An element has the following natural abundances and isotopic masses: \(90.92 \%\) abundance with 19.99 amu, \(0.26 \%\) abundance with 20.99 amu, and \(8.82 \%\) abundance with 21.99 amu. Calculate the average atomic mass of this element.

Isotopic abundance refers to the percentage of each isotope present in a natural sample of an element. To fully understand this concept, one must know that an element can exist in different forms, called isotopes, which have the same number of protons but differ in the number of neutrons. This difference in neutrons results in variations in the atomic mass of the isotopes.

For example, in our exercise, we have three isotopes of an element with different abundances expressed as percentages. These are not mere arbitrary numbers but reflect how often each isotope occurs compared to the others in nature. The isotopic abundances are crucial for calculating the average atomic mass of an element because the more abundant an isotope is, the more it contributes to the element's average atomic mass.

By converting these percentages into decimal form, we prepare them to be used in a calculation that will give us a more accurate representation of the element's average atomic mass based on real-world samples.

The concept of the weighted average plays a pivotal role in calculating an element’s average atomic mass. A weighted average takes into account not only the value of each isotope’s mass but also how abundant each isotope is. This gives us a more precise 'average' that is representative of the actual proportions in which isotopes occur in nature.

In our exercise, after converting the abundances to decimals, each isotope’s mass is multiplied by its respective abundance. By doing this, we are essentially weighting the mass of each isotope by how common it is. The next step combines these weighted masses, summing them together, to get the overall average mass of the element which reflects its isotopic composition. This differs from a simple average, as it wouldn’t appropriately factor in the relative quantities of each isotope.

The atomic mass unit, or amu, is a standard unit of mass that quantifies the weight of atoms. One atomic mass unit is defined as one twelfth of the mass of a carbon-12 atom. It's a very small unit of mass, tailored specifically to express atomic and molecular weights. In our calculations, we use the amu to express the mass of isotopes, which are the distinct forms of elements that vary in neutron count.

Having a consistent unit such as amu allows scientists to compare the masses of different atoms on a relative scale accurately. In the textbook exercise, by expressing isotopic masses in amus, we can perform calculations that will lead us to the weighted average atomic mass of an element, which is also expressed in amus.

The mole concept is a fundamental principle in chemistry that provides a bridge between the atomic scale and the macroscopic world. A mole is defined as the amount of substance that contains as many entities (atoms, molecules, or other particles) as there are atoms in 12 grams of pure carbon-12. This number is also known as Avogadro's number, which is approximately equal to 6.022x10²³ particles.

The mole concept allows us to count atoms by weighing them, since weighing atoms individually is not feasible. When we discuss the average atomic mass of an element, which we calculate from isotopic abundance and masses, we are essentially referring to the mass of one mole of that element. In practice, when you weigh out the average atomic mass expressed in grams, you have one mole of that element, giving you a tangible quantity to work with in experiments.