Question

An element $X$ has three isotopes ${X^{20}},{X^{21}},{X^{22}}$. The percent abundance of ${X^{20}}$ is $90\%$ and its average atomic mass of elements is $20.18$. The percent abundance for ${X^{21}}$ must be:
A.$2\%$
B.$8\%$
C.$10\%$
D.$0\%$

Answer 1

Isotopes are defined as elements that have the same atomic number and different mass number.
-Percentage abundance is the percentage of atoms present in an element

Complete step by step answer:
The word isotopes was first derived from two greek words ‘isos’ and ‘topos’ which means to the same place.
It is defined as the element that has the same atomic number but different mass number.
In other words we can say that they have the same number of protons but different number of neutrons.
There are two different types of isotopes namely: a) radioactive isotopes b) stable isotopes.
A. Radioactive isotopes:
Isotopes that are radioactive are considered as radioactive isotopes.
They have an unstable combination of the number of protons and neutrons.
Since they are radioactive, they decay and emit out alpha, beta and gamma rays.
These isotopes are harmful to human life.
B. Stable isotopes:
Isotopes that are not radioactive are said to be stable isotopes.
They have a stable combination of neutrons and protons.
These isotopes do not cause any harm to human life.
Uses of isotopes:
Isotopes are used in the field of carbon dating, medicinal purposes and in nuclear reactors.
There are around $1000$ isotopes of which some are natural and some are synthetically produced.
The first stable isotope was discovered by Thompson.
The percentage abundance is defined as the percentage of an atom that is present in an element.
Average atomic mass is defined as the sum of mass of its isotopes each multiplied by its natural abundance.
It is given by the formula: Average atomic mass$= \dfrac{{\sum {\% abundance \times mass} }}{{100}}$…..1
Given data:
percent abundance of ${X^{20}} = 90\%$
remaining percent abundance $= 10\%$
Average atomic mass$= 20.18$
Average atomic mass$= \dfrac{{\sum {\% abundance \times mass} }}{{100}}$
Let $x$ and $10 - x$ be the percentage abundance for isotopes ${X^{21}},{X^{22}}$ respectively.
Substituting these values in the formula given in equation 1 we get,
$\Rightarrow 20.18 = \dfrac{{\left( {20 \times 90} \right) + \left( {21 \times x} \right) + \left( {22 \times \left[ {10 - x} \right]} \right)}}{{100}}$
$\Rightarrow 20.18 \times 100 = 1800 + \left( {21x} \right) + \left( {220 - 22{x_{}}} \right)$
On further solving , we get

$\Rightarrow 2018 = 1800 + 21x + 220 - 22x$
$\Rightarrow 2018 = 2020 - x$
$\Rightarrow x = 2\%$

Therefore, the correct answer is option A i.e $2\%$.

Note:
Isotopes have the same atomic number and different mass number whereas isobars have the same mass numbers and different atomic numbers. Isotopes of an element have the same chemical properties and different physical properties.