Avogadro's Number — Question 4

Question 4

An isotope mixture can be inferred from its mean molar mass

Use $N_A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}$, exact. A hypothetical element has two isotopes with assigned molar masses 20.0 and 22.0 g/mol. The measured mixture molar mass is 20.6 g/mol. A sample has mass 4.12 g. Let $x$ be the number fraction of the heavier isotope.

1. Find $x$ and the count of each isotope.

2. Find the heavier isotope's mass fraction and explain why it differs from $x$.

Question 4 - Solution

An isotope mixture can be inferred from its mean molar mass

(a) Method and calculation

$$20(1-x)+22x=20.6\ \Rightarrow\ x=.300.$$$$n=4.12/20.6=.200\,\mathrm{mol},\quad N_{22}=.0600N_A=3.613284456\times10^{22}.$$$$N_{20}=.140N_A=8.430997064\times10^{22}.$$

(b) Interpretation and further calculation

$$m_{22}=.0600(22)=1.32\,\mathrm g,\quad w_{22}=1.32/4.12=.320388.$$$$w_{22}=\frac{xM_{22}}{(1-x)M_{20}+xM_{22}}>x.$$Each heavier-isotope atom contributes more mass than each lighter-isotope atom.

Physical check

The inferred fraction lies between zero and one, and the average molar mass lies between the isotope values.