Question

Two isotopes of an element \(\mathrm{Q}\) are \(\mathrm{Q}^{97}\) (23.4\% abundance) and \(Q^{94}\) (76.6\% abundance). \(Q^{97}\) is \(8.082\) times heavier than \(\mathrm{C}^{12}\) and \(\mathrm{Q}^{94}\) is \(7.833\) times heavier than \(\mathrm{C}^{12}\). What is the average atomic weight of the element Q? (a) \(94.702\) (b) \(78.913\) (c) \(96.298\) (d) \(94.695\)

Short answer

The average atomic weight of element \(\text{Q}\) is approximately \((0.234 \times 8.082 \times 12) + (0.766 \times 7.833 \times 12) = 94.695\), which corresponds to option (d).

Step-by-step solution

Understanding the concept of average atomic weight

The average atomic weight of an element that has multiple isotopes can be calculated by multiplying the atomic weight of each isotope by its relative abundance (expressed as a fraction), and then summing up these products.

Convert the abundances to decimal form

Convert the given percentage abundances of isotopes to decimal form by dividing each by 100. For \(Q^{97}\), it is 23.4%, so in decimal form it is 0.234. Similarly, for \(Q^{94}\), it is 76.6%, which in decimal form is 0.766.

Calculate the atomic weight of each isotope

We can calculate the atomic weight of \(Q^{97}\) by multiplying its given relative weight to that of \(C^{12}\): \((Atomic \ weight \ of \ Q^{97} = 8.082 \times 12\)). Do the same for \(Q^{94}\): \((Atomic \ weight \ of \ Q^{94} = 7.833 \times 12\)).

Calculate the average atomic weight

Finally, combine the information to find the average atomic weight: \(Average \ atomic \ weight = (0.234 \times Atomic \ weight \ of \ Q^{97}) + (0.766 \times Atomic \ weight \ of \ Q^{94})\).

Perform the calculations

Insert the values and perform the calculations: \((0.234 \times 8.082 \times 12) + (0.766 \times 7.833 \times 12)\). This will give the average atomic weight of element \(\text{Q}\).

Key concepts

Relative Atomic Mass

The concept of relative atomic mass is central to chemistry and involves comparing the mass of atoms of an element to a reference, which is usually the carbon-12 isotope. The relative atomic mass is the weighted average of the masses of all the isotopes of an element, with the weights being the natural abundances of the isotopes. To put it simply, it is an average that takes into account the different masses and abundances of an element's isotopes.

For example, if an element has two isotopes with different masses, the relative atomic mass would be closer to the mass of the more abundant isotope, because it contributes more to the average. This concept is crucial when determining the overall atomic mass that appears on the periodic table, and it is also utilised when calculating molar masses for chemical reactions.

Isotope Abundance

Isotope abundance refers to the percentage of a particular isotope in a natural sample of an element. Elements often exist as a collection of isotopes, and these isotopes differ from each other by the number of neutrons in the nucleus. However, not all isotopes are present in the same amounts, so the relative abundance helps us understand which isotopes are more common than others.

For instance, if an isotope has an abundance of 23.4%, this means that out of 100 atoms of the element, about 23 of them would be that particular isotope. Understanding isotope abundance is crucial for accurately calculating the average atomic weight of an element because it determines how much each isotope contributes to the overall weight.

Atomic Weight of Isotopes

The term atomic weight of isotopes describes the mass of a single isotope of an element. Each isotope of an element has a different atomic weight due to the variance in the number of neutrons. When scientists refer to the atomic weight of an isotope, they are usually comparing it to the mass of carbon-12, which is the standard reference with an atomic weight of 12.

For example, an isotope that is expressed as being 8.082 times heavier than carbon-12 will have an atomic weight of 8.082 times 12. This weight is a crucial piece of information needed when computing an element's average atomic weight, as different isotopes will contribute differently based on their individual atomic weights.

Chemical Element Isotopes

Every chemical element can have several isotopes, which are atoms of the element that have the same number of protons but a different number of neutrons. This difference in neutrons results in isotopes of the same element having different physical properties, such as atomic weight, but generally similar chemical properties.

Isotopes play important roles in various scientific fields, including radioactive dating in geology, medical imaging and treatments in medicine, and the study of chemical reaction mechanisms in chemistry. While some isotopes are stable, others are radioactive and can decay over time, releasing energy in the form of radiation. Isotopes of an element can vary widely in abundance and understanding both the stable and radioactive isotopes is vital for multiple applications in science and technology.