Question

According to the quantum-mechanical model for the hydrogen atom, which electron transition produces light with longer wavelength: \(2 p\) to \(1 s\) or \(3 p\) to \(1 s\) ?

Short answer

The transition from \(2p\) to \(1s\) produces light with a longer wavelength compared to the transition from \(3p\) to \(1s\).

Step-by-step solution

Understanding Electron Transitions

In the quantum-mechanical model, when an electron transitions between energy levels, it emits or absorbs a photon with an energy equal to the difference in energy between the two levels. The energy of the photon corresponds to a specific wavelength of light, given by the formula \( E = \frac{hc}{\lambda} \), where \( E \) is the energy, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

Comparing Energy Differences

The energy difference between two energy levels is greater when the electron falls from a higher energy level. In hydrogen, the \(2p\) to \(1s\) transition involves a smaller energy level difference than the \(3p\) to \(1s\) transition.

Determining Wavelength from Energy

Since the energy of the photon is inversely proportional to the wavelength, a larger energy difference results in a shorter wavelength of light, and vice versa. Therefore, the transition involving the smaller energy change, \(2p\) to \(1s\), would result in the emission of light with a longer wavelength compared to the \(3p\) to \(1s\) transition.

Key concepts

Electron Transitions

In the quantum-mechanical model of the atom, an electron transition refers to the electron moving between different energy levels within an atom. These transitions are a core concept in understanding atomic structure and photon emissions. When an electron in a hydrogen atom moves from a higher energy level (like the 3p orbital) to a lower one (like the 1s orbital), energy is released in the form of a photon.

This process is quantified by Niels Bohr's work, which states that each electron orbit corresponds to a certain quantized energy level. An electron can only gain or lose energy in specific amounts to transition between these levels. Therefore, different transitions emit or absorb photons with specific energies, leading to the unique emission spectra that serves as 'fingerprints' for each element.

Photon Energy

Photon energy is directly related to the electromagnetic radiation emitted or absorbed when an electron transitions between energy levels in an atom. This energy is given by the equation \( E = \frac{hc}{\lambda} \), where \( E \) represents the photon energy, \( h \) is Planck's constant (approximately \(6.626 \times 10^{-34} \) Joule-seconds), \( c \) is the speed of light in a vacuum (around \(3 \times 10^8 \) meters per second), and \( \lambda \) is the wavelength of the light.

By understanding the relationship between the energy and wavelength, we can predict the characteristics of the light emitted during electron transitions. A higher photon energy corresponds to a shorter wavelength and typically falls within the ultraviolet or visible range of the electromagnetic spectrum, while lower energy photons have longer wavelengths which may be in the infrared.

Atomic Energy Levels

Atomic energy levels are like the rungs of a ladder within an atom. Each 'rung' corresponds to a specific energy state that an electron can occupy. The quantum-mechanical model of the atom provides us with a map of these energy levels. The levels are labeled with principle quantum numbers (n=1, n=2, n=3, etc.), and sublevels (like s, p, d, f) based on the shape and orientation of the orbitals where electrons are likely to be found.

The difference in energy between these levels determines the energy of the photons absorbed or emitted when an electron transitions from one level to another. In the hydrogen atom, transitions to the first principal energy level (n=1), known as the Lyman series, result in high energy ultraviolet photons, while transitions to the second principal energy level (n=2), known as the Balmer series, produce visible light.

Wavelength of Light

The wavelength of light is an integral aspect to understand when studying the quantum-mechanical model of the hydrogen atom. It is directly proportional to the energy released during electron transitions, as longer wavelengths correspond to lower photon energies and shorter wavelengths correspond to higher energies. Wavelength is commonly measured in nanometers (nm) or angstroms (\(\text{{\r{A}}}\)), and it determines the type of light we observe.

For instance, in the hydrogen atom's emission spectrum, an electron transition from the 2p to the 1s level will emit light in the ultraviolet spectrum, while a transition from 3p to 1s will emit light with a shorter wavelength, which is still usually in the ultraviolet range, but closer to the visible portion of the spectrum. This direct relationship between wavelength and energy allows scientists to calculate the jump in atomic energy levels by simply measuring the wavelength of the emitted light.