Silver (average atomic mass 107.868) has only two isotopes with masses 106.90509 and 108.90470. What is the percentage abundance of the heavier isotope?
The Correct Answer and Explanation is
To find the percentage abundance of the heavier isotope of silver, we need to use the concept of atomic mass and the weighted average of the isotopes.
Given:
- Average atomic mass of silver = 107.868
- Mass of lighter isotope (Ag-107) = 106.90509
- Mass of heavier isotope (Ag-108) = 108.90470
Let the abundance of the heavier isotope (Ag-108) be xxx, and the abundance of the lighter isotope (Ag-107) would therefore be 1−x1 – x1−x because the total abundance must sum to 100% or 1.
The average atomic mass is a weighted sum of the masses of the isotopes, so we can set up the following equation:(106.90509)(1−x)+(108.90470)(x)=107.868(106.90509)(1 – x) + (108.90470)(x) = 107.868(106.90509)(1−x)+(108.90470)(x)=107.868
Expanding the equation:106.90509−106.90509x+108.90470x=107.868106.90509 – 106.90509x + 108.90470x = 107.868106.90509−106.90509x+108.90470x=107.868
Combine like terms:106.90509+(108.90470−106.90509)x=107.868106.90509 + (108.90470 – 106.90509)x = 107.868106.90509+(108.90470−106.90509)x=107.868106.90509+1.99961x=107.868106.90509 + 1.99961x = 107.868106.90509+1.99961x=107.868
Now, isolate xxx:1.99961x=107.868−106.905091.99961x = 107.868 – 106.905091.99961x=107.868−106.905091.99961x=0.962911.99961x = 0.962911.99961x=0.96291x=0.962911.99961=0.4814x = \frac{0.96291}{1.99961} = 0.4814x=1.999610.96291=0.4814
Thus, the abundance of the heavier isotope (Ag-108) is approximately 48.14%.
Therefore, the percentage abundance of the heavier isotope of silver is about 48.14%.
