Solution:

Given:
- An electron in a hydrogen atom is in its ground state.
- The electron absorbs 1.5 times the minimum energy required for it to escape from the atom.

To find the wavelength of the emitted electron, we need to consider the energy levels in the hydrogen atom.

Energy Levels in a Hydrogen Atom:
In a hydrogen atom, the electron can exist in different energy levels. The lowest energy level is called the ground state, and higher energy levels are called excited states. The energy levels are quantized, meaning they can only take specific values.

The energy of an electron in the nth energy level of a hydrogen atom can be calculated using the formula:
E_n = -13.6 eV / n^2

Where:
- E_n is the energy of the electron in the nth energy level.
- n is the principal quantum number representing the energy level (n = 1 for the ground state, n = 2 for the first excited state, and so on).
- -13.6 eV is the ionization energy of hydrogen.

Minimum Energy Required to Escape:
The minimum energy required for an electron to escape from the atom, or ionize the atom, is equal to the ionization energy of hydrogen, which is 13.6 eV.
Therefore, the minimum energy required to escape the atom is 13.6 eV.

Energy Absorbed by the Electron:
The electron absorbs 1.5 times the minimum energy required to escape.
Therefore, the energy absorbed by the electron is 1.5 * 13.6 eV = 20.4 eV.

Energy Conservation:
According to the law of conservation of energy, the energy absorbed by the electron must be equal to the energy emitted when the electron transitions to a lower energy level.

Transition to Ground State:
Since the electron is initially in the ground state, it can only transition to lower energy levels. The electron will transition to the first excited state (n = 2), as it is the closest energy level above the ground state.

To calculate the energy difference between the ground state and the first excited state, we use the formula:
ΔE = E_final - E_initial

Where:
- ΔE is the energy difference.
- E_final is the energy of the final state.
- E_initial is the energy of the initial state.

For the transition from the ground state to the first excited state:
ΔE = E_final - E_initial = E_2 - E_1

Substituting the values into the formula:
ΔE = -13.6 eV / (2^2) - (-13.6 eV / (1^2))
ΔE = -13.6 eV / 4 + 13.6 eV
ΔE = -3.4 eV + 13.6 eV
ΔE = 10.2 eV

Wavelength Calculation:
Using the energy-wavelength relationship:
E = hc/λ

Where:
- E is the energy of the photon.
- h is the Planck's constant (6.626 x 10^-34 J·s or 4.136 x 10^-15 eV·s).
- c is the speed of light (3.0 x