What is the energy in joules, requir ed to shift the electron of the hydrogen atomfrom the first Bohr orbit to the fifth Bohr orbit and what is the wavelength of thelight emitted when the electron retur ns to the ground state? The ground stateelectron energy is –2.18 × 10–11 erg
Question
What is the energy in joules, requir ed to shift the electron of the hydrogen atomfrom the first Bohr orbit to the fifth Bohr orbit and what is the wavelength of thelight emitted when the electron retur ns to the ground state? The ground stateelectron energy is –2.18 × 10–11 erg
Solution
The energy levels of the electron in a hydrogen atom are given by the formula:
E_n = -13.6/n^2 eV
where n is the principal quantum number.
- To find the energy required to shift the electron from the first Bohr orbit (n=1) to the fifth Bohr orbit (n=5), we need to calculate the difference in energy between these two states.
E_1 = -13.6/1^2 = -13.6 eV E_5 = -13.6/5^2 = -0.544 eV
The energy required to shift the electron is the difference between these two values:
ΔE = E_5 - E_1 = -0.544 - (-13.6) = 13.056 eV
To convert this energy to joules, we use the conversion factor 1 eV = 1.6 x 10^-19 J:
ΔE = 13.056 eV * 1.6 x 10^-19 J/eV = 2.089 x 10^-18 J
- When the electron returns to the ground state (n=1), it emits a photon with energy equal to the energy difference between the initial and final states. The wavelength of this photon can be found using the formula:
λ = h*c/E
where h is Planck's constant (6.626 x 10^-34 J*s), c is the speed of light (3 x 10^8 m/s), and E is the energy difference.
E = 13.056 eV * 1.6 x 10^-19 J/eV = 2.089 x 10^-18 J
λ = (6.626 x 10^-34 J*s * 3 x 10^8 m/s) / 2.089 x 10^-18 J = 9.5 x 10^-8 m or 95 nm
So, the energy required to shift the electron from the first to the fifth Bohr orbit is 2.089 x 10^-18 J, and the wavelength of the light emitted when the electron returns to the ground state is 95 nm.
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