Moseley established the concept of atomic number by studying X-rays emitted by the elements. The X-rays emitted by some of the elements have the following wavelengths: $$ \begin{array}{ll} \hline \text { Element } & \text { Wavelength }(\AA) \\\ \hline \mathrm{Ne} & 14.610 \\ \mathrm{Ca} & 3.358 \\ \mathrm{Zn} & 1.435 \\\ \mathrm{Zr} & 0.786 \\ \mathrm{Sn} & 0.491 \\ \hline \end{array} $$

The concept of atomic number, often denoted as 'Z', is fundamental in understanding the properties of elements and their place in the periodic table. It represents the number of protons in an atom's nucleus, which is unique for each element. It directly determines the element's identity and influences its chemical behavior. For example, Neon (Ne) has an atomic number of 10, indicating it has 10 protons in its nucleus.

From the exercise, when examining the atomic numbers of different elements like Calcium (Ca), Zinc (Zn), Zirconium (Zr), and Tin (Sn), we see that these numbers increase stepwise. Moseley used the atomic number as a pivotal part of his law to establish a firm connection between this numerical identifier of elements and the characteristics of the X-rays they emit.

The term X-ray wavelengths describes the distance between successive peaks of an X-ray wave. X-rays themselves are a type of electromagnetic radiation, much like visible light, but with a much shorter wavelength. This characteristic enables them to penetrate substances that light cannot. In the provided exercise, we are given the wavelengths of X-rays emitted by various elements, such as Neon (Ne) at 14.610 Angstroms (Å) and Tin (Sn) at 0.491 Å.

Understanding these wavelengths is critical as they are indicative of the energy levels of electrons within atoms and can be used to identify elements, as per the work of Moseley. The analysis of how X-ray wavelengths vary with atomic numbers leads to insights into the atomic structure and the behavior of electrons within the atom.

Delving into the relationship between atomic number and X-ray wavelengths, Moseley discovered that there was a predictable pattern to the X-rays emitted by atoms when their inner-shell electrons are excited. This relationship turned out to be key to the development of modern atomic theory. As seen in the exercise, when the atomic numbers scale upwards, the corresponding X-ray wavelengths become shorter. This suggests a specific mathematical relationship between these two variables, which Moseley defined and which we often refer to as Moseley's law.

The investigation into this relationship showcased Moseley's ingenuity in linking a physical property (X-ray wavelength) to a fundamental atomic characteristic (atomic number), thereby allowing for the atomic number to be more precisely determined.

In an inversely proportional relationship, as one variable increases, the other decreases at a rate that keeps their product constant. Moseley's law hinges on this kind of relationship between the atomic number and the X-ray wavelengths. Our exercise illustrates this through the equation
\[ A = \frac{k}{Z^2} \]
which implies that the wavelength 'A' is inversely proportional to the square of the atomic number 'Z'. The process of plotting the provided values and calculating a consistent 'k' across several elements, as presented in the exercise, reinforces this inverse proportionality. As an educational concept, understanding inverse relationships is crucial in physics and other scientific disciplines, as it describes how quantities counterbalance each other. This concept in X-ray spectroscopy also demonstrates how advancing our understanding of elements' properties often requires a deep dive into the mathematical relationships governing their behavior.