Question

Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? (a) from $n=3$ to $n=2$, (b) from an orbit of radius $0.846\mathrm{nm}$ to one of radius 0.212 $\mathrm{nm},(\mathbf{c})$ an electron adds to the ${\mathrm{H}}^{+}$ ion and ends up in the $n=2$ shell?

Short answer

In all the given transitions, energy is emitted. For transition (a) from $n=3$ to $n=2$, energy is emitted as the electron moves to a lower energy level. For transition (b) from an orbit of radius 0.846nm to one of radius 0.212nm, energy is emitted due to a move from a higher to lower energy level. Lastly, for transition (c) when an electron adds to the ${\mathrm{H}}^{+}$ ion and ends up in the $n=2$ shell, energy is emitted as the system moves from an ionic state to an atomic state.

Step-by-step solution

Analyze the transition (a) from $n=3$ to $n=2$

The electron is transitioning from a higher energy level ($n=3$) to a lower energy level ($n=2$). According to the Bohr's atomic model, when an electron comes down to a smaller shell, energy is emitted. So, in this transition, energy is emitted.

Analyze the transition (b) from an orbit of radius 0.846nm to one of radius 0.212nm

Here, it is not directly given which energy level the electron is transitioning from or to. However, we know that according to Bohr's model, radius of the $n^{th}$ orbit, $r_n$ is directly proportional to $n^2$ (i.e. $r_n\propto n^2$). Therefore, a transition from a larger orbit (i.e larger $n$) to a smaller orbit corresponds to a transition from a higher energy level to a lower energy level. Therefore, in this transition, similar to the previous one, energy is emitted.

Analyze the transition (c) an electron adds to the ${\mathrm{H}}^{+}$ ion and ends up in the $n=2$ shell?

When an electron is added to a ${\mathrm{H}}^{+}$ ion, it changes from the ionic state to the atomic state. This transition corresponds to a decrease in energy level (from ionization level to $n=2$), and thus, energy will be emitted in this case too. Therefore, for all the given transitions, energy is emitted. Please note that the actual amount of energy emitted or absorbed in such transitions can be calculated using the formula for the energy differences in the Bohr's atomic model, but since this exercise only asks for whether energy is emitted or absorbed, specific calculations are not required.

Key concepts

Energy Emission and Absorption

Energy emission and absorption are fundamental processes in atomic physics, particularly for understanding how electrons move between different energy levels in an atom. Whenever an electron transitions from a higher energy level to a lower one, energy is emitted in the form of photons. Conversely, energy is absorbed when an electron moves from a lower energy level to a higher one.
For instance, in the hydrogen atom, if an electron moves down from the third energy level ( =3 o n=2 ransition), it releases energy. This release of energy happens as a photon is emitted, resulting in the electron losing energy to maintain conservation of energy within the atom.
Remember:

• An electron dropping to a lower energy level => Energy emitted
• An electron jumping to a higher energy level => Energy absorbed

Overall, understanding whether energy is emitted or absorbed during a transition is crucial for explaining phenomena such as the color of light emitted by gases, related to specific electron transitions within atoms.

Bohr's Atomic Model

Bohr's Atomic Model is a pivotal concept in explaining atomic structure and electron behavior in hydrogen-like atoms. Proposed by Niels Bohr in 1913, this model offers a simplified view of the atom as a small nucleus surrounded by electrons in quantized circular orbits.
Key features of Bohr's model include:

• Electrons orbit the nucleus in distinct paths or shells.
• Each orbit corresponds to a specific energy level, represented by a quantum number, .

In Bohr's model, energy levels are quantized, meaning electrons can only exist in these discrete orbits. When an electron transitions between orbits, it does so by absorbing or emitting energy equal to the difference between the energy levels.
Bohr's model was particularly successful in explaining the hydrogen spectrum, accurately predicting the wavelengths of the spectral lines of hydrogen with its quantized energy levels approach.

Electron Energy Levels

Electron energy levels, often visualized as shells in an atom, are the defined regions around the nucleus where electrons are likely to be found. These levels determine the potential energy of electrons based on their position relative to the nucleus.
The principal quantum number, , designates each energy level, with higher values indicating levels farther from the nucleus and often having higher energy.

• =1, 2, 3,... ranges denote different energy levels or shells.

Electrons filling these levels follow the Pauli Exclusion Principle, ensuring that no two electrons have the same set of quantum numbers within an atom.
In a hydrogen atom, the electron transitions between these levels are quantized and direct, contributing to this atom's simple spectroscopic signature. Recognizing how electrons fill and transition between these energy levels is essential for understanding atomic emission and absorption spectra, chemical bonding, and periodic table configurations.