Question

(a) For \(n=4\), what are the possible values of \(l ?\) (b) For \(l=2\), what are the possible values of \(m_{l} ?(c)\) If \(m_{l}\) is 2, what are the possible values for \(l ?\)

Short answer

(a) For n=4, the possible values of l are \(0, 1, 2, 3\). (b) For l=2, the possible values of \(m_{l}\) are \(-2, -1, 0, 1, 2\). (c) If \(m_{l}\) is 2, the possible values for l are \(2, 3, 4, 5, ...\ (l \geq 2)\).

Step-by-step solution

(a) Find the possible values of l for n = 4

Since n = 4, we need to find the possible values of l by using the relationship that l can be any integer value from 0 to n-1. Using this relationship, the possible values for l are: \(0, 1, 2, 3\)

(b) Find the possible values of ml for l = 2

Since l = 2, we need to find the possible values of ml by using the relationship that ml can take integer values from -l to +l, including zero. Using this relationship, the possible values for ml are: \(-2, -1, 0, 1, 2\)

(c) Find the possible values of l for ml = 2

Since ml = 2, the possible values of l must satisfy the relationship that ml can take integer values from -l to +l, including zero. To find the possible values of l, we need to find which values of l can give ml = 2 within the range (-l, +l). We can examine different values of l: For l = 0, the possible values of ml are: 0 (which does not include ml = 2) For l = 1, the possible values of ml are: -1, 0, 1 (which do not include ml = 2) For l = 2, the possible values of ml are: -2, -1, 0, 1, 2 (which include ml = 2) For l = 3, the possible values of ml are: -3, -2, -1, 0, 1, 2, 3 (which include ml = 2) And so on. So, the possible values of l for ml = 2 are: \(2, 3, 4, 5, ...\ (l \geq 2)\)

Key concepts

Principal Quantum Number (n)

In quantum mechanics, the principal quantum number, denoted by \(n\), plays a crucial role in determining the energy level of an electron within an atom. This number is always a positive integer, starting from 1 and increasing infinitely.
The principal quantum number indicates the shell or energy level that an electron occupies within an atom, and it greatly influences the size of the electron cloud. For example,

• If \(n = 1\), the electron is in the first energy level, which is closest to the nucleus.
• A larger \(n\) value means the electron is farther from the nucleus and potentially at a higher energy level.

Each principal quantum number corresponds to a different set of energy levels in an atom. For any given \(n\), the possible values of other quantum numbers, like \(l\) and \(m_l\), are determined subsequently.

Angular Momentum Quantum Number (l)

The angular momentum quantum number, represented by \(l\), defines the shape of the electron's orbital. It is dependent on the principal quantum number \(n\) and can take integer values from 0 to \(n-1\).
For example, if \(n = 4\), then \(l\) can be 0, 1, 2, or 3. Let's explain what these values signify:

• \(l = 0\): The orbital is spherical and is known as an s orbital.
• \(l = 1\): The orbital is shaped like a dumbbell and is referred to as a p orbital.
• \(l = 2\): This corresponds to a d orbital, which has a more complex shape.
• \(l = 3\): Represents an f orbital, with even more intricate shapes.

The angular momentum quantum number also determines the number of angular nodes within an electron's orbital, affecting its stability and energy.

Magnetic Quantum Number (m_l)

The magnetic quantum number, \(m_l\), is intricately associated with the directionality and orientation of an electron's orbital in space, relative to an external magnetic field. It can take on integer values ranging from \(-l\) to \(+l\), including zero. This range is crucial for defining the spatial distribution of the electrons around a nucleus.
For instance, if \(l = 2\), the possible \(m_l\) values are -2, -1, 0, 1, and 2. Here's how they interpret:

• Each \(m_l\) value corresponds to a distinct orientation of the d orbitals in space.
• Different \(m_l\) values do not change the energy level but affect the electron's alignment when an external magnetic field interacts with the atom.

When given a specific \(m_l\) value, the range of \(l\) that includes this \(m_l\) can be determined, such as when \(m_l = 2\), both \(l = 2\) and any higher \(l\) values would allow this magnetic number.