What are the approximate energies of the \(K_{\alpha}\) and \(K_{\beta}\) X-rays for copper?

Moseley's law is a pivotal concept in understanding the behavior of X-rays emitted by elements. English physicist Henry Moseley discovered a relationship between the frequency of X-rays produced by the elements and their atomic numbers. According to this law, the frequency of the X-rays is related to the square of the effective nuclear charge, which accounts for the atomic number (Z) minus the screening constant (B).

For K-series X-rays, the formula is often simplified to \[u = A(Z - B)^{2} \times Ry\]Here, A is a proportionality constant and Ry is the Rydberg constant specifically for hydrogen. The Screening constant B is approximated as 1 for K-series X-rays due to minimal electron screening effect. This law allows us to predict the frequencies of the characteristic X-rays, such as the K-alpha (\(K_{\alpha}\)) and K-beta (\(K_{\beta}\)) transition lines. By knowing the frequency, we can then calculate the energy of these X-rays using Planck's equation.

The atomic number, often denoted by Z, is a fundamental property of an element and is equivalent to the number of protons found in the nucleus of an atom. It uniquely identifies a chemical element and determines its position in the periodic table. The concept of the atomic number is essential in Moseley's law, as it directly affects the emitted X-ray frequencies.

For example, copper has an atomic number of 29. This information allows us to calculate the frequencies and subsequently the energies of the copper K-alpha and K-beta X-rays using Moseley's law. Higher atomic numbers typically result in X-rays with greater frequencies and higher energies, which is a direct consequence of the increased nuclear charge exerted on the electron shells.

Photon energy is the energy carried by a single photon with a certain electromagnetic frequency or wavelength. The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength. The relationship between energy (E) and frequency (\(u\)) is given by Planck's equation:\[E = hu\]where h is Planck's constant (\(6.626 \times 10^{-34} J\cdot s\)). This fundamental relationship makes it possible to calculate the energy of X-rays after determining their frequencies. For instance, in the step-by-step solution we've seen, after applying Moseley's law to identify the frequencies of copper's K-alpha and K-beta X-rays, we used Planck's equation to calculate their energies, enabling us to understand the level of energy that copper's X-rays possess.

The Rydberg constant is a physical constant related to the atomic spectra of various elements. In the context of Moseley's law, the Rydberg constant for hydrogen (Ry) is often used, with a value of approximately \(3.29 \times 10^{15} Hz\). This constant represents the highest frequency (or smallest wavelength) of any photon that can be emitted by a hydrogen atom, corresponding to a quantum leap from the first excited state to the ground state.

When combined with the appropriate constant (A) and the adjusted atomic number Z (taking into account the electronic screening by subtracting the screening constant B), it allows us to calculate the specific frequencies of the X-rays when electrons transition between energy levels. As a result, the Rydberg constant plays a crucial role in determining the energies of X-rays in various elements, including copper in the provided exercise.