Question

The energy of different orbitals in an atom or ion having only one electron, depends on (a) \(n\) only (b) \(n\) and \(l\) only (c) \(n, l\) and \(m\) only (d) \(n, l, m\) and \(s\)

Short answer

The energy of different orbitals in an atom or ion having only one electron, depends on (a) n only.

Step-by-step solution

Understanding Quantum Numbers

Each electron in an atom is described by four quantum numbers. The principal quantum number () determines the energy level of the electron. The azimuthal quantum number (l), also known as the angular momentum quantum number, defines the shape of the orbital. The magnetic quantum number (m) defines the orientation of the orbital in space. Finally, the spin quantum number (s) refers to the spin orientation of the electron.

Energy Levels of a Single Electron System

In the case of a hydrogen-like atom or ion that contains only one electron, the energy level of that electron depends primarily on the principal quantum number (). This is because there are no electron-electron interactions that would cause the energy to split further based on other quantum numbers.

Answer Reasoning

Therefore, for an atom or ion with a single electron, the energy of orbitals depends only on the principal quantum number () and not on the azimuthal (l), magnetic (m), or spin (s) quantum numbers.

Key concepts

Principal Quantum Number

The principal quantum number, represented as '(n)', plays a pivotal role in determining the energy levels within an atom. It is an integer that starts at 1 and increases with increasing energy and distance from the nucleus. Think of it like the floors in a building; the higher the floor, the higher the energy level.

In hydrogen-like atoms, which are atoms with only one electron such as H, He+, Li2+, the principal quantum number solely determines the energy of the electron. This is crucial because it simplifies the way energy can be calculated for these atoms. The principal quantum number essentially defines which shell the electron resides in, affecting its potential energy due to the distance from the positively charged nucleus. A larger '(n)' value indicates that the electron is further away and at a higher energy shell.

Azimuthal Quantum Number

The azimuthal quantum number, indicated by '(l)', provides insight into the shape of the electron orbitals. Orbitals can be visualized as clouds where electrons are likely to be found. The azimuthal quantum number can take on any integer value from 0 to (n)-1, where '(n)' is the principal quantum number.

Commonly, these shapes are known as s, p, d, and f, corresponding to '(l)' values of 0, 1, 2, and 3, respectively. For example, an (l)=0 gives us an 's' orbital which is spherical, while an (l)=1 results in a 'p' orbital which has a dumbbell shape. In multi-electron systems, both '(n)' and '(l)' influence the energy of electrons due to the interactions between them. But, in a single-electron system, like hydrogen-like atoms, '(l)' does not contribute to the energy levels.

Orbital Energy Levels

Orbital energy levels are regions around an atom where the probability of finding an electron is high. These are often visualized as concentric shells, each associated with a principal quantum number '(n)'. Within these shells, we have subshells given by the azimuthal quantum number '(l)', which further define energy.

In multi-electron systems, electrons repel each other, which requires us to consider all quantum numbers (including the azimuthal '(l)', magnetic '(m)', and spin '(s)') to accurately describe energy levels. However, in hydrogen-like atoms with only one electron, the energy of the electron depends only on its principal quantum number '(n)'. This is because there are no repulsive forces from other electrons, leading to a situation where the subshells within a principal energy level have the same energy, known as being 'degenerate'.

Hydrogen-like Atoms

Hydrogen-like atoms are a simple yet fundamental concept in quantum chemistry. They are defined as atoms or ions that contain only one electron. Examples include the hydrogen atom (H), helium ion (He+), and lithium ion (Li2+). In these systems, the absence of electron-electron interactions means that the energy levels only depend on the principal quantum number '(n)'.

This unique feature allows us to use hydrogen-like atoms as models to develop a deeper understanding of atomic structure and quantum mechanics. By studying these simplified systems, we can extrapolate knowledge to more complex atoms. Even though real-world atoms have multiple electrons with complex interactions, hydrogen-like atoms remain a cornerstone in learning about orbital structures and the behavior of electrons at quantum levels.