Question

The atomic weight of bromine is 79.904 . The natural abundance of \({ }^{81} \mathrm{Br}\), atomic weight \(80.916289 \mathrm{u}\), is \(49.31 \%\). What is the atomic weight of the only other natural isotope of bromine?

Short answer

The atomic weight of the other bromine isotope is approximately 78.918.

Step-by-step solution

Understand the given information

We have bromine with an atomic weight of 79.904 u. It has two isotopes: \(^{81}\mathrm{Br}\) with a given atomic weight of 80.916289 u and a natural abundance of 49.31%. The task is to find the atomic weight of the other isotope.

Define variables

Let's denote the atomic weight of the other isotope \( ^{A}\mathrm{Br} \) as \( x \), and its natural abundance as \( 100\% - 49.31\% = 50.69\%. \) Therefore, the formula is given by:\[\text{Average atomic weight} = (80.916289 \times 0.4931) + (x \times 0.5069) = 79.904\]

Set up the equation

The calculation for the average atomic weight of bromine is weighted by the abundance of each isotope:\[79.904 = (80.916289 \times 0.4931) + (x \times 0.5069)\]

Solve the equation

First, calculate \(80.916289 \times 0.4931\):\[39.8764853059\]Place this into the equation:\[79.904 = 39.8764853059 + 0.5069x\]Next, subtract 39.8764853059 from both sides:\[79.904 - 39.8764853059 = 0.5069x\]Calculate the difference:\[40.0275146941 = 0.5069x\]Now, solve for \(x\) by dividing by 0.5069:\[x = \frac{40.0275146941}{0.5069}\]Calculate \(x\):\[x \approx 78.918\]

Interpret the result

The atomic weight of the other natural isotope \(^{A}\mathrm{Br}\) is approximately 78.918.

Key concepts

Understanding Atomic Weight

Atomic weight, often synonymous with atomic mass, refers to the average mass of atoms of an element, calculated using the relative abundance of each isotope present in the element. It is expressed in unified atomic mass units (u), which is approximately equal to the mass of one proton or neutron.

The concept is crucial for chemistry and other sciences because it helps understand how to calculate the mass of large amounts of atoms, which is useful in chemical equations and reactions. For example, if we know the atomic weight of bromine to be 79.904, it represents the weighted average of the different isotopes of bromine based on their natural abundance in a sample.

When an element, like bromine, has isotopes with different masses, the atomic weight calculation is weighted. This means that isotopes with higher natural abundance have a greater influence on the computed atomic weight. Therefore, knowing the atomic weight provides insight into the composition of isotopes present in a natural sample.

Exploring Natural Abundance

Natural abundance refers to the percentage of an isotope compared to the total amount of all isotopes of that element found in nature. This percentage helps determine how common or rare an isotope is within a given sample of the element.

For elements like bromine, which have more than one stable isotope, the natural abundance is crucial in isotopic calculations—such as determining the overall atomic weight. To find the natural abundance of each isotope, we typically look at terrestrial samples, observing the proportions in which these isotopes occur naturally.

In the context of bromine isotopes, we know that \(^{81}\mathrm{Br}\) has a natural abundance of 49.31%. To find the abundance of other isotopes, we subtract the given percentage from 100%. This natural distribution of isotopes is intrinsic to the elemental behavior observed in chemical processes.

Insight into Bromine Isotopes

Bromine, a chemical element with the symbol Br, has two naturally occurring isotopes: \(^{79}\mathrm{Br}\) and \(^{81}\mathrm{Br}\). Each isotope of bromine has a slightly different mass, and therefore different atomic weights should be calculated accordingly when considering the element's total atomic weight.

These isotopes occur in nearly equal proportions. In nature, the isotope \(^{81}\mathrm{Br}\) is approximately 49.31% of bromine's natural store. To find the atomic weight of the other isotope, we looked at this natural abundance balance and utilized the weighted average equation.

• \(^{79}\mathrm{Br}\) has an atomic weight approximated at 78.918 u.


• \(^{81}\mathrm{Br}\) has an atomic weight of 80.916289 u.

Having accurate isotope information allows scientists to perform precise measurements and calculations, which are essential for various applications, such as radiometric dating or nuclear medicine.