Question

Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=2\) to \(n=6\), (b) from an orbit of radius \(4.76 \AA\) to one of radius \(0.529 \AA\), (c) from the \(n=6\) to the \(n=9\) state.

Short answer

In the given electronic transitions, energy is absorbed in cases (a) and (c), and energy is emitted in case (b). Specifically, (a) transitions from \(n=2\) to \(n=6\) (absorbed), (b) transitions from a radius of \(4.76 \AA\) (corresponding to \(n=3\)) to \(0.529 \AA\) (corresponding to \(n=1\)) (emitted), and (c) transitions from \(n=6\) to \(n=9\) (absorbed).

Step-by-step solution

Case (a): Transitions from n=2 to n=6

Here, the electron is transitioning from a lower energy level (n=2) to a higher energy level (n=6). When an electron moves to a higher energy level, energy is absorbed. Therefore, energy is absorbed in this case.

Case (b): Transitions from 4.76 Å to 0.529 Å

First, we need to find the principal quantum numbers corresponding to the given radii using Bohr's formula: \[r = a_0 n^2\] For the initial radius (4.76 Å), we have: \[4.76 = 0.529 n^2\] which gives, \(n^2 = 9\), therefore \(n = 3\). For the final radius (0.529 Å), we have: \[0.529 = 0.529 n^2\] which gives, \(n^2 = 1\), therefore \(n = 1\). Now, the electron is transitioning from a higher energy level (n=3) to a lower energy level (n=1). When an electron moves to a lower energy level, energy is emitted. Therefore, energy is emitted in this case.

Case (c): Transitions from n=6 to n=9

Here, the electron is transitioning from a lower energy level (n=6) to a higher energy level (n=9). When an electron moves to a higher energy level, energy is absorbed. Therefore, energy is absorbed in this case. To summarize, in case (a) and case (c), energy is absorbed, while in case (b), energy is emitted during the electronic transitions.

Key concepts

Energy Absorption

In the context of atomic physics, energy absorption refers to the process whereby an electron absorbs energy in order to jump to a higher energy level or orbit within an atom. This concept is often explored through the transitions of electrons in a hydrogen atom, as this atom only contains a single electron, making it simpler to study.

When an electron moves from a lower energy level (or shell) to a higher one, it needs to gain energy equivalent to the difference between the two energy levels. This process is like climbing a ladder where each rung requires more energy.

• For instance, in the hydrogen atom transition from energy level 2 ( =2) to energy level 6 ( =6), energy is absorbed by the electron because it is moving to a higher state where its potential energy needs to match the energy level difference.
• In these processes, you might often consider the wavelengths involved, as the absorbed energy usually corresponds to specific light wavelengths that excite the electron.

Energy Emission

Conversely, energy emission occurs when an electron falls to a lower energy level, releasing the difference in energy, often as light or another form of electromagnetic radiation.

It's a bit like an electron going down the ladder, giving up energy in the process. This released energy might manifest as visible light or other parts of the electromagnetic spectrum. This fundamental concept is essential for understanding how light emissions, such as those of a neon sign, work.

• In scenario (b) from the exercise, the electron transitions from a higher radius (more diffuse orbit) to a smaller radius, indicating it drops from a higher energy state to a lower one, hence, energy is emitted.
• This emitted light forms a spectrum that can be used to identify elements, making it a powerful tool in spectrometry and chemical analysis.

Bohr Model

The Bohr model is a semi-classical approach to the hydrogen atom. Proposed by Niels Bohr, it describes the atom using quantized orbits for the electron. This model allows us to predict the behavior of an electron in terms of its energy levels and orbit radii.

The Bohr model simplifies the atom to a nucleus surrounded by electrons in distinct circular paths, like planets orbiting the sun. Each of these paths corresponds to a different energy level, and energy absorption or emission occurs when electrons jump between these orbits.

• Bohr's formula for radius, \[ r = a_0 n^2 \]is used to relate the principal quantum number \( n \) to the radius of the orbit, where \( a_0 \) is the Bohr radius.
• This model, though only accurate for hydrogen, sets the stage for understanding atomic structure by introducing quantum concepts that revolutionized physics.

Hydrogen Atom Transitions

Transitions within a hydrogen atom are fundamental for understanding more complex atomic and electronic behaviors. These transitions occur when electrons move between different orbits, and each transition corresponds to specific energy changes according to the principles explored by Bohr.

Real-world applications of these transitions include things like spectral lines in astronomy or the working of hydrogen lamps. The simplicity of hydrogen, due to its single electron, makes these transitions a key area for research and teaching about atomic behavior.

• When electrons transition from one level to another, they emit or absorb quantifiable amounts of energy. These energy changes are significant for understanding how atoms interact with light.
• In hydrogen atoms, these transitions align with specific wavelengths which give rise to spectral lines. For example, Balmer or Lyman series lines result from such electron transitions.