Question

What is the value of \(n\) for a shell if the largest value of \(\ell\) is 5 ?

Short answer

The value of 'n' is 6.

Step-by-step solution

Identify the Relationship Between Quantum Numbers

In quantum mechanics, for a given principal quantum number, denoted as 'n', the azimuthal (or angular momentum) quantum number 'l' can have values ranging from 0 to n-1. The largest value of 'l' for a given shell is therefore n-1.

Set Up the Equation

Given that the largest value of \(\ell\) is 5, we set up the equation relating \(\ell\) with 'n' as follows: \(\ell_{\text{max}} = n - 1\).

Solve for 'n'

To find 'n', we simply add 1 to the largest value of \(\ell\): \(n = \ell_{\text{max}} + 1\). Substituting the value of \(\ell_{\text{max}}\) given as 5, we have \(n = 5 + 1\).

Find the Value of 'n'

Carrying out the addition, we get \(n = 6\).

Key concepts

Principal Quantum Number

The principal quantum number, denoted by the symbol 'n', is fundamental in quantum mechanics because it designates the energy level or shell of an electron within an atom. Like the floors in a building, higher energy levels correspond to larger principal quantum numbers.

For example, if an electron is said to be in the second energy level, this translates to a principal quantum number of 2, or simply put, n=2. The value of 'n' starts at 1, which represents the ground state, or lowest energy level, and increases in integer values as the energy of the atom’s electrons increase.

In understanding the solution to the given problem, knowing that the principal quantum number defines the maximum possible value for the azimuthal quantum number is crucial. Thus, if the largest azimuthal quantum number \(\ell_{\text{max}}\) is 5, then the principal quantum number 'n' must be one greater than that, which is 6.

Azimuthal Quantum Number

Closely associated with the principal quantum number is the azimuthal quantum number, represented by the symbol \(\ell\). This quantum number defines the subshell or the shape of the orbital where electrons reside, and indirectly, their angular momentum.

An easy way to envision this is to think of each energy level being divided into sub-levels with distinct shapes – like different suites on the same floor of a hotel. These shapes are crucial for determining how electrons are distributed around the nucleus and affect the chemical properties of an atom. Each energy level has \(\ell\) values ranging from 0 up to \(n - 1\), which correspond to s, p, d, f, and so on, orbitals in increasing order of energy.

Following the solution provided, when you have the largest \(\ell\) value of 5, this directly points to a 'g' subshell, which is less common and occurs in higher energy levels. By adding 1 to this maximal \(\ell\) value according to the relationship \(\ell_{\text{max}} = n - 1\), we obtain the principal quantum number 'n' for the energy level.

Quantum Mechanics

Quantum mechanics is the branch of physics that studies the behavior of matter and energy at the smallest scales—typically atomic and subatomic levels. At its core, it replaces classical intuitions with probabilistic models and quantized properties.

Unlike the visible world where objects have a well-defined position and speed, quantum particles exist in a haze of probability, with properties like energy, position, and momentum, only taking specific quantized values known as 'quanta.' Fundamental to quantum mechanics are these quantized properties, including the quantum numbers we discuss, which serve as a 'code' to describe the unique state of a subatomic particle, like an electron in an atom.

As a fundamental theory, quantum mechanics is responsible for explaining why atoms have distinct energy levels, why electrons occupy specific orbitals, and how atoms emit or absorb light, all of which are essential to our understanding of chemistry, physics, and even the functioning of high-tech devices.