Question

In a Periodic table the average atomic mass of Magnesium is given as $24.312\text{ u}$.The average value is based on their relative natural abundance on earth. The isotopes and their masses are ${}_{12}^{24}\text{Mg}\left( 23.98504\text{ u} \right), {}_{12}^{25}\text{Mg}\left( 24.98584\text{ u} \right)$ and ${}_{12}^{26}\text{Mg}\left( 25.9859\text{ u} \right)$. The natural abundance of ${}_{12}^{24}\text{Mg}$ is $78.99\%$ by mass. Calculate the abundance of two isotopes.

Answer 1

Atoms that have the same number of protons but different numbers of neutrons are known as isotopes. Isotopes have different atomic masses. The relative abundance of an isotope is the percentage of atoms with a specific atomic mass found in naturally occurring sample
of an element.

Complete step by step answer:
Isotopes are defined as the atoms of the same element having the same number of protons but different number of neutrons. It means all elements have the same atomic number.
As given data in the questions are about isotopes of magnesium which is

ISOTOPES	ABUNDANCE (Y)	ATOMIC MASS (M)
$_{12}\text{M}{{\text{g}}^{24}}$	$\text{ }78.99\text{ }$	$\text{23}\text{.98504}$
$_{12}\text{M}{{\text{g}}^{25}}\text{ }$	$x\text{ }$	$\text{24}\text{.98584}$
$_{12}\text{M}{{\text{g}}^{26}}\text{ }$	$100-78.99-x=21.1-x$	$\text{25}\text{.98259}$
$\sum{Y=100}$

Atomic mass$=24.312$
Average atomic mass$=\dfrac{\sum{Yz}}{\sum{Y}}$
$\Rightarrow 24.332=\dfrac{\left( 78.99\times 23.98504 \right)+\left( x\times 24.98584 \right)+\left( 21.01-x \right)25.98254}{100}$
$\Rightarrow 2431.2=1894.58+24.98584x+545.89-25.98254x$
$\Rightarrow 24.312=2440.47-.99675x$
$\Rightarrow .99675x=2440.47-2431.2$
$\Rightarrow x=\dfrac{9.27}{.99675}=9.30$
Thus $21.01-x=21.01-9.30=11.71$
Therefore,
Relative abundance of $_{12}\text{M}{{\text{g}}^{25}}\text{ }=9.30\%$
Relative abundance of $_{12}\text{M}{{\text{g}}^{26}}\text{ }=11.71\%$

Note:
Basically Relative abundance on an isotope is the percentage of atoms with a specific atomic mass found in a naturally occurring sample of an element.