Problem 78 - General Chemistry: Principles and Modern Applications

Question

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.

Step-by-Step Solution

Answer

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 1: 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 \(20.85\% + 27.54\% + 36.29\% = 84.68\%\).

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

Isotopic Abundance Calculation

Calculating isotopic abundances revolves around determining the percentage that each isotope contributes to the total makeup of an element. With germanium, this involves both its major isotopes and its minor ones. The major isotopes mentioned include:

• \(^{70}\) Ge with a 20.85% abundance
• \(^{72}\) Ge with a 27.54% abundance
• \(^{74}\) Ge with a 36.29% abundance

To determine the contribution of minor isotopes like \(^{73}\) Ge and \(^{76}\) Ge, it's essential to realize that all isotopes combined equate to 100% of the element's natural abundance.

This process begins by first adding up the abundances of the major isotopes, which results in 84.68%. Next, by subtracting this sum from 100%, we get 15.32% – this is the total abundance of the minor isotopes. Although we have this combined percentage, further insights would be needed to split it precisely between the individual minor isotopes.

Precision in Isotopic Data

Precision is key when dealing with isotopic abundances.
With calculations like the ones done for germanium, small inaccuracies in the data can lead to significant variations in results.
When adding or subtracting percentages from isotopic abundances, rounding off can introduce errors.
It becomes necessary to rely on high-precision data to avoid misleading conclusions.

• Even slight differences in the decimal points could skew the calculated results.
• The problem arises particularly when comprehensive data on individual isotopes is unavailable.

Therefore, advanced equipment and methods, which provide more accurate isotopic mass and abundance measurements, are crucial in narrowing down uncertainties.

Germanium Isotopes

Germanium is an element with a fascinating isotopic profile.
The presence of five naturally occurring isotopes – three major and two minor – gives it a complexity that requires careful scientific scrutiny.

Major Isotopes

The major isotopes, \(^{70}\) Ge, \(^{72}\) Ge, and \(^{74}\) Ge, are more commonly found in nature.

• These isotopes together make a significant portion of germanium's natural abundance.
• They are usually the focus when calculating standard isotopic values.

Minor Isotopes

On the other hand, the minor isotopes, \(^{73}\) Ge and \(^{76}\) Ge, while less prevalent, still play a crucial role.

• Their collective abundance needs to be calculated by subtracting the abundance of major isotopes from 100%.
• These minor isotopes can offer insights into geological and environmental processes by their varying abundance.

Understanding the distribution of these isotopes not only enhances comprehension of geochemical processes but also supports applications in industry and science.