Question

Consider the following transitions 1\. \(n=3\) to \(\mathbf{n}=1\) 2\. \(\mathrm{n}=2\) to \(\mathrm{n}=3\) 3\. \(\mathrm{n}=4\) to \(\mathrm{n}=3\) 4\. \(\mathrm{n}=3\) to \(\mathrm{n}=5\) (a) For which of the transitions is energy absorbed? (b) For which of the transitions is energy emitted? (c) Which transitions involve the ground state? (d) Which transition absorbs the most energy? (e) Which transition emits the most energy?

Short answer

Question: Identify the absorption and emission of energy during electron transitions, the ground state involved, and the transitions with the most energy absorbed and emitted. Answer: (a) Transitions with energy absorbed: 2 and 4; (b) Transitions with energy emitted: 1 and 3; (c) Transitions involving the ground state: 1; (d) Transition that absorbs the most energy: 4 (n=3 to n=5); (e) Transition that emits the most energy: 1 (n=3 to n=1).

Step-by-step solution

Determine energy absorbed or emitted during transitions

Electrons can absorb energy when they jump from a lower energy level (smaller n value) to a higher energy level (larger n value) and emit energy when they return to a lower energy level. Let us examine each of the given transitions: 1. n=3 to n=1: Energy is emitted (higher to lower energy level) 2. n=2 to n=3: Energy is absorbed (lower to higher energy level) 3. n=4 to n=3: Energy is emitted (higher to lower energy level) 4. n=3 to n=5: Energy is absorbed (lower to higher energy level) (a) Transitions with energy absorbed: 2 and 4 (b) Transitions with energy emitted: 1 and 3

Identify transitions involving ground state

The ground state is the lowest energy level in which an electron can reside, which corresponds to n=1. Let us identify which transitions involve the ground state: 1. n=3 to n=1: Involves ground state 2. n=2 to n=3: Does not involve ground state 3. n=4 to n=3: Does not involve ground state 4. n=3 to n=5: Does not involve ground state (c) Transitions involving the ground state: 1

Determine transition with maximum energy absorption and emission

The energy difference between levels increases as the energy level difference increases. In other words, the higher the difference between n values, the more energy will be absorbed or emitted during the transition. To find the transition with the most energy absorbed and emitted, compare the differences between n values for each of the transitions: 1. Energy difference = |3-1| = 2 2. Energy difference = |3-2| = 1 3. Energy difference = |4-3| = 1 4. Energy difference = |5-3| = 2 (d) Transition that absorbs the most energy: 4 (n=3 to n=5) with energy difference of 2 (e) Transition that emits the most energy: 1 (n=3 to n=1) with energy difference of 2

Key concepts

Energy Absorption and Emission

The principles of energy absorption and emission are fundamental to understanding electron behavior in atoms. When electrons within an atom receive energy, they can jump from a lower energy state to a higher one, a process known as absorption. Conversely, emission occurs when an electron falls back to a lower energy state, releasing energy in the process.

The energy that is either absorbed or emitted during these electron transitions is often quantified in the form of a photon—a particle representing a quantum of electromagnetic radiation. The energy of such photons fits the equation \[ E = h u \], where \( E \) is the energy of the photon, \( h \) is the Planck constant, and \( u \) is the frequency of the corresponding electromagnetic wave.

Using this equation, and knowing the specific energy states involved, one can determine the characteristics of the absorbed or emitted radiation. These characteristics also correspond with color when considering visible light. For example, when an electron drops from the third energy level to the first, energy is emitted, and this can sometimes be observed as light within the visible spectrum.

Electron Transitions

Electron transitions refer to the movement of an electron between different energy levels within an atom. The energy levels, known as orbits or shells, are indexed by principal quantum numbers, often represented by \( n \). These numbers indicate the relative size and energy of each shell, with the lowest energy state being \( n=1 \), the ground state.

An electron transition from a lower to a higher energy level—e.g., from \( n=2 \) to \( n=3 \)—requires an input of energy, usually in the form of a photon. This transition is described as excitation. Conversely, when an electron falls to a lower energy level, the excess energy is released. The emitted energy is also often in the form of a photon.

Factors Influencing Electron Transition Energies

• Difference in Energy Levels: Larger differences between the initial and final energy levels lead to higher energy photon emission or absorption.
• Atomic Structure: The energy difference between levels can depend greatly on the atomic number and the electron configuration of the particular element.
• External Fields: The presence of external electric or magnetic fields can alter the energy levels themselves and thus affect the energies of transitions.

The study of these transitions, and the energies involved, is crucial for techniques such as spectroscopy, which aids in identifying elements and understanding chemical bonding.

Ground State of Electrons

The ground state of an electron in an atom is its lowest energy state. This is typically represented by the first energy level, with \( n=1 \), where the electron is closest to the nucleus. Being in the ground state means that the electron cannot give up energy to fall to a lower level—there are no lower levels to go.

In the context of a textbook problem, questions about the ground state often relate to identifying situations when an electron is moving to or from this lowest level. For example, a transition from \( n=3 \) to \( n=1 \) involves the ground state and results in emission, as the electron loses energy and falls to this most stable level.

The ground state is significant because it establishes a baseline for comparison of energy levels. All other states are excited states where the electron has more energy than it does in the ground state. The energy gap between the ground state and any excited state determines the specific amount of energy absorbed or emitted during a quantum transition.