Question

For each of the following sublevels, give the \(n\) and \(l\) values and the number of orbitals: (a) \(6 g ;(b) 4 s ;(c) 3 d\)

Short answer

6g: n=6, l=4, 9 orbitals; 4s: n=4, l=0, 1 orbital; 3d: n=3, l=2, 5 orbitals.

Step-by-step solution

Title - Understanding Quantum Numbers

The principal quantum number, denoted as \(n\), represents the main energy level of the atom. It can take any positive integer value (1, 2, 3,...). The azimuthal quantum number, denoted as \(l\), determines the shape of the electron's orbital. For a given value of \(n\), \(l\) can range from 0 to \(n - 1\). The value of \(l\) is associated with the type of sublevel: 0 (s), 1 (p), 2 (d), 3 (f), 4 (g), and so on.

Title - Finding Quantum Numbers for 6g

For the 6g sublevel: \(n = 6\) because the principal quantum number is 6. Since \(g\) corresponds to \(l = 4\). There are \(2l+1 = 2(4)+1 = 9 \) orbitals in the g sublevel.

Title - Finding Quantum Numbers for 4s

For the 4s sublevel: \(n = 4\) because the principal quantum number is 4. Since \(s\) corresponds to \(l = 0\). There are \(2l+1 = 2(0)+1 = 1\) orbital in the s sublevel.

Title - Finding Quantum Numbers for 3d

For the 3d sublevel: \(n = 3\) because the principal quantum number is 3. Since \(d\) corresponds to \(l = 2\). There are \(2l+1 = 2(2)+1 = 5\) orbitals in the d sublevel.

Key concepts

Principal Quantum Number

The principal quantum number, represented by the symbol \(n\), is crucial in determining the main energy level occupied by an electron in an atom. It defines the overall size and energy of the orbital where an electron resides. The larger the \(n\) value, the higher the energy level and the larger the atomic orbital.

Here are some key points about the principal quantum number:

• \( n \) can be any positive integer: 1, 2, 3, and so on.
• A higher \( n \) means a greater average distance of the electron from the nucleus.
• Each \( n \) value defines a shell or energy level. For example, \( n = 1 \) is the first shell, \( n = 2 \) is the second shell, etc.

Understanding the principal quantum number is essential for grasping the overall structure of an atom and how electrons are distributed in different energy levels.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized as \( l \), specifies the shape of the electron's orbital within a particular energy level. This quantum number is also known as the angular momentum quantum number. It provides us information about the sublevels within a given principal energy level.

For a given \( n \), the value of \( l \) ranges from 0 to \( n-1 \). Each \( l \) value corresponds to a specific type of sublevel:

• \( l = 0 \) corresponds to the s-sublevel (spherical shape).
• \( l = 1 \) corresponds to the p-sublevel (dumbbell shape).
• \( l = 2 \) corresponds to the d-sublevel (cloverleaf shape).
• \( l = 3 \) corresponds to the f-sublevel (complex shapes).
• \( l = 4 \) corresponds to the g-sublevel (even more complex shapes).

By understanding the azimuthal quantum number, we can predict not just the energy of the electron's orbital, but also its shape.

Electron Orbitals

Electron orbitals are regions around the nucleus of an atom where electrons are likely to be found. They are defined by the quantum numbers \( n \) and \( l \) as discussed above, and each type of orbital (s, p, d, f, g) has a different shape and capacity to hold electrons.

A few important points about electron orbitals include:

• The s-orbital (\( l = 0 \)) is spherical and holds up to 2 electrons.
• The p-orbital (\( l = 1 \)) is dumbbell-shaped and can hold up to 6 electrons.
• The d-orbital (\( l = 2 \)) has a cloverleaf shape and can hold up to 10 electrons.
• The f-orbital (\( l = 3 \)) has complex shapes and can hold up to 14 electrons.
• The g-orbital (\( l = 4 \)) can hold up to 18 electrons and has even more complex shapes.

The combination of these orbitals helps define the electron configuration of an atom, which is crucial for understanding chemical bonding and properties.