Question

Chlorine is found in nature in two isotopic forms, one of atomic mass 35 amu and one of atomic mass 37 amu. The average atomic mass of chlorine is \(35.453\) amu. What is the percent with which each of these isotopes occurs in nature?

Short answer

The percent abundance of Chlorine-35 in nature is 77.35% and that of Chlorine-37 is 22.65%.

Step-by-step solution

Set up variables for percentage abundances

Let x represent the percentage of Chlorine-35 isotope and y represent the percentage of Chlorine-37 isotope. Since the total percentage is 100%, we have the following equation: x + y = 100

Set up an equation for the average atomic mass

The average atomic mass of chlorine can be found by multiplying the atomic mass of each isotope by its percentage and then adding the two values. So we have the equation: \(35 \cdot x + 37 \cdot y = 35.453 \cdot 100\)

Solve for x and y

We have two linear equations in two variables, x and y. We can either solve this system of equations using substitution or elimination methods. In this solution, we will use the substitution method. First, we solve the equation x + y = 100 for x or y. Let's solve for x: x = 100 - y Now, substitute x in the second equation: \(35 \cdot (100 - y) + 37 \cdot y = 35.453 \cdot 100\)

Solve for y

Now, we have an equation in a single variable y: \(3500 - 35y + 37y = 3545.3\) Simplify and solve for y: \(35y - 37y = 3500 - 3545.3\) \(-2y = -45.3\) \(\ y = 22.65\)

Solve for x

Now we can substitute y back into x = 100 - y: x = 100 - 22.65 x = 77.35

Interpret the results

We found that x = 77.35 and y = 22.65, which means that the percent abundance of Chlorine-35 in nature is 77.35% and that of Chlorine-37 is 22.65%.

Key concepts

Chlorine Isotopes

Chlorine is an element that is naturally found in two isotopic forms, meaning there are two types of chlorine atoms with different numbers of neutrons. Isotopes have the same number of protons but differ in the number of neutrons, resulting in varying atomic masses. For chlorine, these isotopes are:

• Chlorine-35 with an atomic mass of 35 amu
• Chlorine-37 with an atomic mass of 37 amu

These two isotopes exist naturally in different proportions, contributing to the overall atomic mass of chlorine found in the periodic table. Understanding the distribution of these isotopes helps us grasp the concept of isotopic abundance, which indicates how common each isotope is in nature.

Average Atomic Mass

The average atomic mass of an element is a weighted average of the atomic masses of its isotopes. It takes into account both the mass and the relative abundance of each isotope.The formula used to calculate the average atomic mass is:\[ \text{Average Atomic Mass} = (\text{mass of isotope 1} \times \text{abundance of isotope 1}) + (\text{mass of isotope 2} \times \text{abundance of isotope 2}) \]For chlorine, the isotopes are chlorine-35 and chlorine-37. Their contributions to the average atomic mass depend on both their individual masses and how frequently they occur in nature. This is a real-life application of using weighted averages to solve problems in chemistry.

Percent Abundance

Percent abundance is a way of expressing the proportion of each isotope present in a natural sample of an element. It tells us what percentage each isotope contributes to the total composition. To determine the percent abundance, we perform calculations based on the average atomic mass and the masses of the isotopes. Using simple algebra, we calculate:

• The sum of the percentage abundances of both isotopes is 100%
• The contributions of the isotopes to the average atomic mass

In our example, we set up equations to represent these conditions and solve them to find the specific percentages of each isotope, giving us the complete isotopic picture.

Isotopic Composition

Isotopic composition refers to the individual percentage representation of each isotope in a sample of an element. It provides insights into the natural occurrence and distribution of isotopes. Knowing the isotopic composition is essential in fields such as geochemistry and radiometric dating, as it can tell us about processes and timelines involved. For chlorine, understanding its isotopic composition means recognizing that Chlorine-35 is more prevalent than Chlorine-37. Such information is crucial, for instance, in the chemical industry, where precise isotopic knowledge can influence production and application decisions. Overall, mastering isotopic composition helps in the detailed analysis and application of chemical data.