To determine the energy of a photon required for an electronic transition from the n=2 state to the n=5 state in a hydrogen atom, we can use the formula for the energy of a photon:
E = ΔE = hc/λ
Where:
- E is the energy of the photon
- ΔE is the change in energy between the initial and final states
- h is Planck's constant (approximately 6.626 x 10^-34 joule-seconds)
- c is the speed of light (approximately 3 x 10^8 meters per second)
- λ is the wavelength of the photon
The energy difference between two energy levels in a hydrogen atom is given by the Rydberg formula:
ΔE = Rh * (1/n_f^2 - 1/n_i^2)
Where:
- ΔE is the change in energy
- Rh is the Rydberg constant (approximately 2.18 x 10^-18 joules)
- n_f is the final energy level (n=5 in this case)
- n_i is the initial energy level (n=2 in this case)
Substituting the values into the Rydberg formula:
ΔE = Rh * (1/5^2 - 1/2^2)
= Rh * (1/25 - 1/4)
= Rh * (4/100 - 25/100)
= Rh * (-21/100)
≈ -0.0218 * Rh
Now, we can substitute this change in energy value into the energy formula for the photon:
E = hc/λ = -0.0218 * Rh
Rearranging the equation to solve for λ:
λ = hc / E
Substituting the values for h, c, and E:
λ = (6.626 x 10^-34 joule-seconds * 3 x 10^8 meters per second) / (-0.0218 * Rh)
Calculating this expression will give us the wavelength of the photon required for the electronic transition.