Problem 46 - General Chemistry: Principles and Modern Applications

Question

Bromine has two naturally occurring isotopes. One of them, bromine-79, has a mass of 78.918336 u and a natural abundance of \(50.69 \% .\) What must be the mass and percent natural abundance of the other isotope, bromine-81?

Step-by-Step Solution

Answer

The natural abundance of Bromine-81 is \(49.31%\), and the mass is approximately 80.916 u.

Step 1: Understanding the Problem

Bromine has two isotopes, Bromine-79 and Bromine-81. Given are the mass and natural abundance of Bromine-79. The problem is asking for the mass and natural abundance of Bromine-81.

Step 2: Calculate the Natural Abundance of Bromine-81

Since only two isotopes of bromine exist, the sum of their abundances should be equal to 100%. We have the natural abundance of Bromine-79 as \(50.69%\). Therefore, the natural abundance of Bromine-81 would be \(100 - 50.69 = 49.31%\)

Step 3: Determine the mass of Bromine-81

The weight of an element is calculated as the weighted mean of the weights of its isotopes. The weight of Bromine (Br) is given as approximately 79.904 u. So we can create an equation where \(79.904 = 0.5069 * 78.918336 + 0.4931 * x\), where \(x\) signifies the weight of Bromine-81. Solving this equation for \(x\) gives us the weight of Bromine-81.

Key Concepts

Natural Abundance

Natural abundance refers to the percentage of a particular isotope present in a sample of an element. It reflects how common or rare an isotope is in nature. In the case of bromine, there are two main isotopes, bromine-79 and bromine-81.

• Bromine-79 has a natural abundance of 50.69%.
• Since there are only two isotopes, the sum of their abundances must equal 100%.
• Therefore, the natural abundance of bromine-81 must be calculated as 100% minus the percentage of bromine-79.

This principle helps in understanding the relative amounts of different isotopes within a given element. By knowing the natural abundances, you can determine how these isotopes contribute to the overall properties of the element. In our case, since bromine-79 accounts for 50.69%, bromine-81 must make up the remaining 49.31% of natural bromine.

Weighted Mean

The weighted mean is a kind of average that considers the relative importance, or weights, of each component. In isotope calculations, each isotope's weight is its mass, and its importance is given by its natural abundance.
The calculation of the atomic weight of an element like bromine involves a weighted mean of its isotopes:

• We know bromine's average atomic mass is 79.904 u.
• Bromine-79 has a mass of 78.918336 u and contributes 50.69%.
• Bromine-81's mass and contribution make up the remaining percentage.

The formula for the weighted mean in this case would look like: \[79.904 = (78.918336 \times 0.5069) + (x \times 0.4931)\]Solving this equation helps us find the mass of bromine-81, ensuring the average matches the known atomic weight of bromine. This process highlights how the weighted mean gives more accurate information by accounting for varying contributions of different parts.

Bromine Isotopes

Isotopes of an element have the same number of protons but different numbers of neutrons. This results in differences in mass while keeping chemical properties identical. Bromine has two stable isotopes: bromine-79 and bromine-81.

• Both isotopes play a role in the element's chemical identity but have different masses.
• Bromine-79 has a mass of 78.918336 u, and bromine-81 is slightly heavier.
• The distribution between these isotopes leads to variations in natural samples of bromine.

Understanding isotopes is crucial for fields like chemistry and physics since they contribute to an element's overall atomic mass and can affect its behavior. By measuring the masses of these isotopes and knowing their natural abundances, scientists can predict how bromine might behave in different reactions and situations.