Question

Suppose hydrogen atoms absorb energy so that electrons are excited to the \(n=7\) energy level. Electrons then undergo these transitions, among others: (a) \(n=7 \rightarrow\) \(n=1 ;\) (b) \(n=7 \rightarrow n=6 ;\) and \((\text { c) } n=2 \rightarrow n=1\). Which of these transitions produces a photon with (i) the smallest energy, (ii) the highest frequency, and (iii) the shortest wavelength?

Short answer

(i) Transition b; (ii) Transition a; (iii) Transition a.

Step-by-step solution

Understand Energy Level Transitions

In hydrogen, electrons transition between different energy levels, releasing or absorbing a photon in the process. The energy difference between these levels determines the photon's characteristics. Larger energy differences yield higher energy photons, which correlate with higher frequency and shorter wavelength.

Apply the Energy Transition Formula

The energy emitted or absorbed during an electron transition can be calculated using the formula \( E = h u = \frac{hc}{\lambda} = (-13.6 \text{ eV}) \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \), where \( n_i \) and \( n_f \) are the initial and final energy levels, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength. Consider this for each transition.

Determine Energy for Transition a) n=7 to n=1

Calculate the energy for the transition from \( n=7 \) to \( n=1 \). The energy difference can be determined as follows: \( E_1 = (-13.6) \left(\frac{1}{1^2} - \frac{1}{7^2}\right) \text{ eV} \approx 13.05 \text{ eV} \). This is a significant energy change.

Determine Energy for Transition b) n=7 to n=6

Calculate the energy for the transition from \( n=7 \) to \( n=6 \). The energy change is: \( E_2 = (-13.6) \left(\frac{1}{6^2} - \frac{1}{7^2}\right) \text{ eV} \approx 0.19 \text{ eV} \). This is a smaller energy change compared to transition (a).

Determine Energy for Transition c) n=2 to n=1

Calculate the energy for the transition from \( n=2 \) to \( n=1 \). The energy change is computed as: \( E_3 = (-13.6) \left(\frac{1}{1^2} - \frac{1}{2^2}\right) \text{ eV} \approx 10.2 \text{ eV} \). This transition has a high energy change but less than transition (a).

Ranking Based on Energy, Frequency, and Wavelength

The transition with the smallest energy is transition (b), which results in a photon with the least energy. The transition with the highest energy, and thus the highest frequency and shortest wavelength (because \( u \propto E \) and \( \lambda \propto \frac{1}{E} \)), is transition (a).

Key concepts

Photon Emission

Photon emission occurs when an electron transitions from a higher energy level to a lower one in an atom. During this process, the electron loses energy, which is released in the form of a photon. This is a fundamental concept in physics since it explains the behavior of atoms when they interact with light.

• It involves the release of energy that was previously absorbed by the electron.
• The energy of the emitted photon is precisely the difference between the two energy levels involved in the transition.
• The characteristics of the photon, such as its frequency and wavelength, depend directly on this energy difference.

Hence, larger energy level transitions result in photons with higher energy, greater frequency, and shorter wavelength. Understanding these transitions is crucial for interpreting the hydrogen atom spectrum and other atomic spectral lines.

Electron Excitation

Electron excitation refers to the process where an electron in an atom absorbs energy and moves from a lower energy level to a higher one. This absorption of energy can occur in various ways, such as through collisions or by absorbing a photon.

• Once excited, electrons can remain in the higher energy state for a short period before returning to a lower energy level.
• During de-excitation, the absorbed energy is released as a photon, a core principle in photon emission.
• The specific energy levels electrons occupy are determined by the quantum mechanical model of atoms.

This concept is vital for understanding how atoms interact with energy and form the basis of absorption and emission spectra in different elements.

Hydrogen Atom Spectrum

The hydrogen atom spectrum is a key tool for understanding atomic structure and quantum mechanics. It consists of a series of spectral lines, each corresponding to a photon emitted or absorbed by a hydrogen atom during electron transitions.

• The Lyman series involves transitions from higher energy levels to the ground state ( =1), producing ultraviolet lines.
• The Balmer series includes transitions into the second energy level ( =2) and produces visible light.
• Other series, like the Paschen, Brackett, and Pfund, involve transitions to even higher energy levels, resulting in infrared emissions.

These spectral lines provide insight into the energy levels of electrons within hydrogen atoms and confirm the predictions made by quantum mechanics.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles on an atomic and subatomic scale. It provides the framework for understanding electron excitation and photon emission in atoms.

• It describes how electron energy levels are quantized, meaning electrons can only exist in specific energy levels rather than in continuous ranges.
• The mathematical models of quantum mechanics, such as the Schrödinger equation, predict the likely positions of electrons around an atomic nucleus.
• It challenges classical concepts, introducing principles like wave-particle duality and uncertainly in the electrons' position and momentum.

Quantum mechanics is crucial for explaining atomic behavior and the resulting phenomena, such as the hydrogen atom spectrum, providing a detailed comprehension of electron transitions and energy levels.