Question

List all the possible subshells and orbitals associated with the principal quantum number \(n\), if \(n=5\).

Short answer

For \(n = 5\), the subshells are s, p, d, f, and g, with 1, 3, 5, 7, and 9 orbitals respectively, resulting in a total of 25 orbitals.

Step-by-step solution

List All Possible Subshells

For \(n = 5\), subshells are determined by each value of azimuthal quantum number (\(l\)) from 0 to \(n - 1\). Here, the possible values of \(l\) are 0, 1, 2, 3, and 4. These correspond to the subshells s, p, d, f, and g.

Calculate Orbitals for Each Subshell

The number of orbitals in a subshell is given by the formula \(2l + 1\). For s subshell (\(l = 0\)), there's 1 orbital, for p (\(l = 1\)), there are 3 orbitals, for d (\(l = 2\)), there are 5 orbitals, for f (\(l = 3\)), there are 7 orbitals, and for g (\(l = 4\)), there are 9 orbitals.

Present the Final Answer

So for \(n = 5\), we have 5 subshells (s, p, d, f, g) and each have 1, 3, 5, 7, 9 orbitals respectively. Therefore, there are total 25 orbitals.

Key concepts

Principal Quantum Number

The principal quantum number, represented by the symbol \(n\), is a critical component in quantum mechanics. It essentially determines the overall size and energy level of an electron's orbital within an atom. The value of \(n\) is always a positive integer: 1, 2, 3, and so forth, where each number signifies a principal energy level or shell.

• Higher \(n\) values indicate electrons are further from the nucleus, meaning they possess higher energy.
• The principal quantum number also influences the number of possible subshells within an energy level.

For example, when \(n = 5\), it indicates the fifth energy level, and this energy level can contain more complex and numerous subshells compared to lower energy levels.

Azimuthal Quantum Number

The azimuthal quantum number, denoted as \(l\), determines the shape and number of subshells within a given principal quantum number. Its value is dependent on \(n\), where \(l\) can range from 0 up to \(n-1\). Each \(l\) value corresponds to a specific subshell type.

• \(l = 0\) corresponds to the s subshell.
• \(l = 1\) corresponds to the p subshell.
• \(l = 2\) corresponds to the d subshell.
• \(l = 3\) corresponds to the f subshell.
• \(l = 4\) corresponds to the g subshell.

For the principal quantum number \(n = 5\), the possible \(l\) values are 0, 1, 2, 3, and 4, corresponding to the possible subshells that can exist at this energy level.

Orbitals

Orbitals are regions within an atom where there is a high probability of finding electrons. The shape and orientation of an orbital are dictated by the azimuthal quantum number \(l\). Each subshell type has a specified number of orbitals, which is given by the formula \(2l + 1\).

• In the s subshell (\(l=0\)), there is 1 orbital.
• In the p subshell (\(l=1\)), there are 3 orbitals.
• In the d subshell (\(l=2\)), there are 5 orbitals.
• In the f subshell (\(l=3\)), there are 7 orbitals.
• In the g subshell (\(l=4\)), there are 9 orbitals.

Consequently, for \(n = 5\), the orbitals are distributed among the 5 subshells, resulting in 25 orbitals in total.

Subshells

Subshells are subdivisions of electron shells within an atom, and they are specified by the azimuthal quantum number \(l\). Each subshell can hold a specific number of orbitals, which subsequently determine the electron capacity of that subshell.

• The s subshell can hold a maximum of 2 electrons (1 orbital \(\times\) 2 electrons/orbital).
• The p subshell can hold a maximum of 6 electrons (3 orbitals \(\times\) 2 electrons/orbital).
• The d subshell can hold a maximum of 10 electrons (5 orbitals \(\times\) 2 electrons/orbital).
• The f subshell can hold a maximum of 14 electrons (7 orbitals \(\times\) 2 electrons/orbital).
• The g subshell can hold a maximum of 18 electrons (9 orbitals \(\times\) 2 electrons/orbital).

Thus, for \(n = 5\), there are five possible subshells: s, p, d, f, and g. Each subshell has a specific number of orbitals, cumulatively holding a maximum of 50 electrons.