Problem 43 - Chemistry An Atoms First Approach

Question

The mass spectrum of bromine \(\left(\mathrm{Br}_{2}\right)\) consists of three peaks with the following characteristics: $$\begin{array}{|lc|}\hline \text { Mass }(\mathrm{u}) & \text { Relative Size } \\\\\hline 157.84 & 0.2534 \\\159.84 & 0.5000 \\\161.84 & 0.2466 \\\\\hline\end{array}$$ How do you interpret these data?

Step-by-Step Solution

Answer

The three peaks in the mass spectrum of Bromine (\(\mathrm{Br_{2}}\)) can be attributed to the different combinations of its isotopes, \(\mathrm{^{79}Br}\) and \(\mathrm{^{81}Br}\). The first peak (\(157.84\,\mathrm{u}\)) corresponds to the \(\mathrm{^{79}Br - ^{79}Br}\) combination, the second peak (\(159.84\,\mathrm{u}\)) to both \(\mathrm{^{79}Br - ^{81}Br}\) and \(\mathrm{^{81}Br - ^{79}Br}\) combinations, and the third peak (\(161.84\,\mathrm{u}\)) to the \(\mathrm{^{81}Br - ^{81}Br}\) combination. The average atomic mass of \(\mathrm{Br_{2}}\) is calculated as approximately \(159.77\,\mathrm{u}\).

Step 1: Understand Mass Spectrum and Isotopes

A mass spectrum is a distribution of the ions produced in a mass spectrometer (such as a mass spectroscopy) as a function of their mass-to-charge ratio. In this case, the mass spectrum provides the distribution of Bromine (\(\mathrm{Br_{2}}\)) ions. Isotopes are elements with the same number of protons but a different number of neutrons. In the case of Bromine, it consists of isotopes, primarily \(\mathrm{^{79}Br}\) and \(\mathrm{^{81}Br}\). The presence of these isotopes contributes to the different peaks in the mass spectrum.

Step 2: Analyzing the three peaks

There are three peaks in the mass spectrum of \(\mathrm{Br_{2}}\), and their masses are \(157.84\,\mathrm{u}\), \(159.84\,\mathrm{u}\), and \(161.84\,\mathrm{u}\). The corresponding relative sizes are \(0.2534\), \(0.5000\), and \(0.2466\). By looking at the mass values, we can assume that these peaks are formed due to the combination of the isotopes of \(\mathrm{^{79}Br}\) and \(\mathrm{^{81}Br}\) in the Bromine molecule (\(\mathrm{Br_{2}}\)). We will now explain the source of each peak: 1. The first peak, with a mass of \(157.84\,\mathrm{u}\), can be attributed to the \(\mathrm{^{79}Br - ^{79}Br}\) combination. 2. The second peak, with a mass of \(159.84\,\mathrm{u}\), can be caused by the presence of both \(\mathrm{^{79}Br - ^{81}Br}\) and \(\mathrm{^{81}Br - ^{79}Br}\) combinations. 3. The third peak, with a mass of \(161.84\,\mathrm{u}\), can be attributed to the \(\mathrm{^{81}Br - ^{81}Br}\) combination. The relative sizes of the peaks indicate the abundance of the corresponding isotopic combinations.

Step 3: Calculate the average atomic mass of \(\mathrm{Br_{2}}\)

Using the information from the mass spectrum, we can calculate the average atomic mass of the Bromine molecule by multiplying the mass values with their corresponding relative sizes and adding them together: Average atomic mass of \(\mathrm{Br_{2}} = (157.84\,\mathrm{u} \times 0.2534) + (159.84\,\mathrm{u} \times 0.5000) + (161.84\,\mathrm{u} \times 0.2466)\) \(= 39.97\,\mathrm{u} + 79.92\,\mathrm{u} + 39.88\,\mathrm{u}\) \(= 159.77\,\mathrm{u}\) The average atomic mass of \(\mathrm{Br_{2}}\) is approximately \(159.77\,\mathrm{u}\). In conclusion, the given data in the mass spectrum of Bromine helps us understand the presence of isotopes in the molecule and their combinations, and we have interpreted the data by explaining the source of each peak. Additionally, we have calculated the average atomic mass of \(\mathrm{Br_{2}}\), which is approximately \(159.77\,\mathrm{u}\).

Key Concepts

Isotopes

Isotopes are fascinating because, while they belong to the same element, they have differing neutron counts, giving them different mass numbers.
This means isotopes have the same number of protons but a different number of neutrons. This variance results in different physical properties, such as mass.
For example, bromine primarily has two isotopes:

• \(^{79}\textrm{Br} \) with 44 neutrons
• \(^{81}\textrm{Br} \) with 46 neutrons

Even though they weigh different amounts, both isotopes behave identically in chemical reactions.
This characteristic explains the peaks found in the mass spectrum of bromine molecules, which reflect the presence of these isotopes in nature.

Bromine

Bromine is a unique element, famous for forming diatomic molecules, represented as \(\mathrm{Br}_{2}\).
It's part of the halogen group in the periodic table, known for its high reactivity, especially with metals and organic compounds.
In nature, bromine appears in two main isotopic forms: \(^{79}\mathrm{Br} \) and \(^{81}\mathrm{Br} \). These isotopes are almost equally abundant.
The mass spectrum of a bromine molecule shows multiple peaks due to the possible isotopic combinations in the diatomic molecule:

• \(^{79}\mathrm{Br} - ^{79}\mathrm{Br} \)
• \(^{79}\mathrm{Br} - ^{81}\mathrm{Br} \)
• \(^{81}\mathrm{Br} - ^{81}\mathrm{Br} \)

This pattern results in a distinct mass-spectrum signature, aiding in accurate isotopic analysis.
Understanding bromine's isotopes helps decipher its reactions and uses in industries.

Mass-to-Charge Ratio

In mass spectrometry, the mass-to-charge ratio is crucial for identifying atoms or molecules.
The ratio, abbreviated as \(\frac{m}{z}\), assists in determining the particle's identity and the presence of isotopes.
An ion's path through the mass spectrometer is dictated by this ratio, revealing its mass for analysis.
When analyzing bromine, the mass spectrometer captures the mass-to-charge ratio for the isotopic combinations:

• \(157.84\,\mathrm{u} \) for the \(^{79}\mathrm{Br} - ^{79}\mathrm{Br} \) pair
• \(159.84\,\mathrm{u} \) for the \(^{79}\mathrm{Br} - ^{81}\mathrm{Br} \) pair
• \(161.84\,\mathrm{u} \) for the \(^{81}\mathrm{Br} - ^{81}\mathrm{Br} \) pair

This critical aspect allows scientists to analyze and interpret the structure of molecules based on the mass spectrum data.

Atomic Mass Calculation

Calculating atomic mass in a molecule like bromine involves averaging the isotopic masses based on their relative abundances.
This process helps determine the average mass of the molecule as it appears in nature.
Here's how you calculate the average atomic mass of bromine molecules based on the mass spectrum data:
An average value is derived from multiplying each isotopic mass by its relative size and summing the results:

• \(157.84\,\mathrm{u} \times 0.2534\) equals \(39.97\,\mathrm{u} \)
• \(159.84\,\mathrm{u} \times 0.5000\) equals \(79.92\,\mathrm{u} \)
• \(161.84\,\mathrm{u} \times 0.2466\) equals \(39.88\,\mathrm{u} \)

By adding these, one gets a total of \(159.77\,\mathrm{u}\), revealing the average atomic mass of bromine.
This method provides a more accurate representation of how these molecules exist naturally and is foundational in atomic analysis.