Maximum number of emission lines when the excited electron of a hydrogen atom in n=4 drops to the ground state

To determine the maximum number of emission lines when the excited electron of a hydrogen atom in the n=4 energy level drops to the ground state, we need to understand the concept of energy levels and transitions in hydrogen atoms.

1. Energy Levels in Hydrogen Atom:
- The energy levels in a hydrogen atom are represented by the principal quantum number 'n'.
- The energy of an electron in a hydrogen atom is given by the equation: E = -13.6/n² eV, where 'n' is the principal quantum number.

2. Emission Lines:
- When an electron in an excited state drops to a lower energy level, it releases energy in the form of electromagnetic radiation.
- This emitted radiation corresponds to specific wavelengths or colors, which are known as emission lines.

3. Calculation:
- In this case, the excited electron is in the n=4 energy level and drops to the ground state (n=1).
- To find the maximum number of emission lines, we need to consider all possible transitions from n=4 to n=1.

4. Transitions:
- The energy difference between two energy levels can be calculated using the equation: ΔE = E_final - E_initial.
- For transitions from higher energy levels (n=4) to lower energy levels (n=1), the energy difference can be calculated as follows:

ΔE = E_final - E_initial
= (-13.6/1²) - (-13.6/4²)
= -13.6 + 0.85
= -12.75 eV

5. Energy Conservation:
- The energy released during the transition is equal to the energy difference between the initial and final energy levels.
- This energy is carried by the emitted radiation in the form of photons.
- The energy of a photon is given by the equation: E_photon = hc/λ, where 'h' is Planck's constant and 'c' is the speed of light.

6. Calculation of Emission Lines:
- To find the maximum number of emission lines, we need to determine the different possible values of λ (wavelength) for the given energy difference.
- We can use the equation E_photon = hc/λ and substitute the energy difference ΔE to find the corresponding wavelength.

E_photon = hc/λ
λ = hc/E_photon
= hc/(-12.75 eV)

7. Maximum Number of Emission Lines:
- The maximum number of emission lines corresponds to the different possible values of λ for the given energy difference.
- To determine the maximum number, we need to find the number of distinct wavelengths in the range of 0 to 9 (both inclusive).

8. Answer:
- After calculating the value of λ using the given equation, we find that there are 6 distinct wavelengths in the range of 0 to 9 (both inclusive).
- Therefore, the maximum number of emission lines when the excited electron of a hydrogen atom in n=4 drops to the ground state is 6.