Problem 36 - Chemistry The Central Science In Si Units

Question

Bromine has two naturally occurring isotopes, bromine- 79 (atomic mass \(=78.9183 \mathrm{u} ; \quad\) abundance \(=50.69 \%)\) a \(\mathrm{nd}\) bromine- \(81 \quad\) (atomic mass \(=80.9163\) u; abundance = \(49.31 \%)\). Calculate the atomic weight of bromine.

Step-by-Step Solution

Answer

The atomic weight of bromine is approximately 79.904 u.

Step 1: Understand the Problem

We need to calculate the atomic weight of bromine using the two isotopes and their respective masses and natural abundances. The atomic weight of an element is the weighted average of the masses of its isotopes.

Step 2: Formula for Atomic Weight

The atomic weight can be calculated using the formula: \[ \text{Atomic Weight} = (f_1 \times m_1) + (f_2 \times m_2) \] where \(f_1\) and \(f_2\) are the abundances of the isotopes (as decimals) and \(m_1\) and \(m_2\) are the atomic masses of the isotopes.

Step 3: Convert Percent Abundance to Decimal

Convert the percentage abundances of bromine-79 and bromine-81 to decimals by dividing by 100. Thus, \( f_1 = 0.5069 \) for bromine-79 and \( f_2 = 0.4931 \) for bromine-81.

Step 4: Apply the Formula

Substitute the values into the formula: \[ \text{Atomic Weight} = (0.5069 \times 78.9183) + (0.4931 \times 80.9163) \].

Key Concepts

Isotopes

Isotopes are different forms of the same chemical element, each having the same number of protons but a different number of neutrons in their nucleus. This means that while they share similar chemical properties, their physical properties might differ due to the variance in their atomic masses.
Isotopes are naturally present and can be stable or radioactive. Stable isotopes do not change over time, whereas radioactive isotopes decay into other elements. These differences can be essential for various scientific applications, including dating archaeological finds or tracing chemical reactions.
In the case of bromine, it has two stable isotopes, bromine-79 and bromine-81. They are naturally occurring and the only isotopes of bromine that exist in nature in significant amounts.

Isotope Abundance

The concept of isotope abundance refers to how commonly an isotope is found relative to others of the same element. It is expressed as a percentage, indicating how many atoms of the isotope exist compared to the total count of all isotopes of that element. This can also be converted into a decimal for easier calculations.
The abundance of each isotope has a significant impact on the atomic weight of the element. For instance, in bromine's case:

• Bromine-79 has an abundance of 50.69%, which means out of every 100 bromine atoms, about 51 are bromine-79.
• Bromine-81 contributes to 49.31% of naturally occurring bromine.

Knowing these abundances allows scientists to calculate the element's average atomic weight accurately.

Weighted Average

A weighted average is a calculation that takes into account the varying degrees of importance of the numbers in a dataset. In our context, it accounts for both the atomic mass and abundance of an element's isotopes to determine its atomic weight.
To calculate a weighted average, each isotope's atomic mass is multiplied by its fractional abundance (obtained by dividing its percentage abundance by 100). The products are then summed up to get the weighted average. This allows for the atomic weight to reflect the combined effect of all isotopes:

• For bromine, the weighted average is calculated using the formula: \[ \text{Atomic Weight} = (0.5069 \times 78.9183) + (0.4931 \times 80.9163) \]
• Each part of this equation represents the portion of the overall atomic weight contributed by each isotope.

By utilizing a weighted average, we get a precise measure of the atomic weight that reflects the masses and abundances of an element's isotopes.