I need help I keep getting the wrong answer

Question

An unknown element is a mixture of isotopes 151X and 153x. The average atomic mass of X is 152.57amu. What is the percent abundance of 153X?

Answer 1

Typically we use actual masses for a calculation like this rather than mass numbers. The calculation is called a weighted arithmetic mean or a weighted average. When calculating an average of these two masses, we would add them together and divide by 2: 151 + 153 = 304/2 = 152. However, a "normal" average gives equal weight to the numbers which means that for our problem the isotopes would have equal abundance of 50%. This is not the case.

The average atomic mass is closer to 153 than 151 which means that the 153 isotope is more abundant. As a percentage, the abundance of each isotope sum to 100. As decimals, they sum to equal 1. So, the average atomic mass is the weighted average of the isotopes masses which takes into account their abundance:

152.57 amu = 153x + 151(1-x)

x is the abundance for isotope 153 which is what we need for our answer so we must solve for x.

152.57 = 153x + 151 -151x

152.57 = 2x +151

1.57 = 2x

x = 0.785 Multiply this by 100 to get 78.5%

This makes sense. The abundance of the 153 isotope is larger which is why the average mass is closer to 153. The 151 isotope is only 21.5% abundant, although this is not asked for in the problem. You can plug the decimals back into the original equation to check. I hope this helps.

Answer 2

Lets say x1 is the fractional abundance of 151X, and x3 is the fractional abundance of 153X.

The weighted average atomic mass can be written as: x1*151 + x3*153 = 152.57

We also know that only two isotopes exist, so: x1 + x3 = 1

Solving this system of equations should give you the correct answer.