Question

Assume that a hydrogen atom's electron has been excited to the \(n=5\) level. How many different wavelengths of light can be emitted as this excited atom loses energy?

Short answer

The hydrogen atom's electron at the \(n=5\) level can make \(4\) different transitions to lower energy levels (from \(n=5\) to \(n=4\), \(n=3\), \(n=2\), or \(n=1\)), each emitting a distinct wavelength of light. Therefore, \(4\) different wavelengths of light can be emitted.

Step-by-step solution

Identify the Possible Transitions

Since the electron is at the n=5 level, it can make transitions to any of the lower energy levels (n=4, n=3, n=2, or n=1).

Calculate the Number of Transitions

Each transition from the n=5 level to a lower energy level will correspond to a different wavelength of light. We can represent the transitions as follows: 1. From n=5 to n=4 2. From n=5 to n=3 3. From n=5 to n=2 4. From n=5 to n=1 Each of these transitions will emit a different wavelength of light. Thus, the number of different wavelengths emitted from the n=5 level is 4. In general, for an electron at energy level n, it can make (n-1) distinct transitions as it loses energy, resulting in (n-1) different wavelengths of light. Therefore, the answer is \(4\) different wavelengths of light.

Key concepts

Electron Transitions

When an electron in a hydrogen atom absorbs energy, it gets excited from its ground state to a higher energy level. In our problem, the electron jumps to the fifth energy level, or \(n=5\). From this high-energy state, the electron has several pathways to release energy and return to lower energy states. Each drop from \(n=5\) to a lower energy level is known as a transition.
The electron can transition from \(n=5\) to \(n=4\), \(n=3\), \(n=2\), or \(n=1\). Each of these transitions emits a photon, a particle of light, corresponding to specific energy differences between the energy levels.
As the electron transitions, it emits energy in the form of light, and the amount of energy corresponds to the difference in energy levels. This is why different electron transitions give rise to different wavelengths of light. The emitted photon reflects the energy lost by the electron as it moves to a lower energy level in the atomic structure.

Wavelengths of Light

The wavelength of the light emitted during an electron transition depends on the energy difference between the initial and final states. A larger energy difference results in shorter wavelengths, which corresponds to higher energy light. Conversely, smaller energy differences lead to longer wavelengths, signifying lower energy light.
When an electron in hydrogen transitions from \(n=5\) to, for example, \(n=3\), the emitted light represents this energy change. Each of the possible electron transitions from \(n=5\) to a lower level results in a unique wavelength because each transition differs in the energy gap between initial and final states.
Use the formula \( E = \frac{hc}{\lambda} \) to connect the energy difference \(E\) with wavelength \(\lambda\), where \(h\) is Planck’s constant and \(c\) is the speed of light, to find these wavelengths. This relationship helps us understand why different transitions yield different colors of light in the hydrogen emission spectrum.

Excited States

Electrons in their natural or lowest energy state occupy the ground state. When an electron absorbs energy, it moves to an excited state, which is any energy level higher than the ground state. In this case, an electron in a hydrogen atom is excited to the \(n=5\) level.
The excited state is unstable, and the electron naturally seeks to return to a lower energy state, releasing energy as it descends. This released energy manifests as electromagnetic radiation, including visible light, ultraviolet, or other types of waves based on the energy trajectory.

• The electron's jump to higher levels indicates it is in an excited state.
• The more energy the electron absorbs, the higher the energy level it reaches.
• Once in an excited state, the electron may return to lower levels, emitting light of various wavelengths as it does so.

Understanding electron behavior in excited states is essential for explaining phenomena like atomic emission spectra, which is characteristic for every element.

Quantum Energy Levels

Quantum energy levels, also known as electron shells, are specific energies that electrons can occupy in an atom. These are like steps of a ladder around the nucleus, and electrons jump from one level to another by absorbing or emitting quantized energy – meaning they can only move in defined steps.
In the context of hydrogen, these levels are identified by principal quantum numbers \(n\), such as \(n=1\), \(n=2\), etc. The potential energy of electrons increases as they move to higher levels, meaning the electron in the \(n=5\) level has more energy compared to one in the \(n=1\) ground state.
Quantum levels are fundamental because they explain why elements emit or absorb discrete bands of spectral lines and not a continuous spectrum. The transitions we see correspond to the quantized jumps electrons make between these defined levels, which are unique for each element. By studying these spectrum lines, we can gain valuable information about the energy states and structure of atoms.