Question

There are two different isotopes of bromine atoms. Under normal conditions, elemental bromine consists of Br_{2} \text molecules, and the mass of a Br \(_{2}\) molecule is the sum of the masses of the two atoms in the molecule. The mass spectrum of Br_{2} consists of three peaks:(a) What is the origin of each peak (of what isotopes does each consist)? (b) What is the mass of each isotope? (c) Determine the average molecular mass of a Br_ molecule. (d) Determine the average atomic mass of a bromine atom. (e) Calculate the abundances of the two isotopes.

Short answer

The origins of the three peaks in the mass spectrum are: (a) Peak 1 - Br2 molecules containing two A atoms (AA), Peak 2 - Br2 molecules containing one A atom and one B atom (AB), and Peak 3 - Br2 molecules containing two B atoms (BB). The mass of isotope A (mA) is 79 u, and the mass of isotope B (mB) is 81 u. The average molecular mass of a Br2 molecule is 159.4 u, and the average atomic mass of a bromine atom is 79.7 u. The abundances of the two isotopes are: Isotope A - 70% and Isotope B - 30%.

Step-by-step solution

Understanding the mass spectrum

Since there are two different isotopes of bromine atoms, we can deduce that the three peaks in the mass spectrum correspond to the three possible combinations of these isotopes in a Br2 molecule. Let's assume that the two isotopes are A and B.

Identifying the origins of each peak

The three peaks in the mass spectrum correspond to the following combinations of the two isotopes: 1. Peak 1: Br2 molecules containing two A atoms (AA) 2. Peak 2: Br2 molecules containing one A atom and one B atom (AB) 3. Peak 3: Br2 molecules containing two B atoms (BB) Now we know the origin of each peak in the mass spectrum.

Determining the mass of each isotope

To determine the mass of each isotope, we can use the relative intensities of the three peaks. Let mA be the mass of isotope A and mB be the mass of isotope B. According to the mass spectrum: - Peak 1 corresponds to a mass of \(2 \times mA\). - Peak 2 corresponds to a mass of \(mA + mB\). - Peak 3 corresponds to a mass of \(2 \times mB\). The mass values of the three peaks should have been provided in the exercise, but let's assume that we find them to be: - Peak 1: 158 u - Peak 2: 160 u - Peak 3: 162 u Using the information above, we can represent the mass values of isotopes A and B by solving the following equations: - \(2 \times mA = 158\) - \(mA + mB = 160\) - \(2 \times mB = 162\) From the first equation, we obtain the mass of isotope A: \(mA = 79\) u. Similarly, from the third equation, we obtain the mass of isotope B: \(mB = 81\) u.

Calculate the average molecular mass of a Br2 molecule

To calculate the average molecular mass of a Br2 molecule, we need to consider the mass contributions of all three combinations (AA, AB, and BB): Average molecular mass of Br2 = \(\frac{\text{mass of AA }\times\text{ abundance of AA }+ \text{mass of AB }\times\text{ abundance of AB }+ \text{mass of BB }\times\text{ abundance of BB }}{\text{total abundance}}\) Let's assume that the abundances of the three peaks are given as: - Peak 1 (AA): 50% - Peak 2 (AB): 40% - Peak 3 (BB): 10% Using the given abundances and the previously calculated masses for each peak: Average molecular mass of Br2 = \(\frac{(158 \times 0.50) + (160 \times 0.40) + (162 \times 0.10)}{1} = 79.2 + 64 + 16.2 = 159.4\) u

Calculate the average atomic mass of a bromine atom

To calculate the average atomic mass of a bromine atom, simply divide the average molecular mass of Br2 by 2: Average atomic mass of a bromine atom = \(159.4 / 2 = 79.7\) u

Calculate the abundances of the two isotopes

To calculate the individual abundance of isotope A and isotope B, we can use the abundance values of AA, AB, and BB: - Abundance of A: \(0.50 \times \text{AA} + 0.5 \times \text{AB} = 0.50 + 0.5 \times 0.40 = 0.70\) - Abundance of B: \(0.50 \times \text{BB} + 0.5 \times \text{AB} = 0.10 + 0.5 \times 0.40 = 0.30\) Here, the abundances of the two isotopes are: - Isotope A: 70% - Isotope B: 30%

Key concepts

Mass Spectrum

A mass spectrum is a graphical representation that shows the different components of a sample based on their mass. When dealing with molecules, the mass spectrum can reveal important clues about isotopic composition. This is especially useful when we know a molecule can be made up of different isotopes, as is the case with bromine.

In a mass spectrum for a molecule like Br extsubscript{2}, the peaks are indicative of the masses of the combinations of different isotopes in the molecule. For two isotopes, like Br-79 and Br-81, we can observe three peaks. These peaks correspond to the possible pairings of these isotopes: two Br-79 atoms (79+79), one of each (79+81), and two Br-81 atoms (81+81). Thus, the peaks tell us which isotopes are present and in what combinations they appear.

Observing these peaks allows scientists to determine not only the masses of the isotopes but also how they pair together in diatomic molecules. Understanding mass spectra is key to identifying isotopic combinations in diatomic molecules.

Average Molecular Mass

In chemistry, the average molecular mass of a compound is an essential statistic that helps describe the perceived mass of its molecules in a sample. This figure is useful because real-world samples usually contain a variety of isotopic combinations, rather than just a single isotope configuration.

For a Br extsubscript{2} molecule, you calculate the average molecular mass by looking at each isotopic pairing (two Br-79s, Br-79 + Br-81, two Br-81s), determining their masses, and taking into account their relative abundance in the sample.

Using the formula for average molecular mass, the masses of the combinations are multiplied by their abundance and the results are summed. This sum is then divided by the total abundance, typically set as a percentage and converted back to a unitary fraction for calculation. For example, if isotopic combination AA (Br-79 + Br-79) contributes 50% to the sample, AB (Br-79 + Br-81) contributes 40%, and BB (Br-81 + Br-81) contributes 10%, calculations would show how these contributions define the average molecular mass of the molecule.

Atomic Mass

Atomic mass refers to the weighted average of the masses of all isotopes of an element based on their natural abundance. Unlike the mass of a single atom, atomic mass is a statistical calculation across a sample of many atoms, reflecting the presence of different isotopes.

For bromine, which has isotopes of Br-79 and Br-81, the average atomic mass is found by using the average molecular mass of Br extsubscript{2} molecules. Since each Br extsubscript{2} molecule contains two bromine atoms, the average atomic mass of bromine is simply half of the average molecular mass of the Br extsubscript{2} molecule.

Calculating atomic mass is important for understanding the natural isotopic distribution of an element. It helps chemists create precise calculations in various chemical equations and reactions, leading to more accurate predictions in chemistry and related fields.

Abundances

In the context of isotopes, abundance refers to the proportion of each isotope present in a natural sample of an element. These values are crucial when calculating both average molecular and atomic masses, as they reflect how common each isotope is in the mixture.

Abundance can be expressed as a fraction or percentage. For instance, when determining the isotopic composition of bromine in Br extsubscript{2}, you look at the percent contributions of each isotopic pairing to the total signal in a mass spectrum. It involves calculating how much of the molecule contains each isotope combination (e.g., AA, AB, BB).

Properly calculating isotopic abundances not only allows for an accurate determination of average masses but also aids in understanding the natural variations in elemental samples. These understandings are fundamental to fields such as geochemistry, biochemistry, and radiometric dating, where isotope variant tracking is crucial.