Question

Neon has two isotopes, $^{20}{\text{Ne}}$ and $^{22}{\text{Ne}}$. If atomic weight of neon is $20.2$, then the ratio of the relative abundance of the isotopes is:
A. $1:9$
B. $9:1$
C. $1:7$
D. $7:1$

Answer 1

The average atomic mass is determined by adding the product of the mass of each isotope and its relative abundance. We will assume that the relative abundance of one isotope is X, and the relative abundance of the second isotope will be$100 - {\text{X}}$. By putting the values of each in the average atomic mass formula we can determine the value of X that is the relative abundance.

Complete solution:
The average atomic mass is the sum of the product of the mass of isotope and its relative abundance.
The formula of the average atomic mass is as follows:
${\text{Average}}\,{\text{atomic}}\,{\text{mass}}\,{\text{ = }}\,\sum {{\text{isotope}}\,{\text{mass}}\,{\text{ \times }}\,{\text{relative abundance}}} {\text{ of that isotope}}$
For the two isotopes of neon the formula can be written as follows:
${\text{Average}}\,{\text{atomic}}\,{\text{mass}}\,{\text{ = }}\,\,\left( {{\text{mass}}\,{\text{of}}{\,^{20}}{\text{Ne}}\,\, \times \,\,{\text{abundance}}\,{\text{of}}{{\text{ }}^{20}}{\text{Ne}}\,} \right) + \left( {{\text{mass}}\,{\text{of}}{\,^{22}}{\text{Ne}}\,\, \times \,\,{\text{abundance}}\,{\text{of}}{{\text{ }}^{22}}{\text{Ne}}\,} \right)\,$
We assume that the relative abundance of $^{20}{\text{Ne}}$ in the sample is X% and the relative abundance of $^{22}{\text{Ne}}$ in the sample is $100 - {\text{X}}$%.
Mass of $^{20}{\text{Ne}}$ isotope is $20$and the mass of $^{22}{\text{Ne}}$ is $20$.
To determine the value of X, on substituting $20.2$for average atomic mass, $20$for mass of$^{20}{\text{Ne}}$, $22$for mass of$^{22}{\text{Ne}}$, X for relative abundance of $^{20}{\text{Ne}}$ and $100 - {\text{X}}$% for relative abundance of$^{22}{\text{Ne}}$.
$20.2\,{\text{ = }}\,\,\left( {20\,\, \times \,\,\dfrac{{\text{X}}}{{{\text{100}}}}\,} \right) + \left( {22\,\, \times \,\,\dfrac{{100 - {\text{X}}}}{{{\text{100}}}}\,} \right)\,$
$20.2\,{\text{ = }}\,\,\dfrac{{20\,{\text{X}}}}{{{\text{100}}}} + \dfrac{{2200}}{{{\text{100}}}} - \dfrac{{{\text{22X}}}}{{{\text{100}}}}\,\,$
$2020\,{\text{ = }}\,\,20{\text{X}}\,{\text{ + }}\,22{\text{00}}\, - 22\,{\text{X}}\,\,$
${\text{2X = }}\,180$
${\text{X = }}\,90$
So, the relative abundance of $^{20}{\text{Ne}}$ in the sample is $90$%.
We will determine the relative abundance of $^{22}{\text{Ne}}$ as follows:
On substituting $90$% for relative abundance of $^{22}{\text{Ne}}$ in $100 - {\text{X}}$%.
$100 - 90\, = 10$
So, the relative abundance of $^{22}{\text{Ne}}$ in the sample is $10$%.
So, the relative abundance of the isotopes is,
$^{20}{\text{Ne}}$ : $^{22}{\text{Ne}}$
$90:10$
$9:1$
Therefore, option (B) $9:1$ is correct.

Note: The chemical elements that have the same atomic number but different number of neutrons are known as isotopes. The isotopes have the same number of protons and hence represent the same element. The relative abundance of the isotopes is the percent amount of isotopes which is occurring naturally. The total relative abundance of all the isotopes of an element is considered as $100$%.