Calculate the energy of one mole photons (in J/mol) emitted by a hydrogen atom when its electron drops from the n = 6 state to the n = 3 state.

To calculate the energy of one mole of photons emitted by a hydrogen atom when an electron drops from the \( n = 6 \) to the \( n = 3 \) energy level, we will use the following steps:

1. **Energy difference between levels**: 
   The energy levels of a hydrogen atom are given by the formula:
   \[
   E_n = - \frac{13.6 \, \text{eV}}{n^2}
   \]
   where \( n \) is the principal quantum number.

2. **Calculate the energy of the \( n = 6 \) and \( n = 3 \) levels**:
   \[
   E_6 = - \frac{13.6}{6^2} = - \frac{13.6}{36} \, \text{eV} = -0.378 \, \text{eV}
   \]
   \[
   E_3 = - \frac{13.6}{3^2} = - \frac{13.6}{9} \, \text{eV} = -1.511 \, \text{eV}
   \]

3. **Energy difference (in eV)**: 
   The energy of the emitted photon corresponds to the difference between these two energy levels:
   \[
   \Delta E = E_3 - E_6 = -1.511 \, \text{eV} - (-0.378 \, \text{eV}) = -1.133 \, \text{eV}
   \]
   So, the photon emitted has an energy of 1.133 eV (positive value).

4. **Convert to joules**:
   Since 1 electronvolt (eV) = \( 1.602 \times 10^{-19} \, \text{J} \), the energy of one photon is:
   \[
   \Delta E = 1.133 \, \text{eV} \times 1.602 \times 10^{-19} \, \text{J/eV} = 1.815 \times 10^{-19} \, \text{J}
   \]

5. **Energy of one mole of photons**:
   One mole of photons contains \( 6.022 \times 10^{23} \) photons (Avogadro's number), so the energy of one mole of photons is:
   \[
   E_{\text{mole}} = 1.815 \times 10^{-19} \, \text{J} \times 6.022 \times 10^{23} \, \text{photons/mol} = 1.093 \times 10^5 \, \text{J/mol}
   \]

Thus, the energy of one mole of photons emitted when the electron drops from \( n = 6 \) to \( n = 3 \) is approximately \( 1.093 \times 10^5 \, \text{J/mol} \).

Would you like more details or have any questions?

### Relative Questions:
1. How do you calculate the energy for other electron transitions in hydrogen?
2. What is the wavelength of the photon emitted during this transition?
3. How does the Rydberg formula relate to these energy calculations?
4. What is the significance of the energy levels in the hydrogen atom?
5. How do electron transitions affect the electromagnetic spectrum?

### Tip:
Always use Avogadro’s number when converting from the energy of a single photon to the energy of one mole of photons.

Learn how to calculate the energy of one mole of photons emitted by a hydrogen atom during an electron transition from n=6 to n=3. This solution covers quantum mechanics, energy levels, and unit conversions, providing a detailed step-by-step process for high school and college 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