Chapter 8, Problem 14STP

Interpretation Introduction

Interpretation:

The correct average atomic mass is to be calculated.

Concept introduction:

Sum of the masses of the isotopes of element, each multiplied by its fractional natural abundance, is termed as average atomic mass of that element. The average atomic mass can be calculated as shown below.

$M_{\text{avg}}=f_1{M}_1+f_2{M}_2+\mathrm{............}{f}_n{M}_n$

Here, $f$ is the fractional of natural abundance and $M$ is the mass of isotopes.

Answer to Problem 14STP

The correct average atomic mass is 28.09 amu.

Explanation of Solution

The average atomic mass of silicon is calculated by the formula shown below.

$M_{\text{avg}}={\text{Abundance}}_{{}^{\text{28}}\text{Si}}\cdot {\text{Mass}}_{{}^{\text{28}}\text{Si}}+{\text{Abundance}}_{{}^{\text{29}}\text{Si}}\cdot {\text{Mass}}_{{}^{\text{29}}\text{Si}}+{\text{Abundance}}_{{}^{\text{30}}\text{Si}}\cdot {\text{Mass}}_{{}^{\text{30}}\text{Si}}$ The percent abundance of ²⁸Si (mass-27.98 amu), ²⁹Si (mass-28.98 amu) and ³⁰Si (mass 29.97 amu is 92.21%, 4.70% and 3.09% respectively.

Substitute the given data in above equation.

$\begin{array}{c}{M}_{\text{avg}}={\text{Abundance}}_{{}^{\text{28}}\text{Si}}\cdot {\text{Mass}}_{{}^{\text{28}}\text{Si}}+{\text{Abundance}}_{{}^{\text{29}}\text{Si}}\cdot {\text{Mass}}_{{}^{\text{29}}\text{Si}}+{\text{Abundance}}_{{}^{\text{30}}\text{Si}}\cdot {\text{Mass}}_{{}^{\text{30}}\text{Si}}\\ =92.21\%\times 27.98\text{amu}+4.70\%\times 28.98\text{amu}+3.09\%\times 29.97\text{amu}\\ =92.21\left(\frac{1}{100}\right)\times 27.98\text{amu}+4.70\left(\frac{1}{100}\right)\times 28.98\text{amu}+3.09\left(\frac{1}{100}\right)\times 29.97\text{amu}\\ =0.9221\times 27.98\text{amu}+0.0470\times 28.98\text{amu}+0.0309\times 29.97\text{amu}\\ =\text{25}\text{.800358}\text{amu}+1.36206\text{amu}+0.926073\text{amu}\\ M_{\text{avg}}=\text{28}\text{.09}\text{amu}\end{array}$ The average atomic mass of silicon calculated by lab partner is 28.98 amu which is different from the correct average atomic mass. The method of calculation of average atomic mass of silicon calculated by lab partner is shown below.

$\begin{array}{c}{M}_{\text{avg}}=\frac{{\text{Mass}}_{{}^{\text{28}}\text{Si}}+{\text{Mass}}_{{}^{\text{29}}\text{Si}}+{\text{Mass}}_{{}^{\text{30}}\text{Si}}}{3}\\ =\frac{27.98\text{amu}+28.98\text{amu}+29.97\text{amu}}{3}\\ =\frac{86.93\text{amu}}{3}\\ M_{\text{avg}}=28.98\text{amu}\end{array}$

The above method of calculating average atomic mass is wrong. Therefore, the correct average atomic mass is 28.09 amu.

Conclusion

The correct average atomic mass is 28.09 amu.