{u} + 2.696996768 \, \text{u}\) Average atomic mass of "X" = \(47.860051574 \, \text{u}\) #tag_title

Step 1: Identify given information

List all given data and unknowns.

Step 2: Determine relevant principles

Identify applicable chemical laws.

Step 3: Set up and solve

Apply equations and solve.

Step 4: State the answer

The answer is: Average atomic mass of "X" = \(3.6761856 \, \text{u} + 3.42542868 \, \text{u} + 35.369306866 \, \text{u} + 2.69213365 \, \text{u} + 2.696996768 \, \text{u}\) Average atomic mass of "X" = \(47.860051574 \, \text{u}\) #tag_title

Key Concepts

Isotopic Abundance

Understanding isotopic abundance is essential when trying to identify an element or calculate its average atomic mass. Isotopic abundance refers to the percentage of a particular isotope present in a natural sample of an element. Since isotopes of an element have different numbers of neutrons, they vary in mass but not in chemical properties.

For example, if an element has two isotopes, where one isotope makes up 70% of the sample and the other isotope constitutes the remaining 30%, these percentages represent their isotopic abundances. To determine the average atomic mass of an element from isotopic abundances, we multiply the mass of each isotope by its respective abundance (expressed as a decimal), and then sum up these products.

In exercises involving isotopic abundances, it's crucial to remember to convert percentages into decimals by dividing by 100, as was done in the original exercise with element 'X'. This calculation will give a weighted average that reflects the relative masses of the isotopes as they occur in nature, which is important in accurately identifying an element or predicting its behavior.

Atomic Mass Unit (u)

The atomic mass unit, abbreviated as 'u', is a standard unit of mass that quantifies the mass of atoms and molecules. Defined as one twelfth of the mass of a carbon-12 atom, an atomic mass unit is approximately equal to 1.66053906660 × 10^-27 kilograms.

This tiny unit of measurement is essential in chemistry and physics because it allows scientists to express the masses of atoms and subatomic particles on a scale that is convenient for comparison and calculations. The use of atomic mass units makes it much easier to comprehend the very small masses involved in atomic-scale phenomena.

In practice, when calculating the average atomic mass of an element like 'X' from the given problem, scientists use the atomic mass unit to represent the mass of each isotope. This standardization facilitates comparison across different elements and isotopes.

Isotopes of Elements

Isotopes are variants of a particular chemical element. While all isotopes of an element share the same number of protons, they have different numbers of neutrons, resulting in varying atomic masses. The presence of different isotopes leads to the concept of an average atomic mass for an element.

Most elements have several naturally occurring isotopes, and their properties are vital for various scientific and industrial applications. For instance, some isotopes are used in medical imaging and treatment, while others are employed in geological dating techniques.

In the example provided in the question, the element 'X' has five major isotopes. These isotopes are what make it necessary to use their abundance and masses to calculate the average atomic mass. The commonly listed atomic mass of an element on the periodic table is, in fact, this weighted average reflecting both the mass and natural abundance of each of its isotopes. Thus, the isotopic makeup of an element is fundamental to understanding its physical and chemical properties.