Atomic Structure

The problem provides the energy of an electron in an excited state of a hydrogen atom as and asks for the maximum number of possible transitions to lower energy levels.

Concept Used:

The solution involves two key concepts from the Bohr model of the hydrogen atom:

1. Energy Levels of Hydrogen Atom: The energy of an electron in the -th principal energy level (or orbit) is given by the formula: where is the principal quantum number ( ).
2. Number of Spectral Lines: When an electron de-excites from a higher energy level to lower energy levels, it can do so in a series of jumps. The maximum number of distinct spectral lines (or allowed transitions) that can be emitted is given by the formula:

Step-by-Step Solution:

Step 1: Determine the principal quantum number ( ) of the excited state.

We are given the energy of the electron in the excited state, . We can use the energy level formula for the hydrogen atom to find the value of .

Now, we solve for :

To simplify the fraction, we can multiply the numerator and denominator by 100:

Taking the square root of both sides to find :

This means the electron is in the 4th energy level, which is the 3rd excited state.

Step 2: Calculate the maximum number of allowed transitions from this energy level.

Now that we know the electron is in the state, we can use the formula for the maximum number of spectral lines to find the total number of possible transitions to all lower energy levels (n=3, n=2, and n=1).

Substitute into the formula:

Final Computation & Result:

Performing the final calculation:

The maximum number of allowed transitions to lower energy levels is 6.

In a hydrogen-like ion, the energy levels are given by the formula: where: is the energy of the level, is the atomic number, is the principal quantum number (1, 2, 3, etc.).
We are given that the energy difference between the 2nd excitation state (which corresponds to ) and the ground state (which corresponds to ) is 108.8 eV. The energy for the state is: The energy for the state (ground state) is: The energy difference between the 2nd excitation state and the ground state is: We are given that , so:
Thus, the atomic number of the ion is 3.