Introduction

The atomic weight of chlorine (Cl) is 35.5, which is an average of the naturally occurring isotopes Cl-35 and Cl-37. To understand the ratio of these isotopes, we need to analyze their natural abundances. This article will explain the process and provide a detailed method to find this ratio using the concept of weighted averages.

Understanding the Concept

The atomic weight of a given element is the average of all its naturally occurring isotopes, weighted by their abundances. Chlorine has two naturally occurring isotopes: Cl-35 and Cl-37. The atomic masses of these isotopes are 35 and 37, respectively.

Formulating the Problem

Let:

Cl-35 is in abundance of (x). Cl-37 is in abundance of (y).

We know the following:

The average atomic weight of chlorine is 35.5. The atomic weight of Cl-35 is 35 and Cl-37 is 37. The sum of the abundances must equal 1: (x y 1).

Solving for the Abundances

The total mass of chlorine is given by the weighted sum of its isotopes:

[35x 37y 35.5]

Substituting (y 1 - x) into the equation, we get:

[35x 37(1 - x) 35.5]

Simplifying the equation:

[35x 37 - 37x 35.5]

[-2x 37 35.5]

[-2x 35.5 - 37]

[-2x -1.5]

[x 0.75]

Now, substituting back to find (y):

[y 1 - x 1 - 0.75 0.25]

Interpreting the Results

The abundances of the isotopes are:

Cl-35: 75% or 0.75 Cl-37: 25% or 0.25

The ratio of Cl-35 to Cl-37 can be calculated as:

[text{Ratio} frac{x}{y} frac{0.75}{0.25} 3]

Therefore, the ratio of Cl-35 to Cl-37 in ordinary chlorine is 3:1.

Verification Using a Different Approach

Let's verify this using a different method:

Let the percent abundance of Cl-35 be (x) Then the percent abundance of Cl-37 is (100 - x)

The atomic weight equation becomes:

[35.5 frac{x times 35}{100 - x times 37}{100}]

Solving for (x):

[35.5 frac{35x - 37x}{100 - x}]

[35.5 frac{-2x}{100 - x}]

[35.5(100 - x) -2x]

[3550 - 35.5x -2x]

[3550 33.5x]

[x frac{3550}{33.5} 75%]

Therefore, the percent abundance of Cl-35 is 75% and Cl-37 is 25%.

Conclusion

Using the concept of weighted averages and natural abundances, we determined that the ratio of Cl-35 to Cl-37 in ordinary chlorine is 3:1. This method can be applied to other elements to determine the ratio of their isotopes based on their atomic weights and natural abundances.