Consider an electron in the ground state of hydrogen.
(a)What is the least amount of energy needed to excite the ground state electron in Joules?
(b) Is an electron in the ground state of hydrogen capable of absorbing a photon with energy of 1.38e-18 Joules.
dE = 2.180E-18(1/1 - 1/2^2)
Solve for dE for and electron moving from n = 1 to n = 2.
1.38E-18 = 2.180E-18 (1/1 - 1/x^2)
Solve for x. If x can be rounded (reasonably) to a whole number, yes.
Thank you very much.
(a) To determine the least amount of energy needed to excite the ground state electron in hydrogen, we need to first find the energy difference between the ground state and the first excited state.
The energy levels in hydrogen are given by 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.
The ground state corresponds to n = 1, so the energy of the ground state (E1) can be calculated as:
E1 = -13.6 eV / 1^2
Using the conversion factor 1 eV = 1.6e-19 Joules, we can convert the energy to Joules:
E1 = -13.6 eV / 1^2 * 1.6e-19 J/eV
E1 ≈ -2.18e-18 J
Therefore, the least amount of energy needed to excite the ground state electron in hydrogen is approximately 2.18e-18 Joules.
(b) An electron in the ground state of hydrogen can absorb a photon with energy of 1.38e-18 Joules if the energy of the photon matches the energy difference between the ground state and an excited state.
In order to determine if this is possible, we need to calculate the energy difference between the ground state and all the excited states. If the energy of the photon matches any of these energy differences, then the electron can absorb the photon.
Using the same formula as before, the energy levels for the excited states can be calculated:
E2 = -13.6 eV / 2^2
E3 = -13.6 eV / 3^2
E4 = -13.6 eV / 4^2
...
Converting these energies to Joules, we have:
E2 ≈ -0.845e-18 J
E3 ≈ -0.378e-18 J
E4 ≈ -0.212e-18 J
...
Since none of the energy differences match the given energy of 1.38e-18 Joules, an electron in the ground state of hydrogen is not capable of absorbing a photon with energy of 1.38e-18 Joules.
To answer part (a) of the question, you need to understand the concept of energy levels in hydrogen atoms and the ground state of an electron.
(a) The ground state of a hydrogen atom is the lowest possible energy level that an electron can occupy. To excite the electron from the ground state, it needs a certain amount of energy. This energy is known as the ionization energy or the least amount of energy required to remove the electron completely.
The ionization energy of hydrogen is given by the Rydberg formula:
E = -13.6 eV / n^2
Where E is the ionization energy, -13.6 eV is the ionization energy of hydrogen, and n is the principal quantum number of the energy level.
For the ground state, n = 1. Plugging this value into the formula, we get:
E = -13.6 eV / 1^2
E = -13.6 eV
Now we convert this value from electron volts (eV) to joules (J) using the conversion factor 1 eV = 1.602 x 10^-19 J:
E = -13.6 eV x (1.602 x 10^-19 J / 1 eV)
E = -13.6 x (1.602 x 10^-19 J)
E ≈ -2.18 x 10^-18 J
Note that the negative sign indicates the release of energy when transitioning from the excited state to the ground state. So the least amount of energy needed to excite the ground state electron is approximately 2.18 x 10^-18 J.
For part (b) of the question:
(b) To determine if an electron in the ground state of hydrogen can absorb a photon with an energy of 1.38e-18 Joules, we compare the energy of the photon to the ionization energy calculated in part (a).
If the energy of the photon is equal to or greater than the ionization energy, the electron can absorb the photon and get excited to a higher energy level. If the energy of the photon is less than the ionization energy, the electron cannot absorb the photon and will remain in the ground state.
In this case, the energy of the photon is 1.38e-18 Joules, which is greater than the ionization energy of approximately 2.18 x 10^-18 J. Therefore, the electron in the ground state of hydrogen is capable of absorbing a photon with an energy of 1.38e-18 Joules.