Question

An excited hydrogen atom with an electron in the \(n=5\) state emits light having a frequency of \(6.90 \times 10^{14} \mathrm{s}^{-1} .\) Determine the principal quantum level for the final state in this electronic transition.

Short answer

The final principal quantum level in this electronic transition is \(n_f = 4\).

Step-by-step solution

Rydberg Formula

The Rydberg formula for the hydrogen atom is given by: \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) \] where \(R_H\) is the Rydberg constant, \(n_i\) and \(n_f\) represent the initial and final states, respectively. To solve for the final state n_f, we will first convert the given frequency to the wavelength and then use the formula to find \(n_f\). 2. Calculate the wavelength from the given frequency

Wavelength Calculation

The relationship between frequency (\(v\)), wavelength (\(\lambda\)) and the speed of light (\(c\)) is given as: \[ \lambda = \frac{c}{v} \] Given the frequency \(v = 6.90 \times 10^{14} \mathrm{s}^{-1}\) and the speed of light \(c = 3.00 \times 10^8 \mathrm{m}\cdot\mathrm{s}^{-1}\), the wavelength can be calculated as: \[ \lambda = \frac{3.00 \times 10^8 \mathrm{m}\cdot\mathrm{s}^{-1}}{6.90 \times 10^{14} \mathrm{s}^{-1}} = 4.35 \times 10^{-7} \mathrm{m} \] 3. Substitute values in the Rydberg formula.

Substituting Values

Our known values are \(\lambda = 4.35 \times 10^{-7} \mathrm{m}\), the Rydberg constant \(R_H = 1.097 \times 10^7 \mathrm{m}^{-1}\), and the initial quantum level \(n_i=5\). Substitute these values into the Rydberg formula: \[ \frac{1}{4.35 \times 10^{-7} \mathrm{m}} = 1.097 \times 10^7 \mathrm{m}^{-1} \left( \frac{1}{5^2} - \frac{1}{n_f^2} \right) \] 4. Solve for the final quantum level

Solving for Final Quantum Level

Re-write the formula to isolate \(n_f\): \[ \frac{1}{n_f^2} = \frac{1}{5^2} - \frac{1}{4.35 \times 10^{-7} \mathrm{m} \times 1.097 \times 10^7 \mathrm{m}^{-1}} \] Calculate \(\frac{1}{n_f^2}\): \[ \frac{1}{n_f^2} = \frac{1}{25} - \frac{1}{4.761 \times 10^{-7}} \] Solve for \(n_f\): \[ n_f^2 = \frac{1}{(\frac{1}{25} - \frac{1}{4.761 \times 10^{-7}})^{-1}} \] \[ n_f^2 \approx 15.97 \] \[ n_f = \sqrt{15.97} \approx 3.99 \] 5. Conclusion

Final Answer

As the quantum levels have to be whole numbers and the square root found is closer to 4, the final principal quantum level in this electronic transition is \(n_f = 4\).

Key concepts

Rydberg formula

The Rydberg formula is essential for understanding how light interacts with atoms, particularly hydrogen. It predicts the wavelengths of the spectral lines of hydrogen, allowing us to determine the energy changes as electrons move between energy levels.
The formula is \[\frac{1}{\lambda} = R_H \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right)\]In this formula,

• \(\lambda\) represents the wavelength of the light emitted during the transition.
• \(R_H\) is the Rydberg constant for hydrogen, known to be approximately \(1.097 \times 10^7\ \mathrm{m}^{-1}\).
• \(n_i\) and \(n_f\) are the initial and final principal quantum numbers, respectively.

By using this formula, we can calculate the wavelength of light or, conversely, determine which energy levels an electron transitions between when it emits light.

principal quantum number

Each electron in an atom is defined by a set of quantum numbers, among which the principal quantum number is paramount. The principal quantum number, denoted as \(n\), determines the energy level an electron inhabits in an atom.

• \(n\) is always a positive integer (1, 2, 3, ...).
• The value of \(n\) indicates the average distance of the electron from the nucleus.
• Energy levels increase as \(n\) increases, meaning electrons at higher \(n\) values have more energy.

In our problem, we began with an electron in the \(n=5\) state. By calculating the wavelength and using the Rydberg formula, we deduced it ended in the \(n=4\) state. Understanding \(n\) is critical in quantum mechanics, as it provides a foundation for predicting electron behavior in atoms.

frequency to wavelength conversion

Frequency and wavelength are intertwined characteristics of waves. In our exercise, we needed to convert frequency into wavelength to use it in the Rydberg formula.
The relationship between the frequency \(v\), wavelength \(\lambda\), and the speed of light \(c\) is given by the formula:\[\lambda = \frac{c}{v}\]where

• \(c\) is the speed of light, approximately \(3.00 \times 10^8\ \mathrm{m/s}\).
• \(v\) is the frequency of the light, given as \(6.90 \times 10^{14}\ \mathrm{s^{-1}}\) in the problem.

By using this relation, we could determine that the wavelength \(\lambda\) is \(4.35 \times 10^{-7}\ \mathrm{m}\). This conversion is crucial, as the Rydberg formula requires wavelength as input to solve for the principal quantum number.

electronic transition

An electronic transition occurs when an electron in an atom moves between energy levels. This transition involves a change in the electron's energy, resulting in the emission or absorption of light.

• During emission, an electron moves from a higher energy level (higher \(n\)) to a lower one.
• The difference in energy between these levels determines the frequency or wavelength of the emitted light.

In the given problem, an electron transitioned from the \(n=5\) state to a lower energy level. By using spectral data, like the frequency of emitted light, we calculated the final energy level as \(n=4\). Understanding electronic transitions is key to interpreting atomic spectra and the behavior of electrons under excitation.

quantum mechanics

Quantum mechanics is the branch of physics that deals with the behavior of very small particles, like electrons, under atomic dimensions. It regards particles as having both wave-like and particle-like characteristics. A core principle of quantum mechanics is quantization:

• Energy levels in a hydrogen atom are quantized, meaning electrons can only exist in specific energy states determined by their quantum numbers.
• The transitions between these states release or absorb energy in discrete amounts, known as quanta.

This framework illustrates why only certain wavelengths are emitted by hydrogen atoms. For students, capturing the concepts of quantum mechanics involves understanding these intricate behaviors and mathematical formulations, such as the Rydberg formula, that describe them. Quantum mechanics provides a detailed explanation of the fundamental interactions governing atomic and subatomic systems.