Question

The average atomic mass of a sample of an element \[X\] is \[16.2\,u\] . What are the percentages of isotopes \[_8^{16}X\]and \[_8^{18}X\] in the sample?

Answer 1

Hint:The average atomic mass of an element which is measured in the atomic mass unit (i.e., amu, also known as D (i.e., Daltons)) is the atomic mass of an element. The atomic mass is weighed by using the average of all isotopes of that element in which the mass of each isotope is multiplied by the abundance of that particular isotope respectively.


Complete step by step solution:
The atoms which are having the same number of protons but a different number of neutrons is called an isotope. The isotopes have a different number of atomic masses of the same element. The relative abundance of an isotope is a percentage of the atom with a specific atomic mass found in nature of an element.
The relative abundance of an isotope is the percentage of an atom having a specific atomic mass which is found naturally.
The average atomic mass is calculated by summing all the masses of isotopes of that particular element, therefore multiplied by each of its natural abundance on Earth.
Mathematically we can write as,
\[ = \,\dfrac{{\sum\limits_{i = 1}^n {(mas{s_{(i)}})\, \times \,\,(percentage\,isotopic\,abundance{e_{(i)}})} }}{{100}}\]
The average atomic mass of the given sample of an element \[X\] is \[16.2\,u\]
Let the percentages of isotope \[_8^{16}X\] be \[a\% \] and \[_8^{18}X\] be \[\left( {100 - a} \right)\% \]
Based on the given data, the average atomic mass of element;
\[
   = \,\,\dfrac{{16\,\, \times \,a\,}}{{100}}\, + \,\dfrac{{18\,\, \times \,(100 - a)}}{{100}} \\
    \\
 \]
The given average atomic mass of the element= \[16.2\,u\]
Therefore,
\[
  \dfrac{{16\,\, \times \,a\,}}{{100}}\, + \,\dfrac{{18\,\, \times \,(100 - a)}}{{100}}\, = 16.2 \\
  \, \\
 \]
\[\,\dfrac{{16\,a\, + \,1800\, - 18a\,}}{{100}}\, = \,16.2\]
\[16\,a\, + \,1800\, - 18a\,\, = \,1620\]
\[1800\, - 2a\, = \,1620\]
\[ - 2a\, = \,1620\, - 18\]
\[ - 2a\, = \, - 180\]
\[a\, = \,\dfrac{{ - 180}}{{ - 2}}\]
After solving the equation,
\[a\, = \,90\]
We get,

The percentage of isotope \[_8^{16}X\] \[ = \,a\% \]\[ = \,90\% \]
And another percentage of isotope \[_8^{18}X\] \[ = \,\left( {100 - a} \right)\% \, = \,(100 - 90)\% \]\[ = \,10\% \]


Note:The combined mass of all the protons and neutrons are known as atomic mass. The average atomic mass of an element is the mean product of all isotopes of the atom and percent isotopic abundance. Isotopes have the same chemical properties but have different physical properties of an element.