Problem 52 - Chemistry: The Central Science

Question

Give the values for \(n, l\), and \(m_{l}\) for (a) each orbital in the \(2 p\) subshell, (b) each orbital in the \(5 d\) subshell.

Step-by-Step Solution

Answer

(a) For the 2p subshell: \(n=2\), \(l=1\), and \(m_l= -1, 0, +1\) (representing 3 orbitals: 2p -1, 2p 0, and 2p +1). (b) For the 5d subshell: \(n=5\), \(l=2\), and \(m_l= -2, -1, 0, +1, +2\) (representing 5 orbitals: 5d -2, 5d -1, 5d 0, 5d +1, and 5d +2).

Step 1: Identify the main quantum number (n) for the 2p subshell

The main quantum number (n) corresponds to the energy level, which is indicated by the number in the subshell notation. For the 2p subshell, the main quantum number is 2.

Step 2: Identify the azimuthal quantum number (l) for the p subshell

The azimuthal quantum number (l) represents the subshell, defined as: s = 0 p = 1 d = 2 f = 3 For the p subshell, the azimuthal quantum number is 1 (l=1).

Step 3: Identify the magnetic quantum numbers (ml) for each orbital in the 2p subshell

The magnetic quantum number (ml) corresponds to the orbital within the subshell and ranges from -l to +l. In the p subshell where l=1, the possible values for ml are: -1, 0, and +1. So there are 3 orbitals in the p subshell: 2p -1, 2p 0, and 2p +1. (b) Orbital Quantum Numbers for the 5d Subshell

Step 1: Identify the main quantum number (n) for the 5d subshell

The main quantum number (n) corresponds to the energy level, which is indicated by the number in the subshell notation. For the 5d subshell, the main quantum number is 5.

Step 2: Identify the azimuthal quantum number (l) for the d subshell

The azimuthal quantum number (l) represents the subshell, defined as: s = 0 p = 1 d = 2 f = 3 For the d subshell, the azimuthal quantum number is 2 (l=2).

Step 3: Identify the magnetic quantum numbers (ml) for each orbital in the 5d subshell

The magnetic quantum number (ml) corresponds to the orbital within the subshell and ranges from -l to +l. In the d subshell where l=2, the possible values for ml are: -2, -1, 0, +1, and +2. So there are 5 orbitals in the d subshell: 5d -2, 5d -1, 5d 0, 5d +1, and 5d +2.

Key Concepts

Quantum Number n

The quantum number 'n', known as the principal quantum number, represents the electron's energy level or shell in an atom. It also indicates the relative distance of an electron from the nucleus.

For instance, in a 2p subshell, 'n' would be 2, suggesting that the electrons are located in the second energy level. As 'n' increases, so does the energy of the electron and its average distance from the nucleus. Only positive integers (1, 2, 3, ...) are allowed for the principal quantum number.

Azimuthal Quantum Number l

The azimuthal quantum number 'l' is second in the hierarchy of quantum numbers and defines the shape of the electron's orbital. It is also synonymous to the angular momentum quantum number.

For any given principal quantum number 'n', 'l' can range from 0 to 'n-1'. This explains why, for the 2p subshell, where 'n' is 2, 'l' can only be 1. The values of 'l' correspond to different types of orbitals: 0 (s), 1 (p), 2 (d), and 3 (f). Each of these orbitals has a unique shape and denotes a particular subshell within an energy level.

Magnetic Quantum Number ml

The magnetic quantum number 'ml' describes the orientation of the electron's orbital in space and is derived from the azimuthal quantum number 'l'.

For each value of 'l', 'ml' ranges from '-l' to '+l', including zero. Hence, every subshell will have (2l + 1) possible values for 'ml'. For example, the p subshell, with 'l' equal to 1, gives 'ml' values of -1, 0, and +1, corresponding to three unique p orbitals. These values can be interpreted as the spatial orientation of the orbitals around the nucleus.

Electron Subshells

Electron subshells refer to groups of orbitals within an electron shell that have the same value of the azimuthal quantum number 'l'. They are crucial in understanding the electron configuration within an atom.

The subshells are named s, p, d, and f corresponding to 'l' values of 0, 1, 2, and 3 respectively. Each subshell type can hold a distinct number of electrons: s (2 electrons), p (6 electrons), d (10 electrons), and f (14 electrons). This is due to the number of orbitals within each subshell; for example, the d subshell contains five orbitals, each of which can hold two electrons (one with spin up and one with spin down), giving a total of 10 electrons in the d subshell.