Chapter 2, Problem 4CO

Interpretation Introduction

Interpretation:

Calculate the atomic weight of an element from the masses and abundances of its isotopes.

Concept Introduction:

Isotopes:

Atoms of same element have the same number of proton but different number of neutrons in its nucleus.

Ex: ${}_{\text{'}}^1{\text{H, }}_{\text{'}}^2{\text{H, }}_{\text{'}}^3\text{H}$

These all are isotopes of $\text{H}-$ atom

Due to presence of different isotopes of an element we have to calculate average atomic mass, which is known as atomic weight of the element.

Answer to Problem 4CO

Solution: Average atomic mass (atomic weight)

$={\text{ f}}_{\text{ '}}{\text{ M}}_1+{\text{f}}_2{\text{ M}}_2+{\text{f}}_3{\text{ M}}_3+\mathrm{......}{\text{ f}}_n{\text{ M}}_n$

${\text{f}}_{\text{'}}:$ Fraction of ${\text{M}}_1$ isotopes (fractional abundances)

${\text{M}}_1:$ Atomic mass of first isotopes of the element

${\text{f}}_2$ : Fraction of ${\text{M}}_2$ isotopes

${\text{M}}_2:$ Atomic mass of second isotope of the element, and so on

Explanation of Solution

Due to isotopes of an element we cannot define atomic mass of the element. Each isotope is present in different amounts.

Hence we have to define average atomic mass.

Average atomic mass

$=\frac{\text{Total mass of elements}}{\text{ No}\text{. of atoms}}$

Let $\times$ atoms

$\begin{array}{l}\underset{\_}{\begin{array}{l}{\text{Fraction of M}}_1{\text{ is X}}_1\%\\ {\text{Fraction of M}}_2{\text{ is X}}_2\%\\ {\text{Fraction of M}}_3{\text{ is X}}_3\%\end{array}}\\ {\text{ Fraction of M}}_n{\text{ is X}}_n\%\end{array}$

Average atomic mass

$=\frac{\text{Total mass of element}}{\text{ No}\text{. of atoms}}$

$=\frac{\left({\text{M}}_1{X}_1\right)+\left({\text{M}}_2{X}_2\right)+\left({\text{M}}_3{X}_3\right)+-\left({\text{M}}_n{X}_n\right)}{\left(X_1+X_2+X_3+\mathrm{...}{X}_n\right)}$

$={\text{M}}_{1\left(\frac{X_1}{X_1+X_2+X_3+\mathrm{....}{X}_n}\right)}+{\text{M}}_{\text{ 2 }\left(\frac{X_2}{X_1+X_2+X_3+\mathrm{....}{X}_n}\right)}+{\text{M}}_{n\left(\frac{X_n}{X_1+X_2+X_3+\mathrm{....}{X}_n}\right)}$

$={\text{M}}_1{\text{f}}_1+{\text{M}}_2{\text{f}}_2+\mathrm{....}+\text{Mnfn}$

$\begin{array}{l}\underset{\_}{\begin{array}{l}{\text{Where f}}_1{\text{: abundance fraction of M}}_1\\ {\text{ f}}_2{\text{: abundance fraction of M}}_2\end{array}}\\ {\text{ f}}_3:{\text{ abundance fraction of M}}_3\end{array}$

Hence average atomic mass (atomic weight)

$={\text{ M}}_1{\text{f}}_1+{\text{M}}_2{\text{f}}_2+\mathrm{....}+{\text{M}}_n{\text{f}}_n$

Where ${\text{M}}_1:$ Atomic mass of first isotopes

$X_1$ : Abundance fraction of first isotopes

${\text{M}}_2$ : Atomic mass of second isotopes

$X_2$:

Abundance fraction of second isotopes.

Conclusion

Average atomic mass (atomic weight)

$={\text{ f}}_{\text{ '}}{\text{ M}}_1+{\text{f}}_2{\text{ M}}_2+{\text{f}}_3{\text{ M}}_3+\mathrm{......}{\text{ f}}_n{\text{ M}}_n$

${\text{f}}_{\text{'}}:$ Fraction of ${\text{M}}_1$ isotope (fractional abundances)

${\text{M}}_1:$ Atomic mass of first isotopes of the element

${\text{f}}_2$ : Fraction of ${\text{M}}_2$ isotope

${\text{M}}_2:$ Atomic mass of second isotope of the element and so on