Chapter 2, Problem 27QAP

Silicon $\left(\text{average atomic mass}=\text{28}.0\text{855 amu}\right)$ has three isotopes. Their masses are 27.9769 amu, 28.9765 amu, and 29.9738 amu. The abundance of the heaviest isotope is 2.96%. Estimate the abundances of the first two isotopes.

Interpretation Introduction

Interpretation:

To estimate the abundances of first two isotopes of silicon.

Concept introduction:

Average atomic mass = It may be defined as the sum of all the masses of its known isotopes, each multiplied with its natural abundance (in percentage) divided by 100. Mathematically,

$\begin{array}{l}\text{Average}\text{atomic}\text{mass}\text{=}\frac{\sum _{\text{i=1}}^{\text{n}}{\text{(mass)}}_{\text{i}}{\text{(\%}\text{abundance)}}_{\text{i}}}{\text{100}}\\ \text{where}\\ \text{i}\text{=}\text{1,}\text{2,}\text{3,}\text{4}\mathrm{.......}\end{array}$

Answer to Problem 27QAP

Abundances of first two isotopes of silicon are 92.09 % and 4.95 % respectively.

Explanation of Solution

According to question there are three isotopes of silicon (Si) such that:

Mass of heaviest isotope = 29.9738 amu

Abundance of heaviest isotope = 2.96 % (given)

Mass of first isotope = 27.9769 amu

Abundance of first isotope = Let it be ‘k’ %.

Mass of second isotope = 28.9765 amu

Since sum of abundances must be equal to 100 %. Therefore,

Abundance of second isotope = [100- (k + 2.96)] = (97.04- k) %

Making tabular data as:

Isotopes	Mass of isotopes(amu)	% abundance
Heaviest isotope	29.9738	2.96
First isotope	27.9769	k
Second isotope	28.9765	(97.04 - k)

Average atomic mass of silicon (Si) = 28.0855 amu.

Substituting the values in formula to get average atomic mass as:

$\begin{array}{l}\text{Average}\text{atomic}\text{mass}\text{=}\frac{\sum _{\text{i=1}}^{\text{n}}{\text{(mass)}}_{\text{i}}{\text{(\%}\text{abundance)}}_{\text{i}}}{\text{100}}\\ \text{Here}\text{i}\text{=}\text{1,2}\text{and}\text{3}\text{,}\text{where}\text{1}\text{,}\text{2}\text{and}\text{3}\text{represents}\text{first}\text{,}\text{second}\text{and heaviest}\text{isotope}\text{respeectively}\\ \\ \text{28}\text{.0855}\text{=}\frac{\text{(27}\text{.9769}\text{×}\text{(k))}\text{+}\text{(28}\text{.9765×(97}\text{.04}\text{-k))}\text{+}\text{(29}\text{.9738×2}\text{.96)}}{\text{100}}\\ \Rightarrow \text{28}\text{.0855}\text{=}\frac{2900.60204\text{-}\text{0}\text{.9996k}}{\text{100}}\\ \\ \Rightarrow \text{2808}\text{.55}\text{-}\text{2900}\text{.60204}\text{=}\text{-}0.9996\text{k}\\ \Rightarrow \text{k}\text{=}\frac{\text{92}\text{.05201}}{0.9996}\text{=}92.0884\text{\%}\end{array}$

Hence abundance of first isotope = k = 92.09 %

Abundance of second isotope = 97.04 − k = 97.04 − 92.09 = 4.95 %

Conclusion

Abundances of first two isotopes of silicon are 92.09 % and 4.95 % respectively.