Question

If \(\lambda_{C u}\) is the wavelength of \(K_{\alpha} X\) -ray line of copper (atomic number 29 ) and \(\lambda_{M o}\) is the wavelength of the \(K_{\alpha} X\) -ray line of molybdenum (atomic number 42), then the ratio \(\lambda_{C u} / \lambda_{M o}\) is close to (A) \(1.99\) (B) \(2.14\) (C) \(0.50\) (D) \(0.48\)

Short answer

The ratio \(\lambda_{Cu} / \lambda_{Mo}\) is closest to option (B) 2.14.

Step-by-step solution

Understand the Moseley's Law

Moseley's Law relates the atomic number (Z) of an element to the frequency (v) of its X-rays. The law can be expressed as: \( v = A(Z - B)^2 \), where A and B are constants, and Z is the atomic number of the element. For K-alpha lines, we can consider the B term to be approximately the same for closely related elements such as copper and molybdenum. The frequency is inversely proportional to the wavelength (\( v = c/\lambda \)), therefore the law can be reformulated in terms of wavelengths as: \( \frac{1}{\lambda} = A'(Z - B)^2 \).

Calculate the Wavelength Ratio

Using the modified Moseley's Law, for two different elements we can write the ratio of their wavelengths as: \( \frac{\lambda_{Cu}}{\lambda_{Mo}} = \frac{[A'(Z_{Mo} - B)^2]^{-1}}{[A'(Z_{Cu} - B)^2]^{-1}} = \frac{(Z_{Mo} - B)^2}{(Z_{Cu} - B)^2} \), where \(Z_{Cu}\) is the atomic number of copper (29) and \(Z_{Mo}\) is for molybdenum (42). Since B is constant and nearly the same for both elements given their proximity in the periodic table, it can be factored out.

Simplify the Ratio of Atomic Numbers

Substitute the atomic numbers into the ratio: \( \frac{\lambda_{Cu}}{\lambda_{Mo}} = \frac{(42 - B)^2}{(29 - B)^2} \). As we don't know the exact value of B but know that it is relatively small compared to the atomic numbers of copper and molybdenum, the subtraction by B will result in atomic numbers that are relatively close to their actual values. Therefore, for the purposes of estimation, we can ignore B and approximate the ratio to: \( \frac{\lambda_{Cu}}{\lambda_{Mo}} \approx \frac{42^2}{29^2} \).

Calculate and Compare the Options

Perform the squaring and division of the numbers: \( \frac{\lambda_{Cu}}{\lambda_{Mo}} \approx \frac{1764}{841} \approx 2.10 \). Comparing this approximation with the given options, the closest value is option (B) 2.14. Therefore, we infer that option (B) is the answer to the question.

Key concepts

K-alpha X-ray Line

The K-alpha X-ray line is a specific type of X-ray radiation that is significant in the field of physics and materials science. When an electron from an outer shell of an atom fills a vacancy in the K-shell (the innermost electron shell), it emits energy in the form of an X-ray photon. This emitted radiation is known as the K-alpha line. The wavelength of the K-alpha line is characteristic of the element from which it is emitted because the energy levels of electrons are unique to each element.

This characteristic X-ray radiation is often used in techniques like X-ray fluorescence (XRF) to identify and analyze the composition of materials. Since the energies of the K-alpha lines for different elements are distinct, they serve as a 'fingerprint' to uniquely identify each element's presence in a sample.

Atomic Number

The atomic number of an element, denoted by 'Z', is a fundamental property in chemistry and physics that defines the identity of an atom. It represents the number of protons in the nucleus of an atom. For example, copper has an atomic number of 29, which means that a copper atom contains 29 protons. Similarly, molybdenum has an atomic number of 42, indicating 42 protons in its nucleus.

The atomic number is not just a counting term; it deeply influences the chemical behavior and the physical properties of an element as well. Electrons orbit the nucleus in discrete energy levels, and the atomic number helps determine how these electrons are arranged. It is this arrangement that affects how an element reacts chemically with others and also determines the X-ray spectra, including the aforementioned K-alpha X-ray line.

Wavelength Ratio Calculation

Understanding how to calculate the wavelength ratio between two different X-ray lines is essential in exercising Moseley's Law. With Moseley's Law, we can connect the X-ray wavelength to the atomic number of an element. In practice, when comparing the wavelengths of the K-alpha X-ray lines of two different elements, the law lets us factor out the constants and focus on how the square of the difference in atomic numbers affects the wavelengths.

To simplify the calculations, variables that have a constant value, like the term 'B' in Moseley's Law for closely related elements, can often be ignored or approximated. This simplification is based on the assumption that such constants have a negligible effect in comparison to the large atomic numbers involved. For educational purposes, students should note that while simplifications can aid understanding, they might introduce some degree of estimation, and precision can be essential in professional scientific contexts.