# 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?

Aug 15, 2017

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)=$107 x + 10900 - 109 x = 10798 , 3$

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

x=50,85%

Aug 15, 2017

sf(Ag107=50.85%)

#### Explanation:

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

$\textsf{\left(107 \times x\right) + \left(109 \times y\right) = 107.983 \text{ } \textcolor{red}{\left(1\right)}}$

We can also say:

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

From $\textsf{\textcolor{red}{\left(2\right)}}$ we can say:

$\textsf{x = \left(1 - y\right)}$

Substitute this value for x into $\textsf{\textcolor{red}{\left(1\right)} \Rightarrow}$

$\textsf{107 \left(1 - y\right) + 109 y = 107.983}$

$\textsf{107 - 107 y + 109 y = 107.983}$

$\textsf{2 y = 107.983 - 107}$

$\textsf{2 y = 0.983}$

$\textsf{y = 0.4915}$

From $\textsf{\textcolor{red}{\left(1\right)} \Rightarrow}$

$\textsf{x = 1 - 0.4915 = 0.5085}$

In percentages:

sf(Ag107=50.85%)

sf(Ag109=49.15%)

Aug 16, 2017

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
$0.5085 \times 107 + 0.4915 \times 109 = 107.983$ ; $54.4095 + 53.5735 = 107.983$ ; $107.983 = 107.983$ ; CORRECT!