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Chapter 2: Problem 78

Germanium has three major naturally occurring isotopes: \(^{70}\) Ge \((69.92425 \mathrm{u}, 20.85 \%),^{72} \mathrm{Ge}(71.92208 \mathrm{u},\) \(27.54 \%),^{74} \mathrm{Ge}(73.92118 \mathrm{u}, 36.29 \%) .\) There are also two minor isotopes: \(^{73}\) Ge \(\left(72.92346 \text { u) and }^{76} \mathrm{Ge}\right.\) (75.92140 u). Calculate the percent natural abundances of the two minor isotopes. Comment on the precision of these calculations.

Short Answer

Expert verified
The two minor isotopes of Germanium, \(^{73}\) Ge and \(^{76}\) Ge, together account for approximately 15.32% of the natural abundance of Germanium. As the calculation depends on the given data about other isotopes, precision of the result is tied to the precision of the given data.

Step by step solution

01

Calculate the total abundance of major isotopes.

Given that there are three major isotopes of Germanium denoted as \(^{70}\) Ge, \(^{72}\) Ge and \(^{74}\) Ge with corresponding abundances of 20.85%, 27.54% and 36.29% respectively, they can be added up to provide the total abundance of major isotopes. In this case, it will be \(20.85\% + 27.54\% + 36.29\% = 84.68\%\).
02

Determine the total abundance of minor isotopes.

Knowing that all the isotopes combined give 100% abundance, the total abundance of major isotopes calculated in step 1 can be subtracted from 100% to find the total abundance of minor isotopes. From the calculation, \(100\% - 84.68\% = 15.32\%\), which refers to the total abundance of the two minor isotopes combined.
03

Assemble the calculated results.

Given that the two minor isotopes of Germanium are \(^{73}\) Ge and \(^{76}\) Ge, they collectively make up 15.32% of the natural abundance of Germanium. We cannot break this value down between the two isotopes without additional information because isotopic abundances cannot be found solely from the atomic masses of the isotopes. So, we conclude that \(^{73}\) Ge and \(^{76}\) Ge together account for 15.32% of the natural abundance.
04

Comment on the Precision of These Calculations.

These calculations should be considered rough approximations due to round-off errors in the data and the assumption that the abundance percentages provided in the problem are precise. Also, they would be more accurate if the individual natural abundances of \(^{73}\) Ge and \(^{76}\) Ge had been directly provided.

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