Question

The atomic mass of silver found in nature is $ Ar\left( Ag \right)=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

Hint: We know that isotope is the property of any element to have the same atomic number but different mass numbers. For calculating the mass numbers of isotopes, an average atomic mass is used, that takes the average of the mass of each isotopic form. Silver has isotopes of $ Ag\left( 107 \right),\text{ }Ag\left( 109 \right) $ mass.

Complete answer:
Isotopes of an element are the forms of the same element that have the same atomic number but they differ in their mass number. So, average atomic mass is used to calculate their atomic masses. We have been given isotopes of silver, with their atoms that are in abundance. This is basically a ratio math problem. We combine known percentages of 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. We have to calculate the atomic mass of silver, given that if x is the fraction of $ Ag\left( 107 \right) $ and y is the fraction of $ Ag\left( 109 \right) $ then we can write:
 $ \left( 107\times x \right)+\left( 109\times y \right)=107.983 $ …………(i) also $ x+y=1 $ …………(ii)
From equation (ii) we get; $ x=\left( 1-y \right) $
Substitute this value for x into (i);
 $ 107\left( 1-y \right)+109y=107.983\Rightarrow 107-107y+109y=107.983 $
On further solving we get; $ 2y=107.983-107\Rightarrow 2y=0.983 $
 $ \therefore y=0.4915 $
From equation (i) we get; $ x=1-0.4915=0.5085 $
In percentages: $ Ag\left( 107 \right)=50.85% $ and $ Ag\left( 109 \right)=49.15% $
Therefore natural silver is $ Ag\left( 107 \right)=50.85% $ .

Note:
Remember that the property of an element to have different isotopes is used in the phenomena of radioactivity. The isotopes in abundance are either their percentage divided by hundred, or their number of atoms upon total number of atoms present. This is because each isotope will contribute its part by its total abundance.