https://file.aiquickdraw.com/mathbot/file/1729091146316j6tmnmlg.jpg

The image illustrates an electron transition in a hydrogen atom from energy level \( n = 6 \) to \( n = 2 \). The question asks for the energy change rank and the wavelength (\( \lambda \)) associated with this transition.

### Step 1: Energy change during the transition
In hydrogen, the energy associated with each level is given by the formula:
\[
E_n = -13.6 \, \text{eV} \times \frac{1}{n^2}
\]
The energy difference between two levels \( n_i \) (initial) and \( n_f \) (final) is:
\[
\Delta E = E_f - E_i = 13.6 \, \text{eV} \times \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)
\]
For the transition from \( n = 6 \) to \( n = 2 \):
\[
\Delta E = 13.6 \, \text{eV} \times \left( \frac{1}{2^2} - \frac{1}{6^2} \right)
= 13.6 \, \text{eV} \times \left( \frac{1}{4} - \frac{1}{36} \right)
\]
\[
= 13.6 \, \text{eV} \times \left( \frac{9}{36} - \frac{1}{36} \right) 
= 13.6 \, \text{eV} \times \frac{8}{36}
= 13.6 \times 0.222 = 3.02 \, \text{eV}
\]
So the energy change (\( \Delta E \)) is approximately **3.02 eV**.

### Step 2: Wavelength of the emitted photon
The energy of a photon is related to its wavelength by the equation:
\[
E = \frac{hc}{\lambda}
\]
Where:
- \( h = 6.626 \times 10^{-34} \, \text{Js} \) (Planck's constant)
- \( c = 3 \times 10^8 \, \text{m/s} \) (speed of light)

Rearranging for \( \lambda \):
\[
\lambda = \frac{hc}{E}
\]
Substituting the values for \( E = 3.02 \, \text{eV} \), but converting it to joules (\( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \)):
\[
\lambda = \frac{(6.626 \times 10^{-34}) \times (3 \times 10^8)}{3.02 \times 1.602 \times 10^{-19}}
\]
\[
\lambda \approx 4.11 \times 10^{-7} \, \text{m} = 411 \, \text{nm}
\]
So the wavelength of the emitted photon is approximately **411 nm**.

### Summary:
- **Energy change rank:** 3.02 eV
- **Wavelength (\( \lambda \)):** 411 nm

Would you like further details on these calculations or any clarifications?

Here are five related questions for exploration:
1. What happens if the electron falls to \( n = 1 \) instead of \( n = 2 \)?
2. How does the energy difference vary with different transitions in a hydrogen atom?
3. Why does the energy of an electron become more negative as \( n \) decreases?
4. How do energy levels change for atoms other than hydrogen?
5. What is the physical significance of the wavelength of emitted radiation?

**Tip:** The energy levels in hydrogen are quantized, and the energy difference between higher levels (e.g., \( n = 6 \) to \( n = 5 \)) is much smaller than between lower levels (e.g., \( n = 3 \) to \( n = 2 \)).

A. n = 6 to n = 2, energy change rank = _____ λ = ______

Calculate the energy change and wavelength for the electron transition from n = 6 to n = 2 in a hydrogen atom. Learn the key formulas and concepts such as energy level differences and photon wavelength calculations. Suitable for high school physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation