To solve this, we can use the Rydberg formula to determine the change in energy when an electron in a hydrogen atom transitions from a higher energy level (excited state) back to the ground state.

Step 1: Calculate the Change in Energy ($\Delta E$)

The energy difference between two energy levels in a hydrogen atom is given by:

$\Delta E = -13.6 \, \text{eV} \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right)$

where:

• $n_{\text{initial}} = 4$ (the excited state)
• $n_{\text{final}} = 1$ (the ground state)
• $-13.6 \, \text{eV}$ is the energy of the electron in the ground state of the hydrogen atom.

Substitute the values:

$\Delta E = -13.6 \, \text{eV} \left( \frac{1}{1^2} - \frac{1}{4^2} \right)$

$\Delta E = -13.6 \, \text{eV} \left( 1 - \frac{1}{16} \right)$

$\Delta E = -13.6 \, \text{eV} \times \frac{15}{16}$

$\Delta E = -12.75 \, \text{eV}$

The negative sign indicates energy is released as the electron drops to a lower energy level.

Step 2: Calculate the Frequency of the Emitted Light

The relationship between the energy of a photon and its frequency is given by:

$\Delta E = h \nu$

where:

• $h$ is Planck's constant ($h = 6.626 \times 10^{-34} \, \text{J} \cdot \text{s}$),
• $\nu$ is the frequency of the emitted light.

First, convert $\Delta E$ from eV to joules:

$\Delta E = -12.75 \, \text{eV} \times 1.602 \times 10^{-19} \, \text{J/eV} = -2.043 \times 10^{-18} \, \text{J}$

Now, solve for the frequency $\nu$:

$\nu = \frac{\Delta E}{h}$

$\nu = \frac{2.043 \times 10^{-18} \, \text{J}}{6.626 \times 10^{-34} \, \text{J} \cdot \text{s}}$

$\nu = 3.08 \times 10^{15} \, \text{Hz}$

Final Answers:

• Change in energy: $12.75 \, \text{eV}$
• Frequency of emitted light: $3.08 \times 10^{15} \, \text{Hz}$

Would you like a breakdown of the steps or have any questions about the calculations?

---

Additional Related Questions:

1. What is the wavelength of the emitted photon in this transition?
2. How much energy is absorbed when an electron transitions from $n=1$ to $n=4$?
3. What are the energy values for the $n=2$ and $n=3$ levels in a hydrogen atom?
4. How would the frequency change if the electron only fell to $n=2$ before reaching $n=1$?
5. What is the difference in energy between $n=4$ to $n=3$ compared to $n=4$ to $n=1$?

Tip:

Remember that higher frequency light corresponds to shorter wavelengths and higher energy.