Problem 43 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A mixture of iron and an unknown material are bombarded with electrons. The wavelength of the \(\mathrm{K}_{\alpha}\) lines are \(194 \mathrm{pm}\) for iron and \(229 \mathrm{pm}\) for the unknown. What is the unknown material?

Step-by-Step Solution

Answer

The unknown material is cobalt (Z = 27).

Step 1: Understand the K_alpha Line

The \( \mathrm{K}_\alpha \) line represents the X-ray emission due to electron transitions from the L shell to the K shell in atoms. This line is characteristic of every element due to its unique energy level differences.

Step 2: Use Moseley's Law

Moseley's Law allows us to relate the frequency of the \( \mathrm{K}_\alpha \) X-ray lines to the atomic number \( Z \) of the element: \( \sqrt{u} = a (Z - b) \), where \( u \) is the frequency, \( a \) and \( b \) are constants. For a given \( \lambda \) (wavelength), use \( c = \lambda u \) to find the frequency \( u \).

Step 3: Calculate the Frequency for Iron

For iron, \( \lambda = 194 \, \mathrm{pm} = 194 \times 10^{-12} \, \mathrm{m} \). Calculate the frequency \( u \) using \( u = \frac{c}{\lambda} \) where \( c = 3 \times 10^8 \, \mathrm{m/s} \).

Step 4: Solve for Iron's Atomic Number Relationship

Plug the frequency of iron into Moseley's Law to find the relationship \( \sqrt{u_{\mathrm{Fe}}} = a (Z_{\mathrm{Fe}} - b) \) with the known atomic number for iron (Z = 26).

Step 5: Calculate the Frequency for Unknown Material

For the unknown material, \( \lambda = 229 \, \mathrm{pm} = 229 \times 10^{-12} \, \mathrm{m} \). Use the same calculation to determine the frequency \( u \).

Step 6: Determine the Atomic Number of Unknown

Use the frequency found for the unknown material in the relationship from Step 4. Rearrange Moseley's Law to solve for \( Z_{\text{unknown}} \), using the frequency calculated in Step 5.

Step 7: Identify the Unknown Material

With the calculated \( Z_{\text{unknown}} \), cross-reference periodic table data to find the element that matches this atomic number.

Key Concepts

X-ray emission

X-ray emission occurs when high-energy electrons strike a material, causing a specific interaction with the atoms of the material. This process begins with the ejection of one of the inner shell electrons, typically from the K shell, creating a vacancy. To fill this vacancy, an electron from a higher energy level, such as the L shell, drops down, releasing the energy difference as X-ray radiation.

• This transition releases photons in the form of X-rays.
• These X-rays possess specific wavelengths and frequencies that are characteristic of the element.
• The K series of X-ray lines are among the most significant, with the K-alpha line resulting from transitions specifically from the L shell to the K shell.

Understanding the X-ray emission is crucial because it holds the fingerprints of the atomic structure of the element emitting them. Since each element has a unique atomic number, its electron configurations and thus its X-ray emissions are also unique. This makes X-ray spectroscopy an invaluable tool in element identification.

atomic number

The atomic number, denoted as \( Z \), is a fundamental property that represents the number of protons found in the nucleus of an atom of the element. It determines the identity of the element and dictates its position in the periodic table. Here's why the atomic number is vital:

• The atomic number defines the chemical and physical properties of the element.
• In X-ray emission, it directly influences the energy levels of the electron shells.
• A higher atomic number typically means greater energy differences between shells, leading to higher frequency X-ray emissions.

Moseley's Law takes advantage of this atomic number relationship by linking it to the frequency of X-ray emissions from the K-alpha lines. This relationship helps scientists determine unknown elements by measuring their X-ray frequencies and calculating their atomic numbers.

K-alpha line

The K-alpha line is one of the most prominent and widely studied X-ray emission lines. It appears when an electron transitions from the second energy level (L shell) to the first energy level (K shell) in an atom.

• The energy released during this transition is specific and unique to each element.
• It serves as a crucial component in identifying substances through X-ray spectroscopy.
• K-alpha lines are used in applications ranging from material studies to forensic analysis.

This line provides insights into the electronic structure of atoms, as it reflects the distinct energy organization between the atomic shells. By studying these emissions, one can glean insights into not only the identity of elements but also the specific electron transitions occurring within them.

frequency calculation

Frequency calculation for X-ray emissions is performed using the relationship between the speed of light \( c \), wavelength \( \lambda \), and frequency \( u \). The fundamental equation that relates these is:
\[ u = \frac{c}{\lambda} \]
Where:

• \( u \) is the frequency of the X-ray emission.
• \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \mathrm{m/s} \).
• \( \lambda \) is the wavelength of the emitted X-ray.

For the exercise at hand, the K-alpha X-ray emission for iron and the unknown element are provided with specific wavelengths. By substituting these values into the equation, we can calculate the frequency for each element. These frequencies, when plugged into Moseley's Law, help in determining the atomic numbers of the elements, enabling the identification of unknown materials.