I am confused on how to even start this problem...
A ground state H atom absorbs a photon of wavelength 94.91 nm, and its electron attains a higher energy level. The atom then emits two photons: one of wavelength 1281 nm to reach an intermediate level, and a second to reach the ground state.
a) What higher level (n=?) did the electron reach?
b) What intermediate level (n=?) did the electron reach?
c) What was the wavelength of the second photon emitted?
Thanks
duplicate post.
To solve this problem, we need to understand the concept of energy levels in hydrogen atoms and how photon absorption and emission relate to the transition between these energy levels.
The energy levels in a hydrogen atom are given by the formula:
En = -13.6 eV / n^2,
where En is the energy level, n is the principal quantum number, and -13.6 eV is the ionization energy of the hydrogen atom.
a) To find the higher energy level (n=?) that the electron reached, we need to determine the initial and final energy levels. The electron absorbed a photon of wavelength 94.91 nm. We can convert this wavelength to energy using the equation:
E = hc / λ,
where E is the energy, h is the Planck's constant (6.626 x 10^-34 J⋅s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength in meters.
Substituting the values, we get:
E = (6.626 x 10^-34 J⋅s * 3.00 x 10^8 m/s) / (94.91 x 10^-9 m) = 2.201 x 10^-17 J.
Now, we can equate this energy to the difference between the initial and final energy levels:
ΔE = -13.6 eV / n_i^2 - (-13.6 eV / n_f^2).
Let's assume the initial energy level is n_i and the final energy level is n_f:
2.201 x 10^-17 J = (13.6 eV / n_i^2) - (13.6 eV / n_f^2).
We can solve this equation for n_f by rearranging the terms:
2.201 x 10^-17 J * n_i^2 = 13.6 eV - (13.6 eV / n_f^2).
Substituting the known value of n_i as 1 (ground state energy level), we can rearrange the equation:
2.201 x 10^-17 J = 13.6 eV - (13.6 eV / n_f^2) / (1^2).
Now we can solve for n_f:
(13.6 eV / n_f^2) = 13.6 eV - 2.201 x 10^-17 J.
n_f^2 = (13.6 eV) / (13.6 eV - 2.201 x 10^-17 J).
n_f^2 = 1 + (2.201 x 10^-17 J) / (13.6 eV - 2.201 x 10^-17 J).
Taking the square root and solving, we find:
n_f ≈ 2.9985.
So, the electron reached a higher energy level of n=3.
b) To find the intermediate level (n=?) that the electron reached, we use a similar approach. The electron emits a photon with a wavelength of 1281 nm. We can convert this wavelength to energy as we did before:
E = (6.626 x 10^-34 J⋅s * 3.00 x 10^8 m/s) / (1281 x 10^-9 m) = 1.540 x 10^-17 J.
Using the same equation as before, we have:
ΔE = -13.6 eV / n_f^2 - (-13.6 eV / n_i^2).
Substituting the known values of n_f = 3 and n_i = 1, we find:
1.540 x 10^-17 J = (13.6 eV / 3^2) - (13.6 eV / 1^2).
Simplifying and solving, we get:
13.6 eV / 3^2 = 1.540 x 10^-17 J + 13.6 eV.
(13.6 eV/ 3^2) - 13.6 eV = 1.540 x 10^-17 J.
(13.6 eV - 13.6 eV)/9 = 1.540 x 10^-17 J.
0 eV/9 = 1.540 x 10^-17 J.
Since 0 eV is not equal to 1.540 x 10^-17 J, it means there is no intermediate level in this transition.
c) Finally, to find the wavelength of the second photon emitted when the electron reaches the ground state, we can use the same energy conversion equation:
E = hc / λ.
Rearranging, we get:
λ = (hc) / E.
Substituting the values, we have:
λ = (6.626 x 10^-34 J⋅s * 3.00 x 10^8 m/s) / (13.6 eV / 1^2).
Simplifying and converting electron volts to joules (1 eV = 1.602 x 10^-19 J), we find:
λ = (6.626 x 10^-34 J⋅s * 3.00 x 10^8 m/s) / (13.6 eV * (1.602 x 10^-19 J / eV)).
λ = 2.754 x 10^-7 m.
Therefore, the wavelength of the second photon emitted when the electron reaches the ground state is approximately 275.4 nm.
I hope this helps!