Question

: Three isotopes of an element have mass numbers \[{\text{ M}},\;\left( {{\text{M}} + 1} \right)\;{\text{and}}\;\left( {{\text{M}} + 2} \right){\text{ }}\]. If the mean mass number is \[{\text{ }}\left( {{\text{M}} + 0.5} \right)\,\;\] then which of the following ratios may be accepted for \[{\text{ M}},\;\left( {{\text{M}} + 1} \right)\;{\text{and}}\;\left( {{\text{M}} + 2} \right){\text{ }}\] in that order?
A.1 : 1 : 1
B.4 : 1 : 1
C.3 : 2 : 1
D.2 : 1 : 1

Answer 1

Hint: We need to calculate the ratio by mass of the respective isotope in which they exist. We will make an equation using a, b and c as variable abundance and then simplifying the equation we will get the ratio.

Complete step by step answer:
Let us assume the elements with mass number \[{\text{ M}},\;\left( {{\text{M}} + 1} \right)\;{\text{and}}\;\left( {{\text{M}} + 2} \right){\text{ }}\]are present in the ratio ‘a’, ‘b’ and ‘c’ respectively. The mean or average is calculated by adding the quantities and then dividing by total number of quantities. We will multiply the respective mass with their respective ration and divide it by sun of ratio to equate it to the given value of mean mass. We will get the following equation:
\[{\text{ }}\dfrac{{{\text{a}} \times {\text{M}} + {\text{b}} \times \;\left( {{\text{M}} + 1} \right)\; + {\text{c}} \times \;\left( {{\text{M}} + 2} \right)}}{{{\text{a}} + {\text{b}} + {\text{c}}}} = {\text{M}} + 0.5\]
Solving the above equation we will get:
\[{\text{ aM}} + {\text{b}}\;{\text{M}} + {\text{b}} + {\text{cM}} + 2{\text{c}} = {\text{aM}} + 0.5{\text{a}} + {\text{bM}} + 0.5{\text{b}} + {\text{cM}} + 0.5{\text{c}}\]
We can cancel the same terms on the left hand side and the right hand side and write the equation in terms of variables by solving as:
\[0.5{\text{b}} + 1.5{\text{c}} = 0.5{\text{a}}\]
We will multiply the above equation with 2 to convert them into whole number:
\[{\text{ b}} + 3{\text{c}} = {\text{a }}\]
If we can see that for one part of ‘b’ and ‘c’ that is if we put ‘b’ and ‘c’ as 1 then we will get the value of a as 4. So the ration between ‘a’, ‘b’ and ‘c’ will be 4 : 1 : 1.

Hence, the correct option is B.

Note:
Isotopes are those elements which have the same atomic number but different mass number. Hydrogen has 3 isotopes names as protium, deuterium and tritium with mass number 1, 2 and 3 respectively and the atomic number is 1 for all.