Problem 41 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Estimate the wavelength for an \(n=2\) to \(n=1\) transition in iron \((Z=26)\).

Step-by-Step Solution

Answer

The estimated wavelength for the \(n=2\) to \(n=1\) transition in iron is approximately \(1.80 \times 10^{-10} \text{ m}\).

Step 1: Understand the context

The exercise involves calculating the wavelength of a photon emitted during an electronic transition in an atom. For this, we need to use the Rydberg formula for hydrogen-like atoms, where the atomic number, \(Z=26\), is for iron.

Step 2: Identify the Rydberg Formula

The Rydberg formula for calculating the wavelength of light emitted during a transition between two energy levels \(n_1\) and \(n_2\) for hydrogen-like atoms is: \[\frac{1}{\lambda} = RZ^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\] where \(R\) is the Rydberg constant (\(1.097 \times 10^7 \text{ m}^{-1}\)), \(Z\) is the atomic number (26 for iron), \(n_1\) is the final state (1), and \(n_2\) is the initial state (2).

Step 3: Plug in the known values

Plug the known values into the Rydberg formula: \[\frac{1}{\lambda} = 1.097 \times 10^7 \times 26^2 \left(\frac{1}{1^2} - \frac{1}{2^2}\right)\]Simplify the expression inside the parentheses: \[\frac{1}{1} - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}\]

Step 4: Calculate the frequency (\(\frac{1}{\lambda}\))

Multiply the values to find \(\frac{1}{\lambda}\): \[\frac{1}{\lambda} = 1.097 \times 10^7 \times 676 \times \frac{3}{4}\]First, calculate \(676 \times \frac{3}{4} = 507\), then \[1.097 \times 10^7 \times 507 = 5.562 \times 10^{9} \text{ m}^{-1}\]

Step 5: Calculate the wavelength \(\lambda\)

Invert the result to calculate the wavelength: \[\lambda = \frac{1}{5.562 \times 10^9} \approx 1.80 \times 10^{-10} \text{ m}\]This is the wavelength of the photon emitted during the transition from \(n=2\) to \(n=1\) in iron.

Key Concepts

Wavelength Calculation

When figuring out the wavelength of light emitted during an electronic transition in an atom like iron, we often resort to the Rydberg formula. This is particularly useful for understanding transitions in hydrogen-like atoms. If the exercise involves an iron atom with an atomic number of 26, the Rydberg formula plays an essential role.

• The Rydberg formula is: \(\frac{1}{\lambda} = RZ^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\).
• Here, \(R\) is the Rydberg constant, \(Z\) is the atomic number, \(n_1\) is the final energy level, and \(n_2\) is the initial energy level.

Plugging in the values specific for the transition from \(n=2\) to \(n=1\) in iron, and with \(R = 1.097 \times 10^7 \text{ m}^{-1}\), the formula helps us arrange everything we need for calculating wavelength.
By simplifying the equation, you get the value for \(\lambda\), which represents the wavelength of the emitted photon.

Photon Emission

Photon emission is a fascinating process that occurs when an electron drops from a higher energy level to a lower one within an atom. In our context of a transition in iron, this concept is crucial for calculating the emitted photon's wavelength.
During the transition from \(n=2\) to \(n=1\), energy is released in the form of a photon. This emitted photon's energy determines the wavelength we calculate using the Rydberg formula.

• The energy difference between the two levels converts into photon energy, influencing the wavelength.
• A larger difference results in a photon with a shorter wavelength, as observed in high atomic number atoms like iron.

Understanding photon emission is key to grasping how transitions lead to different wavelengths across various elements. This principle allows us to explore fields like spectroscopy, crucial for analyzing atomic and molecular structures.

Electronic Transition

Electronic transitions are the shifts electrons make between different energy levels within an atom. Particularly, an electron in iron moving from an energy level of \(n=2\) to \(n=1\) results in photon emission.
The significance of electronic transitions lies not only in energy release but also in the characteristics of the emitted light.

• These changes between levels are quantized, meaning electrons can only occupy certain energy states.
• The atomic number greatly impacts these transitions, especially in elements like iron with high atomic numbers, resulting in more energy release, and thus, light with shorter wavelengths.

Comprehending electronic transitions helps us understand why different elements emit specific colors of light when energized, which is central to many applications in both scientific research and practical technological developments.