Question

The atomic mass of silver found in nature is Ar (Ag) = 107,983. This silver is made of 107Ag and 109Ag Isotopes. Calculate the proportion of mass of 107Ag isotope which is in natural silver atoms?

Answer 1

`w(107Ag)=50,85%`

Explanation:

x: 107Ag isotope mass %
100-x: 109Ag isotope mass %

Ar (Ag)= `\frac {107x+109 × (100-x)}{100}=107,983` |×100

Ar (Ag)=`107x+10900-109x=10798,3`

`107x-109x=10798,3-10900`
`-2x=-101,7` |`÷2`

`x=50,85%`

Answer 2

`sf(Ag107=50.85%)`

Explanation:

If x is the fraction of Ag107 and y is the fraction of Ag109 then we can write:

`sf((107xxx)+(109xxy)=107.983" "color(red)((1)))`

We can also say:

`sf(x+y=1" "color(red)((2))`

From `sf(color(red)((2)))` we can say:

`sf(x=(1-y))`

Substitute this value for x into `sf(color(red)((1))rArr)`

`sf(107(1-y)+109y=107.983)`

`sf(107-107y+109y=107.983)`

`sf(2y=107.983-107)`

`sf(2y=0.983)`

`sf(y=0.4915)`

From `sf(color(red)((1))rArr)`

`sf(x=1-0.4915=0.5085)`

In percentages:

`sf(Ag107=50.85%)`

`sf(Ag109=49.15%)`

Answer 3

50.85%
Similar to the others, but posted by request of Austte.

Explanation:

This is basically a ratio math problem. We combine known percentages of have several different masses into a weighted-average mass by multiplying each mass by the weight percent and adding them up. We can use the same equation to work from the weighted average back to the percentages.

`%_1 xx Mass_1 + %_2 xx Mass_2 = Mass_avg`
`%_1 xx 107 + %_2 xx 109 = 107.983` (we know that the final ratio must be close to even, with slightly more Mass107 because the average is just under the average value of the two masses).

With only two isotopes, we also know that `%_107 = 1 - %_109` so our equation becomes:
`(1 - %_109) xx 107 + %_109 xx 109 = 107.983` ; `107 - %_109 xx 107 + %_109 xx 109 = 107.983` ; `107 + %_109 xx (109 - 107) = 107.983`
`107 + %_109 xx (2) = 107.983` ; `%_109 xx (2) = 0.983` ; `%_109 = 0.4915` Thus `%_107 = 1 – 0.4915 = 0.5085`

CHECK
`%_1 xx Mass_1 + %_2 xx Mass_2 = Mass_avg`
`0.5085 xx 107 + 0.4915 xx 109 = 107.983` ; `54.4095 + 53.5735 = 107.983` ; `107.983 = 107.983` ; CORRECT!