The atomic weight of chlorine is 35.5. It has two isotopes of atomic weight 35 and 37. What is the percentage of the heavier isotope in the sample?
Understand the Problem
The question is asking for the percentage of the heavier isotope of chlorine in a sample, given the atomic weight of chlorine and the weights of its isotopes. The problem requires applying the concept of averages and proportions to solve.
Answer
The percentage of the heavier isotope is $25\%$.
Answer for screen readers
The percentage of the heavier isotope in the sample is $25%$.
Steps to Solve
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Define Variables for Isotope Weights Let the percentage of the heavier isotope ($^{37}\text{Cl}$) be $x$%. Then the percentage of the lighter isotope ($^{35}\text{Cl}$) will be $(100 - x)$%.
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Set Up the Average Atomic Weight Equation We know the average atomic weight can be expressed as: $$ 35.5 = \frac{35(100 - x) + 37x}{100} $$
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Multiply Through by 100 to Eliminate the Denominator To simplify the equation, multiply the entire equation by 100: $$ 3550 = 35(100 - x) + 37x $$
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Distribute and Combine Like Terms Expanding the equation gives: $$ 3550 = 3500 - 35x + 37x $$ This simplifies to: $$ 3550 = 3500 + 2x $$
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Isolate the Variable $x$ Subtract 3500 from both sides: $$ 50 = 2x $$ Now divide by 2: $$ x = 25 $$
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Determine the Percentage of the Heavier Isotope Thus, the percentage of the heavier isotope ($^{37}\text{Cl}$) is: $$ x = 25% $$
The percentage of the heavier isotope in the sample is $25%$.
More Information
This calculation shows how isotopic abundance affects the average atomic weight. The heavier isotope's weight contributes more significantly to the average because it is greater than the lighter isotope.
Tips
- Incorrectly setting up the equation: Make sure to correctly represent the percentages of both isotopes in the average atomic weight equation.
- Forgetting to distribute terms: Always be careful to include all parts of the equation when expanding.
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