Naturally occurring thallium consists of two stable isotopes, Tl-203 and Tl-205 (atomic masses = 203.0 and 205.0, respectively) and has an average atomic mass of 204.4. What is the percentage of Tl-205?
Let us say we have `X%` of Ti-205. Then the rest `(100-X)%` is the Ti-203 percentage.
It is given that average atomic mass of Ti is 204.4.
Mass of Ti-205 in isotope` = X/100xx205`
Mass of Ti-203 in isotope `= (100-X)/100xx203`
`X/100xx205+(100-X)/100xx203 = 204.4`
`205X+20300-203X = 20440`
`X = 70`
So the percentage of Ti-205 in the isotope is 70%.
This problem can be interpreted through an equation:
`203x + 205y = 204.4`
Note that "x" and "y" are the percentages and must equal 1.0 (100%).
`x+y = 1.0`
Since we are looking for the percentage of Tl-205, we need to make the equation in relation to "y."
`x = 1.0 - y`
Plug in the value for "x" into the main equation:
`203(1.0-y) + 205y = 204.4`
With only one variable, you can now solve for "y."
`203 (1.0-y) + 205y = 204.4`
`203 - 203y + 205y = 204.4`
`2y = 1.4`
`y = 0.7`
The abundance of Tl-205 is 70%.