Calculate the Wavelength of Emitted Light in a Bohr Model Transition

📚 Understanding the Bohr Model

The Bohr model, proposed by Niels Bohr in 1913, provides a simplified yet insightful understanding of atomic structure and electron behavior. It postulates that electrons orbit the nucleus in specific, quantized energy levels. When an electron transitions between these energy levels, it emits or absorbs energy in the form of light. The wavelength of this light is inversely proportional to the energy difference between the levels.

📜 History and Background

Prior to Bohr's model, classical physics failed to explain the discrete spectral lines observed in atomic emission spectra. Bohr combined classical mechanics with quantum concepts introduced by Max Planck and Albert Einstein. His model successfully predicted the Rydberg formula for hydrogen and laid the foundation for quantum mechanics.

✨ Key Principles and Formulas

• ⚛️ Energy Levels: Electrons occupy specific energy levels, denoted by the principal quantum number $n$ (n = 1, 2, 3,...).
• ⚡ Energy Transitions: When an electron transitions from an initial energy level ($n_i$) to a final energy level ($n_f$), energy is either emitted (if $n_i > n_f$) or absorbed (if $n_i < n_f$).
• 📊 Energy Change: The energy change ($\Delta E$) during a transition is given by: $\Delta E = -R_H(\frac{1}{n_f^2} - \frac{1}{n_i^2})$ where $R_H$ is the Rydberg constant ($R_H \approx 2.18 \times 10^{-18} J$).
• 🌈 Wavelength Calculation: The wavelength ($\lambda$) of the emitted light is related to the energy change by: $\Delta E = \frac{hc}{\lambda}$ where $h$ is Planck's constant ($h \approx 6.626 \times 10^{-34} J\cdot s$) and $c$ is the speed of light ($c \approx 3.00 \times 10^8 m/s$). Rearranging to solve for wavelength: $\lambda = \frac{hc}{\Delta E}$

🧮 Step-by-Step Calculation

1. 🔢 Determine Initial and Final Energy Levels: Identify $n_i$ and $n_f$ from the problem.
2. ➗ Calculate the Energy Change: Use the Rydberg formula to find $\Delta E$. Be mindful of the sign; a negative value indicates energy emission.
3. 📏 Calculate the Wavelength: Use the formula $\lambda = \frac{hc}{\Delta E}$ to find the wavelength. Ensure consistent units (Joules for energy, meters for wavelength). Take the absolute value of $\Delta E$ for this calculation.
4. 📈 Convert to Nanometers (Optional): If needed, convert the wavelength from meters to nanometers (1 nm = $10^{-9}$ m).

🧪 Example Problem

Calculate the wavelength of light emitted when an electron transitions from $n = 3$ to $n = 2$ in a hydrogen atom.

1. Identify Energy Levels: $n_i = 3$, $n_f = 2$
2. Calculate Energy Change: $\Delta E = -R_H(\frac{1}{n_f^2} - \frac{1}{n_i^2}) = -2.18 \times 10^{-18} J(\frac{1}{2^2} - \frac{1}{3^2})$ $\Delta E = -2.18 \times 10^{-18} J(\frac{1}{4} - \frac{1}{9}) = -2.18 \times 10^{-18} J(\frac{9-4}{36}) = -2.18 \times 10^{-18} J(\frac{5}{36})$ $\Delta E \approx -3.03 \times 10^{-19} J$
3. Calculate Wavelength: $\lambda = \frac{hc}{|\Delta E|} = \frac{(6.626 \times 10^{-34} J\cdot s)(3.00 \times 10^8 m/s)}{3.03 \times 10^{-19} J}$ $\lambda \approx 6.56 \times 10^{-7} m$
4. Convert to Nanometers: $\lambda \approx 6.56 \times 10^{-7} m \times \frac{10^9 nm}{1 m} = 656 nm$

Therefore, the wavelength of the emitted light is approximately 656 nm (red light).

🌍 Real-world Applications

• 💡 Spectroscopy: Analyzing the wavelengths of light emitted or absorbed by substances to identify their composition and structure.
• 🔆 Lasers: Generating coherent light at specific wavelengths for various applications, including medical procedures, telecommunications, and industrial processes.
• 🔭 Astronomy: Studying the light from stars and galaxies to determine their temperature, composition, and distance.

📝 Practice Quiz

1. ❓ Calculate the wavelength of light emitted when a hydrogen electron transitions from $n=4$ to $n=2$.
2. ❓ An electron in a hydrogen atom emits light with a wavelength of 486 nm. What transition ($n_i$ to $n_f$) caused this emission?
3. ❓ What is the energy (in Joules) of a photon with a wavelength of 500 nm?

(Answers: 1. 486 nm, 2. n=4 to n=2, 3. 3.97 x 10^-19 J)

🎓 Conclusion

Calculating the wavelength of emitted light in a Bohr model transition involves understanding energy levels, energy changes, and the relationship between energy and wavelength. By applying the Rydberg formula and related equations, one can accurately predict the spectral lines observed in atomic emission spectra. This knowledge is crucial in various scientific fields, enabling us to probe the fundamental properties of matter and the universe.