Question

Boron (B) has two naturally occurring isotopes: boron-10 (abundance \(=19.8 \%\) , mass \(=10.013\) amu) and boron-11 (abundance \(=80.2 \%,\) mass \(=11.009\) amu). Calculate the atomic mass of boron.

Short answer

The atomic mass of boron can be calculated using the weighted average of its isotopes' atomic masses and abundances. Convert the percentage abundances to decimals and use the formula: Weighted average atomic mass = (Boron-10 abundance × Boron-10 mass) + (Boron-11 abundance × Boron-11 mass). The atomic mass of boron is approximately \(10.809 \text{ amu}\).

Step-by-step solution

Convert percentage abundances to decimals

First, we need to convert the percentage abundances of each isotope into decimals by dividing them by 100. Boron-10 abundance: \(19.8 \% = \frac{19.8}{100} = 0.198\) Boron-11 abundance: \(80.2 \% = \frac{80.2}{100} = 0.802\)

Calculate the weighted average atomic mass

Now, we'll calculate the weighted average atomic mass using the decimal abundances we found in Step 1 and the given atomic masses for each isotope. Weighted average atomic mass = (Boron-10 abundance × Boron-10 mass) + (Boron-11 abundance × Boron-11 mass) = \((0.198 × 10.013 \text{ amu}) + (0.802 × 11.009 \text{ amu})\)

Calculate the result

Using the values from Step 2, calculate the weighted average atomic mass of boron. = \((0.198 × 10.013 \text{ amu}) + (0.802 × 11.009 \text{ amu})\) = \(1.982574 \text{ amu} + 8.826418 \text{ amu}\) = \(10.808992 \text{ amu}\)

Round the result

Finally, we will round the atomic mass of boron to a suitable number of decimal places. In this case, three decimal places are appropriate. Atomic mass of Boron = \(10.809 \text{ amu}\) The atomic mass of boron is approximately \(10.809 \text{ amu}\).

Key concepts

Isotopic Abundance

When we discuss isotopic abundance, we're referring to the relative amount of each isotope of an element present in a natural sample. Isotopes are variants of a particular chemical element which differ in neutron number, and consequently in mass. However, they have the same number of protons, which means they display very similar chemical behavior.

For instance, as in the case of boron, it exists naturally with two isotopes: boron-10 and boron-11. If we're given percentages, such as 19.8% for boron-10 and 80.2% for boron-11, this means that in a given sample of boron, 19.8% of the atoms will be boron-10, while the remaining 80.2% will be boron-11. To use these percentages for calculations, we convert them into decimal form by dividing by 100. It's these decimal values that play a key role in calculating the average atomic mass of an element.

Weighted Average

The average atomic mass of an element isn't just a simple average of the masses of its isotopes; rather, it's a weighted average. This takes into account the relative abundance of each isotope, ensuring that more common isotopes have a greater impact on the final average.

In our boron example, to compute this weighted average, each isotope's mass is multiplied by its decimal abundance. These products are then summed to yield the average atomic mass of the element. This approach reflects the fact that the mass of a boron sample will be more influenced by the more plentiful boron-11 isotope than the less abundant boron-10.

Chemical Elements

Chemical elements are the building blocks of matter, each distinguished by a unique number of protons in the atomic nucleus, known as the atomic number. While an element's isotopes may vary in mass due to different numbers of neutrons, they retain identical chemical properties because they have the same number of protons and electrons.

The mass recorded on the periodic table for each element is not that of a single isotope, but rather the weighted average mass of all the element's naturally occurring isotopes, as we've seen with the calculation for boron. This mass is critical for a wide range of applications, from chemical reaction stoichiometry to the study of atomic structures in physics.