Potential Energy of a Hydrogen Atom in the Ground State

To understand the energy levels of a hydrogen atom, we need to consider its potential energy. The potential energy of a hydrogen atom is determined by the attractive force between the electron and the proton in the nucleus.

In the ground state of a hydrogen atom, the electron is in its lowest energy level, closest to the nucleus. We can assign this energy level a value of zero potential energy, as per the given information.

Energy in the First Excited State

The first excited state of a hydrogen atom corresponds to the electron being in a higher energy level compared to the ground state. To determine its energy, we can use the Bohr model of the hydrogen atom, which provides a simplified representation of the electron's energy levels.

According to the Bohr model, the energy of the electron in any energy level is given by the formula:

E = -13.6 eV / n^2

Where E is the energy, n is the principal quantum number, and -13.6 eV is a constant representing the ionization energy of hydrogen.

For the first excited state, n = 2. Plugging this value into the formula, we get:

E = -13.6 eV / 2^2
E = -13.6 eV / 4
E = -3.4 eV

Therefore, the energy of a hydrogen atom in its first excited state is -3.4 eV.

Explanation

The energy levels of a hydrogen atom are quantized, meaning they can only take on certain discrete values. The energy of an electron in the atom depends on its distance from the nucleus, which is determined by the principal quantum number (n). As n increases, the energy level becomes higher, and the electron is further away from the nucleus.

In the ground state, the electron is in the lowest energy level (n = 1), resulting in a potential energy of zero. When the electron absorbs energy, it can transition to a higher energy level, such as the first excited state (n = 2). This transition corresponds to the electron moving further away from the nucleus, resulting in a higher potential energy.

The Bohr model provides a simplified explanation of the energy levels in a hydrogen atom. However, it should be noted that the actual energy levels are more accurately described by quantum mechanics. Nonetheless, the Bohr model serves as a useful tool to understand the concept of energy levels and transitions in atoms.

In Summary

- The potential energy of a hydrogen atom in the ground state is zero.
- The energy in the first excited state is -3.4 eV.
- The energy levels in a hydrogen atom are determined by the principal quantum number (n).
- The Bohr model provides a simplified representation of the energy levels, with the formula E = -13.6 eV / n^2.
- The transition from the ground state to the first excited state involves the electron moving to a higher energy level and gaining potential energy.