6.1. Problem 1: Chlorine’s Average Atomic Mass

Given: ● Chlorine-35, mass = 34.9688527 mass-units, abundance = 75.78% ● Chlorine-37, mass = 36.9659026 mass-units, abundance = 24.22% Compute: The average atomic mass of chlorine.

This problem involves calculating the average atomic mass of chlorine, which is a weighted average of the masses of its isotopes. To do this, we need to know the mass and the natural abundance (percentage) of each isotope. Here, we have two isotopes of chlorine: Chlorine-35 and Chlorine-37. We will convert their abundances from percentages to fractions and then use these fractions to calculate the average atomic mass based on their respective masses.

Think of the average atomic mass as a way of calculating a student's grade based on various assessments. If a student scores differently on two exams but spends more time on one exam, we weight the final grade reflects both performance levels proportionally to the importance of each exam. Similarly, in our atomic mass calculation, the percentages act like weights, influencing how much each isotope contributes to the overall average.

1. Convert percentages to fractions: 75.78% → 0.7578; 24.22% → 0.2422.

To convert percentages to fractions, we simply divide the percentage by 100. For Chlorine-35, we take 75.78% and convert it to decimal form by dividing by 100, resulting in 0.7578. For Chlorine-37, we do the same with 24.22%, resulting in 0.2422. This step is important because we will use these decimal forms as multipliers in the average calculation.

Converting percentages to fractions is like converting a recipe. If a recipe calls for percentages of ingredients (like 75% flour), you would convert that to a usable amount for a specific total quantity. For example, for 100 grams of total ingredients, you would need 75.78 grams of flour, just like we convert percentages to fractions for our calculations.

2. Multiply each isotope’s mass by its fraction: • 0.7578 × 34.9688527 = 26.5073 mass-units • 0.2422 × 36.9659026 = 8.9458 mass-units
3. Add them: 26.5073 + 8.9458 = 35.4531 mass-units.

In this step, we calculate the contribution of each isotope to the average atomic mass. We take the mass of each isotope and multiply it by its respective fraction (the decimal we calculated earlier). For Chlorine-35, we multiply its mass (34.9688527) by its fraction (0.7578), yielding 26.5073 mass-units. For Chlorine-37, we do a similar calculation, yielding 8.9458 mass-units. Finally, we sum these contributions to obtain the average atomic mass of chlorine, which is approximately 35.4531 mass-units.

Imagine you're mixing two different colored paints, and you want to find the average color. You would calculate how much each paint contributes to the final color based on how much paint you have of each color. Similarly, we are weighting how much each isotope contributes to the overall atomic mass based on its abundance.

Therefore, the average atomic mass of chlorine is about 35.45 mass-units.

In conclusion, we see that the average atomic mass for chlorine is 35.4531 mass-units, which we round off to 35.45 mass-units for simplicity. This value reflects the combined contributions of both Chlorine-35 and Chlorine-37 based on their natural abundances.

Think of the average atomic mass as the collective average score for a group of students. The average score tells you how the students performed overall, incorporating everyone’s contributions relative to how much they participated. The average atomic mass does the same by showing how the isotopes of chlorine contribute to what we recognize as chlorine's mass.

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