Question

Concerning the concept of subshells and orbitals, (a) How many subshells are found in the \(n=3\) level? (b) What are the names of the subshells in the \(n=3\) level? (c) How many orbitals have the values \(n=4\) and \(\ell=3 ?\) (d) How many orbitals have the values \(n=3, \ell=2\) and \(m_{\ell}=-2 ?\) (e) What is the total number of orbitals in the \(n=4\) level?

Short answer

The answers to the questions are: (a) 3 subshells, (b) They are 3s, 3p, 3d, (c) 7 orbitals, (d) 1 orbital, (e) 16 orbitals.

Step-by-step solution

Determination of Number of Subshells

In quantum mechanics, the number of subshells in a principal energy level 'n' is equal to the value of 'n'. Therefore for \(n=3\), the number of subshells is 3.

Naming of Subshells

The subshells in an energy level 'n' are usually designated by the letters s, p, d, f, corresponding to the azimuthal quantum number values of 0, 1, 2, 3 respectively. Hence, for \(n=3\), the subshells are: 3s, 3p and 3d.

Number of Orbitals for Given Quantum Numbers

The number of possible orbitals for a given azimuthal quantum number '\ell' is given by '2\ell+1'. For the case \(n=4\), \(\ell=3\), the number of orbitals is \(2*3+1 = 7\).

Specific Orbital for Given Quantum Numbers

If specific quantum numbers are given like \(n=3, \ell=2\) and \(m_{\ell}=-2\), it means we are talking about a specific orbital. Therefore, there is 1 orbital corresponding to these values.

Total Number of Orbitals in a Given Level

The total number of orbitals in a principle energy level 'n' is given by \(n^2\). So, for \(n=4\), the total number of orbitals is 16.

Key concepts

Understanding Subshells

Subshells are an integral part of quantum mechanics and provide insight into the structure of atomic orbitals. These subshells represent the division of electron shells around an atom, specifically based on the azimuthal quantum number, often denoted as \( \ell \). Each subshell corresponds to a specific value of \( \ell \) and represents a particular shape of electron cloud. In a given energy level \( n \), the number of subshells is equal to \( n \) itself.
For example, when \( n=3 \), we have three subshells:

• 3s which corresponds to \( \ell = 0 \)
• 3p corresponding to \( \ell = 1 \)
• 3d corresponding to \( \ell = 2 \)

These letters (s, p, d, f, ...) indicate the shape and orientation of the subshells, which play an important role in determining how electrons are distributed around the nucleus.

Exploring Orbitals

Once you've identified subshells, orbitals become the next key component to understand. Orbitals are specific regions within subshells where the probability of finding an electron is high. Each orbital can hold a maximum of two electrons and is defined by a set of quantum numbers.
The azimuthal quantum number \( \ell \) determines the number of orbitals within a subshell, calculated by the formula \( 2\ell + 1 \). For instance, if \( \ell = 3 \) as seen in the 3d subshell, there are \( 2 \times 3 + 1 = 7 \) orbitals. This calculation is vital for understanding the electron configuration within an atom. Consequently, each unique set of quantum numbers defines a specific orbital. For example, when \( n=3, \ell=2 \), and \( m_{\ell}=-2 \), it points to one unique orbital in the 3d subshell. The orbital's distinctiveness highlights how electrons are organized in energy levels, offering a precise map of an atom's electron arrangement.

The Role of Quantum Numbers

Quantum numbers are the backbone of quantum mechanics, describing the characteristics and behavior of electrons in atoms. They specify the unique "address" of each electron, comprising four varieties: the principal quantum number \( n \), azimuthal quantum number \( \ell \), magnetic quantum number \( m_{\ell} \), and the spin quantum number \( m_{s} \).

• Principal Quantum Number (\( n \)): Indicates the energy level and size of the orbital. For example, \( n=4 \) reveals that it is the fourth energy level.


• Azimuthal Quantum Number (\( \ell \)): Determines the subshell and shapes of orbitals, offering values from 0 to \( n-1 \). In \( n=3 \), possible \( \ell \) values are 0, 1, and 2, corresponding to the 3s, 3p, and 3d subshells.


• Magnetic Quantum Number (\( m_{\ell} \)): Specifies the orientation of an orbital within a subshell, range from \( -\ell \) to \( +\ell \). For \( \ell = 2 \), \( m_{\ell} \) can be \(-2, -1, 0, 1, \) or \( 2 \).


• Spin Quantum Number (\( m_{s} \)): Represents the electron's spin direction, typically \( +1/2 \) or \( -1/2 \).

Together, these quantum numbers provide a comprehensive framework for predicting electron configurations, enabling a deeper understanding of atomic structure and behavior.