How many different photons (photons of different frequencies) can be emitted as a hydrogen atom in the fifth excited state falls back directly or indirectly to the ground state?
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To determine the number of different photons that can be emitted as a hydrogen atom in the fifth excited state falls back to the ground state, we need to consider the energy levels involved.
The energy of an electron in a hydrogen atom can be calculated using the equation:
E = (-13.6 eV)/n^2
where E is the energy, -13.6 eV is the ionization energy of hydrogen, and n is the principal quantum number.
In the ground state of hydrogen (n=1), the energy is -13.6 eV.
In the fifth excited state (n=6), the energy can be calculated as:
E = (-13.6 eV)/6^2 = -13.6 eV/36 = -0.3778 eV
To find the energy difference between the fifth excited state and the ground state, we subtract the energy of the ground state from the energy of the fifth excited state:
E difference = (-0.3778 eV) - (-13.6 eV) = 13.2222 eV
Now, let's find the corresponding wavelength (λ) of this energy difference using the equation:
E = (hc)/λ
where E is the energy difference, h is Planck's constant (6.626 x 10^-34 J*s), c is the speed of light (3.0 x 10^8 m/s), and λ is the wavelength.
First, we need to convert the energy difference from electron volts (eV) to joules (J):
E = 13.2222 eV * 1.602 x 10^-19 J/eV = 2.116 x 10^-18 J
Now, we can rearrange the equation to solve for λ:
λ = (hc)/E
λ = (6.626 x 10^-34 J*s * 3.0 x 10^8 m/s) / (2.116 x 10^-18 J) = 9.848 x 10^-7 m = 984.8 nm
The corresponding wavelength is approximately 984.8 nm.
Since the energy level transitions in hydrogen are quantized, the emitted photons must have energies that match the energy difference between the excited state and the ground state. In this case, the only photon that can be emitted is the one with a wavelength of approximately 984.8 nm.
Therefore, the number of different photons (photons of different frequencies) that can be emitted as a hydrogen atom in the fifth excited state falls back directly or indirectly to the ground state is 1.