Question

Are the following statements true for the hydrogen atom only, true for all atoms, or not true for any atoms? a. The principal quantum number completely determines the energy of a given electron. b. The angular momentum quantum number, \(\ell\), determines the shapes of the atomic orbitals. c. The magnetic quantum number, \(m_{\ell}\), determines the direction that the atomic orbitals point in space.

Short answer

a. True for hydrogen atom only. b. True for all atoms. c. True for all atoms.

Step-by-step solution

a. Principal quantum number and electron energy

For hydrogen atom, the energy of a given electron depends only on the principal quantum number (n). This is due to the fact that in a hydrogen atom, there's only one electron and the energy levels are directly connected to the principal quantum number. However, for multi-electron atoms, the energy of an electron depends not only on the principal quantum number (n) but also on the angular momentum quantum number (ℓ). This is because in multi-electron atoms, electron-electron interactions and additional forces come into play, which affects the energy levels. Therefore, the statement "The principal quantum number completely determines the energy of a given electron" is true for hydrogen atom only.

b. Angular momentum quantum number and atomic orbital shapes

The angular momentum quantum number (ℓ) determines the shapes of the atomic orbitals, regardless of the type of atom (hydrogen or multi-electron atoms). This is because (ℓ) takes values from 0 to n-1 and describes the type of orbital, such as s, p, d, and f orbitals, which have distinct shapes. Therefore, the statement "The angular momentum quantum number, ℓ, determines the shapes of the atomic orbitals" is true for all atoms.

c. Magnetic quantum number and atomic orbital direction

The magnetic quantum number (\(m_{\ell}\)) determines the orientation of the atomic orbitals in space. It takes values from -ℓ to +ℓ, including 0, and corresponds to the number of possible orientations of the orbitals. This property is consistent for all atoms, regardless of whether they are hydrogen or multi-electron atoms. Therefore, the statement "The magnetic quantum number, \(m_{\ell}\), determines the direction that the atomic orbitals point in space" is true for all atoms.

Key concepts

Hydrogen Atom

In the world of chemistry and physics, the hydrogen atom is often used as a model to understand larger and more complex systems. This is because the hydrogen atom is the simplest atom, consisting of just one proton and one electron. The simplicity of the hydrogen atom allows scientists to study fundamental principles without the interference of additional electrons or more complex interactions.

One significant aspect of the hydrogen atom is that its electron's energy is solely dependent on the principal quantum number, denoted by \(n\). This means that the energy levels of a hydrogen atom can be uniquely determined just by knowing the value of \(n\). In hydrogen, there are no additional electrons to create interactions that could alter these energy levels. Thus, the study of hydrogen atoms provides valuable insights into the principles of quantum mechanics that apply broadly to other atoms as well.

Atomic Orbitals

Atomic orbitals are regions around an atom's nucleus where electrons are likely to be found. They are defined by quantum numbers, which describe the energy, shape, and orientation of these regions. The angular momentum quantum number, represented as \(\ell\), is crucial in determining the shape of an orbital.

Orbitals come in different shapes such as:

• s orbitals: spherical in shape, these are found in all principal energy levels.
• p orbitals: dumbbell-shaped, appear in levels starting from \(n=2\).
• d orbitals: more complex shapes, found in levels starting from \(n=3\).
• f orbitals: even more complex, starting from \(n=4\).

For any atom, whether it's hydrogen or a multi-electron system, the nature and shape of these orbitals are determined by \(\ell\). These orbitals provide a visual and mathematical model for understanding how electrons occupy space around a nucleus.

Electron Energy

The energy of an electron within an atom is a fundamental concept in quantum chemistry. For hydrogen atoms, the electron energy is neatly defined by the principal quantum number \(n\). Each increase in \(n\) corresponds to a higher energy level where the electron can be found.

However, when it comes to multi-electron atoms, the picture becomes more complicated. In these atoms, the electron energy is affected not only by \(n\) but also by the angular momentum quantum number \(\ell\). This is because the presence of multiple electrons leads to additional interactions, such as electron-electron repulsions, which influence the energy states. Understanding these interactions is key to mastering atomic theory and predicting atom behavior.

Multi-Electron Atoms

Unlike hydrogen, multi-electron atoms contain more than one electron, leading to complexities in their atomic structure. These complexities arise from the interactions between multiple electrons, which are not present in a hydrogen atom.

In multi-electron atoms, the energy levels are influenced by both the principal quantum number \(n\) and the angular momentum quantum number \(\ell\). These additional interactions modify how electrons are distributed in the atomic orbitals. Furthermore, the magnetic quantum number \(m_{\ell}\) remains important for determining the spatial orientation of each orbital, regardless of the number of electrons in the atom.

Understanding multi-electron atoms is crucial for grasping how different elements on the periodic table form bonds and participate in chemical reactions. The complexities introduced by multiple electrons are key to the diverse chemical behaviors observed in nature.