Problem 109 - Chemistry: An Atoms-Focused Approach

Question

There are three stable isotopes of magnesium. Their masses are \(23.9850,24.9858,\) and 25.9826 amu. If the average atomic mass of magnesium is 24.3050 amu and the natural abundance of the lightest isotope is \(78.99 \%,\) what are the natural abundances of the other two isotopes?

Step-by-Step Solution

Answer

Answer: The natural abundances of the second and third stable isotopes of magnesium are 13.46% and 7.55%, respectively.

Step 1: Understand the average atomic mass formula

The average atomic mass of an element is calculated by taking the sum of the products of the atomic mass of each isotope and their respective natural abundances. In this case, for magnesium, the formula is: \(24.3050 = 23.9850 * abundance_1 + 24.9858 * abundance_2 + 25.9826 * abundance_3\) We are given the natural abundance of the lightest isotope, \(abundance_1 = 78.99\% = 0.7899\). We can now plug this value into the formula and simplify the equation to find the natural abundances of the other two isotopes.

Step 2: Plug in known values and simplify

Replace \(abundance_1\) with 0.7899 in the formula: \(24.3050 = 23.9850 * 0.7899 + 24.9858 * abundance_2 + 25.9826 * abundance_3\) Now, simplify the equation: \(24.3050 = 18.9818 + 24.9858 * abundance_2 + 25.9826 * abundance_3\) Next, rearrange the equation to isolate the unknown variables: \(5.3232 = 24.9858 * abundance_2 + 25.9826 * abundance_3\)

Step 3: Set up a second equation

Since the sum of all natural abundances is equal to 100% or 1, we can write a second equation relating the natural abundances of the three isotopes: \(1 = 0.7899 + abundance_2 + abundance_3\) Now, rearrange the equation to isolate the unknown variables: \(0.2101 = abundance_2 + abundance_3\)

Step 4: Solve the equation system

Now we have a system of two equations with two unknowns: 1. \(5.3232 = 24.9858 * abundance_2 + 25.9826 * abundance_3\) 2. \(0.2101 = abundance_2 + abundance_3\) To solve this system of equations, we can use substitution or elimination. In this case, we will use the substitution method. Solve for one of the unknowns in the second equation, for example, \(abundance_2\): \(abundance_2 = 0.2101 - abundance_3\) Now, substitute this expression for \(abundance_2\) into the first equation: \(5.3232 = 24.9858 * (0.2101 - abundance_3) + 25.9826 * abundance_3\) Next, expand and simplify the equation: \(5.3232 = 5.2477 - 24.9858 * abundance_3 + 25.9826 * abundance_3\) Now rearrange the equation and solve for \(abundance_3\): \(0.0755 = abundance_3\) Now, substitute this value for \(abundance_3\) in the expression for \(abundance_2\): \(abundance_2 = 0.2101 - 0.0755 = 0.1346\)

Step 5: Convert to percentages and present the solution

Now that we have the natural abundances for the other two isotopes, we can express them as percentages: \(abundance_2 = 0.1346 * 100\% = 13.46\%\) \(abundance_3 = 0.0755 * 100\% = 7.55\%\) So, the natural abundances of the other two isotopes are \(13.46\%\) and \(7.55\%\).