In a periodic table the average atomic mass of magnesium is given as 24.312 u. The average value is based on their relative natural abundance on earth. The three isotopes and their masses are `""_12^24Mg` (23.98504u), `""_12^25 Mg` (24.98584u) and `""_12^26Mg` (25.98259u). The natural abundance of `""_12^24 Mg` is 78.99% by mass. Calculate the abundances of other two isotopes.

#### Solution

Average atomic mass of magnesium, *m* = 24.312 u

Mass of magnesium isotope `""_12^24 Mg`, *m*_{1} = 23.98504 u

Mass of magnesium isotope`""_12^25 Mg`,* m*_{2 }= 24.98584 u

Mass of magnesium isotope `""_12^26 Mg`,* m*_{3 }= 25.98259 u

Abundance of `""_12^24 Mg`, *η*_{1}= 78.99%

Abundance of `""_12^25 Mg`, *η*_{2 }= *x*%

Hence, abundance of `""_12^26 Mg`, *η*_{3 }= 100 − *x* − 78.99% = (21.01 − *x*)%

We have the relation for the average atomic mass as:

`m = (m_1eta_1 + m_2eta_2 + m_3eta_3)/(eta_1 + eta_2 + eta_3)`

`24.312 = (23.98504 xx 78.99 + 24.98584 xx x + 25.98259 xx (21.01 - x))/100`

`2431.2 = 1894.5783096 + 24.98584x + 545.8942159 - 25.98259 x`

0.99675x = 9.2725255

:. x ~~ 9.3 %

And 21.01 - x = 11.71 %

Hence, the abundance of `""_12^25Mg` is 9.3% and that of `""_12^26 Mg` is 11.71%.