Question

An unknown element \(Q\) has two known isotopes: \({ }^{60} Q\) and \({ }^{63} Q\). If the average atomic mass is \(61.5 \mathrm{u}\), what are the relative percentages of the isotopes?

Short answer

The relative percentages of the isotopes { }^{60} Q and { }^{63} Q are 50% each.

Step-by-step solution

Assign variables to unknowns

Let the relative abundance of { }^{60} Q be denoted as x and that of { }^{63} Q as (1-x), since the total abundance must add up to 100%.

Set up the equation using the average atomic mass

The average atomic mass is calculated as the weighted average of the isotopes. So we set up the equation: (60 * x) + (63 * (1-x)) = 61.5.

Solve for x

Simplify and solve the equation to find the value of x: 60x + 63 - 63x = 61.5, which simplifies to -3x = -1.5. Solving for x gives x = 0.5 or 50%.

Calculate the percentage of the second isotope

Since the abundance of the second isotope is (1-x), we calculate it as 1 - 0.5 = 0.5 or 50%.

Key concepts

Average Atomic Mass

The average atomic mass of an element is a weighted average of the masses of its isotopes, which involves taking into account not only the isotopic masses but also their relative abundances in nature. To visualize this, imagine you have a bag of candies, with two flavors: one lighter and one heavier. If you have an equal number of each, the average weight of a candy will be halfway between the two. But if one flavor is more common, the average weight will be closer to that of the more abundant flavor. Similarly, the average atomic mass of an element is akin to the average weight of a 'bag' of atoms, where the 'flavors' are the different isotopes, and their 'weight' is their isotopic mass.

For a practical understanding, if an element has two isotopes, and we denote their isotopic masses as `m1` and `m2`, and their relative abundances as `x` and `(1-x)` respectively, the average atomic mass `M` can be calculated using the formula: \[ M = (m1 \times x) + (m2 \times (1-x)) \].

Isotopic Mass

Isotopic mass refers to the mass of a specific isotope of an element. Atoms of the same element can have different numbers of neutrons, thus leading to different isotopes. These isotopes have nearly identical chemical properties but differ in mass and sometimes in nuclear stability. The isotopic mass is usually expressed in atomic mass units (u), where one atomic mass unit is defined as one twelfth the mass of a carbon-12 atom.

Understanding isotopic mass is crucial in the calculation of average atomic mass. An example would be carbon, which has two stable isotopes, \( ^{12}C \) and \( ^{13}C \), with isotopic masses of approximately 12 u and 13 u, respectively. It's these individual masses that, when averaged according to their relative abundance in nature, contribute to the overall atomic mass of carbon that we find on the periodic table.

Chemical Elements

Chemical elements are pure substances consisting entirely of one type of atom, characterized by a specific number of protons in the nucleus, known as the atomic number. For instance, all atoms of oxygen have 8 protons. While the atomic number remains constant for a given element, the number of neutrons can vary, resulting in different isotopes. The presence of different isotopes means that the element can exhibit a variety of atomic masses.

Elements are represented on the periodic table and are the building blocks of molecules and compounds. Understanding each element's isotopic composition is a critical part of chemistry, as it influences many of the element's physical properties and behaviors, including its average atomic mass.

Relative Abundance

Relative abundance quantifies how common each isotope of an element is compared to the total amount of the element in a given sample or the environment. It is usually expressed as a percentage. In the case of our unknown element `Q`, there are two isotopes with a relative abundance that should add up to 100%. If one isotope is found to be more abundant than the other, this will heavily influence the average atomic mass of the element.

The concept shines when we are trying to determine an unknown element's isotopic composition, like our exercise example. Knowing the average atomic mass and isotopic masses allows us to calculate the relative abundances of the isotopes, provided we have at least one isotopic mass and the relative abundance of its isotopes. This concept is vital in fields such as geochemistry, medicine, and forensics.