Problem 66 - Chemistry: Principles and Reactions

Question

Consider the following transitions 1\. \(\mathrm{n}=3\) to \(\mathrm{n}=1\) 2\. \(\mathbf{n}=2\) to \(\mathbf{n}=\mathbf{3}\) 3\. \(\mathbf{n}=4\) to \(\mathbf{n}=3\) 4\. \(\mathbf{n}=3\) to \(\mathbf{n}=5\) (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?

Step-by-Step Solution

Answer

Answer: The transition that absorbs the most energy is n=3 to n=5, and the transition that emits the most energy is n=3 to n=1.

Step 1: Transition Energy Direction for Each Pair

First, let's check the direction of energy for each transition: 1. n=3 to n=1: Energy is emitted (electron moves to a lower energy level) 2. n=2 to n=3: Energy is absorbed (electron moves to a higher energy level) 3. n=4 to n=3: Energy is emitted (electron moves to a lower energy level) 4. n=3 to n=5: Energy is absorbed (electron moves to a higher energy level)

Step 2: Energy Absorbed Transitions

Transitions in which energy is absorbed are: 2. n=2 to n=3 4. n=3 to n=5

Step 3: Energy Emitted Transitions

Transitions in which energy is emitted are: 1. n=3 to n=1 3. n=4 to n=3

Step 4: Ground State Transitions

The ground state is the lowest energy level of an atom (n=1). So, transitions involving the ground state are: 1. n=3 to n=1

Step 5: Highest Energy Absorption

To find which transition absorbs the most energy, compare the energy levels in the absorbed transitions: 2. n=2 to n=3: Energy difference of 1 level 4. n=3 to n=5: Energy difference of 2 levels Since transition 4 has a larger energy difference, it absorbs the most energy.

Step 6: Highest Energy Emission

To find which transition emits the most energy, compare the energy levels in the emitted transitions: 1. n=3 to n=1: Energy difference of 2 levels 3. n=4 to n=3: Energy difference of 1 level Since transition 1 has a larger energy difference, it emits the most energy.

Key Concepts

Energy Absorption and Emission

Understanding the process of energy absorption and emission is fundamental in studying atomic physics. It all begins with electrons within an atom occupying different energy levels. When an electron jumps from a lower energy level to a higher one, it must absorb energy from its surroundings; this is known as energy absorption. Conversely, energy emission occurs when an electron falls from a higher energy level to a lower one, releasing energy, typically in the form of a photon.

The amount of energy involved in these transitions is quantized, meaning it can only occur in fixed amounts. This discrete nature is crucial when analyzing atomic spectra, as unique patterns of energy emission or absorption can be observed for each element. This is akin to a fingerprint for the element, allowing scientists to identify the composition of distant stars or to study the properties of various elements on Earth.

Electron Energy Levels

Electrons within an atom are not scattered randomly; they occupy specific electron energy levels. These levels are determined by the principles of quantum mechanics and can be visualized as concentric shells or orbitals around the nucleus of an atom. Each level is denoted by a principal quantum number, usually represented as 'n'.

The lowest energy level, closest to the nucleus, is n=1, and higher levels correspond to larger 'n' values. Electrons tend to occupy the lowest possible energy levels first—a principle called the Aufbau principle. Energy levels are not all spaced equally apart; the difference between them decreases as 'n' increases. When electrons jump between these levels due to energy absorption or emission, the energy involved is precisely the difference in energies of those levels.

Quantum Mechanical Model

The quantum mechanical model of the atom is a robust framework explaining the behavior of electrons in atoms. Rooted in the principles of quantum theory, it proposes that electrons have both particle-like and wave-like characteristics. Unlike the earlier Bohr model that depicted electrons in fixed orbits, the quantum model describes electrons in terms of probability densities that define regions called atomic orbitals.

Inside these orbitals, the exact position of an electron cannot be determined (Heisenberg's uncertainty principle), but the likelihood of finding an electron in a given region can. The energy of an electron is quantized and is a direct result of the shape and size of the orbital it occupies. The model is essential for explaining the complex behaviors of atoms, especially when bonding to form molecules, and has been fundamental in the development of electronic materials and pharmaceuticals.

Ground State Atom

An atom in its most stable, lowest energy configuration is said to be in its ground state. This means that all of its electrons are in the lowest available energy levels, starting with n=1 and filling up subsequent levels in accordance with the Pauli exclusion principle and Hund's rule. An atom in the ground state has the least amount of internal energy and is not excited.

When an atom absorbs energy, for instance when exposed to heat or light, its electrons can transition to higher energy levels, causing the atom to become 'excited'. The ground state of an atom is pivotal in determining the atom's properties such as ionization energy, chemical reactivity, and spectral lines when it returns to this lowest state from an excited state by emitting energy.