Problem 98 - Chemistry: The Central Science in SI Units, Global Edition

Question

There are two different isotopes of bromine atoms. Under normal conditions, elemental bromine consists of \(\mathrm{Br}_{2}\) molecules, and the mass of a \(\mathrm{Br}_{2}\) molecule is the sum of the masses of the two atoms in the molecule. The mass spectrum of \(\mathrm{Br}_{2}\) consists of three peaks: $$ \begin{array}{lc} \hline \text { Mass (u) } & \text { Relative Size } \\ \hline 157.836 & 0.2569 \\ 159.834 & 0.4999 \\ 161.832 & 0.2431 \\ \hline \end{array} $$ (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 \(\mathrm{Br}_{2}\) molecule. (d) Determine the average atomic mass of a bromine atom. (e) Calculate the abundances of the two isotopes.

Step-by-Step Solution

Answer

The origin of the peaks is as follows: 1. Peak 1: Isotope 1 + Isotope 1 2. Peak 2: Isotope 1 + Isotope 2 (or Isotope 2 + Isotope 1) 3. Peak 3: Isotope 2 + Isotope 2 The masses of the isotopes are: - Isotope 1: \(78.918\,u\) - Isotope 2: \(80.916\,u\) The average molecular mass of a Br2 molecule is \(159.808\, u\). The average atomic mass of a Bromine atom is \(79.904\,u\). The abundances of the isotopes are approximately: - Isotope 1: 50.00% - Isotope 2: 50.00%

Step 1: Identify the possible combinations of the isotopes in Br2 molecules

Since Br2 is a diatomic molecule and there are two different isotopes of bromine, the three possible Br2 combinations are: 1. Isotope 1 + Isotope 1 (same isotopes) 2. Isotope 1 + Isotope 2 (mixed isotopes) 3. Isotope 2 + Isotope 2 (same isotopes)

Step 2: Understand the origin of the peaks

The mass spectrum shows 3 peaks; each peak corresponds to one of the Br2 combinations mentioned in step 1. The mass of each Br2 molecule is the sum of the masses of the two Bromine atoms within it. So, each peak shows the mass of each possible combination of the two different isotopes: 1. Peak 1: Isotope 1 + Isotope 1 2. Peak 2: Isotope 1 + Isotope 2 (or Isotope 2 + Isotope 1) 3. Peak 3: Isotope 2 + Isotope 2

Step 3: Find the isotopic masses

Let the mass of Isotope 1 be X and the mass of Isotope 2 be Y. Based on the given information, we can create a system of equations: \(X + X = 157.836\) \(X + Y = 159.834\) \(Y + Y = 161.832\) Solving this system of equations, we have: \(X = 78.918 \, u\) (mass of Isotope 1), and \(Y = 80.916 \, u\) (mass of Isotope 2)

Step 4: Determine the average molecular mass of Br2 molecules

To find the average molecular mass of Br2 molecules, we can use the sum of the products of the mass of each combination and their relative sizes: \( Average \, Molecular \, mass = 157.836 \times 0.2569 + 159.834 \times 0.4999 + 161.832 \times 0.2431 = 159.808 \, u \)

Step 5: Calculate the average atomic mass of a Bromine atom

Since the mass of a Br2 molecule is the sum of the masses of the two Br atoms in it, we can calculate the average atomic mass of Br by dividing the average molecular mass by 2: \( Average \, Atomic \, Mass = \frac{159.808 \, u}{2} = 79.904 \, u \)

Step 6: Compute the abundances of the two isotopes

If A and B represent the relative sizes for isotopes 1 and 2 contributing to the mass of Br2 molecules, we can relate them with the peak values: \(A = B = 0.4999\) (the mixed isotopes) The sum of A and B in terms of the relative size of the isotopes can be expressed as: \(A + B = 1\) \(0.4999 + 0.4999 = 0.9998 \approx 1\) So, the abundances of Isotope 1 and Isotope 2 are approximately: Isotope 1: 50.00% Isotope 2: 50.00%

Key Concepts

Atomic Mass

Atomic mass refers to the average mass of atoms of an element, calculated using the relative abundance of each isotope. It's typically measured in atomic mass units (u). Each isotope of an element has its specific mass, and the atomic mass is the weighted average of these isotopes. In the case of bromine, the atomic mass is derived by considering the masses and relative abundances of its two isotopes.
To find the atomic mass of bromine, you can take the average of the isotopic masses, considering their relative abundance in nature. This results in an atomic mass of approximately 79.904 u.
Understanding atomic mass helps in chemical calculations and understanding the properties of elements.

Mass Spectrum

A mass spectrum is a chart that shows the mass-to-charge ratio of ions. It's a tool used to identify different isotopes of elements and to measure their relative abundance. In the mass spectrum of bromine, three distinct peaks are observed. Each peak corresponds to a different combination of bromine isotopes in the diatomic molecule \( \mathrm{Br}_{2} \).

• The first peak (157.836 u) represents \( \mathrm{Br}_{2} \) with two of Isotope 1 in it.
• The second peak (159.834 u) corresponds to a mix of Isotope 1 and Isotope 2.
• The third peak (161.832 u) shows two Isotope 2 atoms.

By analyzing the mass spectrum, we can gain insights into the isotopic composition of an element.

Diatomic Molecule

A diatomic molecule is composed of two atoms. In the case of bromine, the molecule \( \mathrm{Br}_{2} \) consists of two bromine atoms. Diatomic molecules can be homonuclear (same element) like Br₂ or heteronuclear (different elements) like CO.Bromine's diatomic nature affects its physical and chemical properties. The mass of a diatomic molecule is the sum of the masses of its two atoms. Therefore, the mass spectrum of bromine (\( \mathrm{Br}_{2} \) molecules) gives us peaks that relate to different isotopic combinations.This concept is crucial when studying molecular structures and reactions involving gases.

Molecular Mass

Molecular mass is the sum of the masses of all atoms in a molecule. In the case of the diatomic molecule \( \mathrm{Br}_{2} \), it is the sum of the masses of two bromine atoms. To find the average molecular mass, you can consider the different isotopic combinations and their relative abundances.For bromine:

• A combination of two Isotope 1 atoms gives a mass of 157.836 u.
• One Isotope 1 and one Isotope 2 give 159.834 u.
• Two Isotope 2 atoms give 161.832 u.

The average molecular mass of \( \mathrm{Br}_{2} \) is calculated by weighting these combinations by their relative sizes, resulting in an average value of 159.808 u.Molecular mass is key for determining the stoichiometry in reactions and understanding molecular properties.