Problem 112 - Chemistry The Molecular Science

Question

Criticize these statements. If a statement is incorrect, rewrite it so that it is correct. (a) The energy of a photon is inversely proportional to its frequency. (b) The energy of the hydrogen electron is inversely proportional to its principal quantum number \(n\). (c) Electrons start to enter the fourth energy level as soon as the third level is full. (d) Light emitted by an \(n=4\) to \(n=2\) transition has a lower frequency than that from an \(n=5\) to \(n=2\) transition.

Step-by-Step Solution

Answer

(a) Directly proportional, (b) Inversely proportional to \( n^2 \), (c) Before third is full, (d) Higher frequency.

Step 1: Evaluate Statement (a)

The statement claims the energy of a photon is inversely proportional to its frequency. The energy of a photon is actually given by the formula \( E = h u \), where \( E \) is the energy, \( h \) is Planck's constant, and \( u \) is the frequency. Therefore, the energy is directly proportional to its frequency, not inversely. The correct statement is: "The energy of a photon is directly proportional to its frequency."

Step 2: Evaluate Statement (b)

The statement suggests the energy of a hydrogen electron is inversely proportional to its principal quantum number \(n\). For the hydrogen atom, the energy levels are given by \( E_n = - \frac{13.6 \, eV}{n^2} \). Thus, the energy is inversely proportional to \( n^2 \), not just \( n \). The correct statement is: "The energy of a hydrogen electron is inversely proportional to the square of its principal quantum number \( n \)."

Step 3: Evaluate Statement (c)

The statement implies electrons start to fill the fourth energy level only after the third level is completely filled. However, electron configuration follows the Aufbau principle, which states that lower energy orbitals fill before higher energy ones, but this doesn't mean the third shell must be full before the fourth begins to fill. For instance, 3d orbitals are higher in energy than 4s, so 4s fills before 3d. The corrected statement is: "Electrons start to enter the fourth energy level before the third energy level is full, adhering to the order of orbital energies."

Step 4: Evaluate Statement (d)

The statement compares the frequency of light emitted during transitions from \( n=4 \) to \( n=2 \) versus \( n=5 \) to \( n=2 \). The energy difference between these levels is related directly to the frequency of emitted light by \( \Delta E = h u \). A larger energy change implies a higher frequency. The \( n=5 \) to \( n=2 \) transition involves a larger energy change than \( n=4 \) to \( n=2 \). Therefore, the frequency of light from \( n=5 \) to \( n=2 \) is higher than from \( n=4 \) to \( n=2 \). The correct statement is: "Light emitted from an \( n=4 \) to \( n=2 \) transition has a higher frequency than the light from an \( n=5 \) to \( n=2 \) transition."

Key Concepts

Photon Energy

Photon energy is a fundamental concept in quantum mechanics. It is the energy carried by a photon, which is a particle of light. The energy of a photon is determined by its frequency, and the relationship between the energy \( E \) and frequency \( u \) is given by the formula: \[ E = h u \] where \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \) Joule-seconds. This means that the more frequently a photon vibrates (the higher its frequency), the more energy it possesses.

• Higher frequency means greater energy.
• Lower frequency means less energy.

Understanding photon energy is crucial as it affects how light interacts with matter, influencing phenomena such as photoelectric effect and the behavior of electronic devices.

Electron Configuration

Electron configuration describes how electrons are distributed in an atom's energy levels or orbitals. Electrons fill atomic orbitals based on specific rules, the most notable being the Aufbau principle. This principle states that electrons fill lower-energy orbitals before filling higher-energy ones.

• The sequence is based on increasing energy levels.
• Exceptions occur due to orbital hybridization and differing energy levels (like 4s filling before 3d).

Understanding electron configuration helps predict an element's chemical behavior and reactivity. For example, in transition metals, the d-block electrons play significant roles in forming bonds and determining properties.

Hydrogen Atom Energy Levels

The hydrogen atom serves as a simple model for understanding atomic energy levels, mainly because it consists of a single electron. The energy levels in a hydrogen atom can be described by the formula: \[ E_n = - \frac{13.6 \, \text{eV}}{n^2} \] where \( n \) is the principal quantum number. As \( n \) increases, the energy levels become less negative, indicating that the electron is further from the nucleus and has absorbed energy.

• Energy is quantized, meaning it can only exist in specific levels.
• Larger \( n \) values correspond to higher, less negative energy levels.

This concept is foundational in explaining the energy changes that happen during electronic transitions, which lead to the production of spectral lines.

Spectral Lines

Spectral lines are unique signatures of elements and are created when electrons transition between energy levels in an atom. When an electron moves from a higher energy level to a lower one, it emits energy in the form of light, resulting in a spectral line. The frequency of this light depends on the energy difference between the two levels.

• Larger energy differences produce higher frequency (and energy) light.
• Smaller differences produce lower frequency light.

These spectral lines are characteristic of each element, much like a fingerprint, and are used extensively to identify chemical compositions in stars and other celestial bodies, as well as in laboratory settings. The Balmer series is a well-known group of spectral lines of hydrogen, visible as transitions ending or starting at the \( n=2 \) energy level.