Problem 160 - Introductory Chemistry Atoms First

Question

Which subshell has the lowest energy? (a) \(4 s\) (b) \(3 p\) (c) \(2 p\) (d) \(3 s\) (e) \(2 s\)

Step-by-Step Solution

Answer

The subshell with the lowest energy is (e) \(2s\).

Step 1: List the given subshells and their quantum numbers

: First, let's list the given subshells and their corresponding quantum numbers (n, l): (a) \(4s: n = 4, l = 0\) (s corresponds to l = 0) (b) \(3p: n = 3, l = 1\) (p corresponds to l = 1) (c) \(2p: n = 2, l = 1\) (d) \(3s: n = 3, l = 0\) (e) \(2s: n = 2, l = 0\)

Step 2: Calculate the (n + l) values for each subshell

: Now, calculate the (n + l) values for each subshell: (a) \(4s: n + l = 4 + 0 = 4\) (b) \(3p: n + l = 3 + 1 = 4\) (c) \(2p: n + l = 2 + 1 = 3\) (d) \(3s: n + l = 3 + 0 = 3\) (e) \(2s: n + l = 2 + 0 = 2\)

Step 3: Identify the subshell with the lowest (n + l) value

: Among the given subshells, the one with the lowest (n + l) value is: (e) \(2s: n + l = 2\)

Step 4: Check for any subshells with the same (n + l) value

: There are no other subshells with the same (n + l) value as the \(2s\) subshell. Therefore, there's no need to compare their n values.

Step 5: Write the answer

: The subshell with the lowest energy is: (e) \(2s\)

Key Concepts

Quantum Numbers

Quantum numbers are crucial to understanding the arrangement of electrons in atoms. These numbers define the unique quantum state of an electron and help determine its energy and position within an atom. There are four significant quantum numbers:

• Principal quantum number (n): Represents the main energy level or shell in which the electron resides. The values of n are positive integers (1, 2, 3, ...). As n increases, the electron's energy and its average distance from the nucleus also increase.
• Azimuthal quantum number (l): Also known as the angular momentum quantum number, l determines the shape of the electron's orbital or subshell. It can take integer values from 0 to (n-1) for each principal quantum number n. For example, if n = 3, then l can be 0, 1, or 2.
• Magnetic quantum number (m_l): Specifies the orientation of the orbital in space. It can take integer values from -l to +l, including zero.
• Spin quantum number (m_s): Represents the spin of the electron, which can either be +1/2 or -1/2.

Quantum numbers create a roadmap for electrons, offering insight into their arrangement and energy levels.

n+l Rule

The n+l rule, also known as the Madelung rule, is a useful guideline for determining the order in which atomic orbitals are filled as atoms increase in size. This rule states that the subshells fill in order of increasing (n + l) values. The value of n is the principal quantum number and l is the azimuthal quantum number corresponding to the subshell type:

• s subshell: l = 0
• p subshell: l = 1
• d subshell: l = 2
• f subshell: l = 3

When two subshells have the same (n + l) value, the one with the lower n value fills first. For example, in the exercise provided, the subshell 2s has an (n + l) value of 2, which is lower than all others, making it the lowest energy subshell.

Atomic Subshells

Within each electron shell, there are subshells distinguished by different values of the azimuthal quantum number (l). These subshells define the shape and type of orbitals:

• s subshell: Has a spherical shape and contains one orbital. This is l = 0.
• p subshell: Shaped like a dumbbell and contains three orbitals. This represents l = 1.
• d subshell: More complex in shape, containing five orbitals. This corresponds to l = 2.
• f subshell: Even more complex structures with seven orbitals, indicated by l = 3.

Each orbital in a subshell can hold up to two electrons with opposite spins. As electrons fill these subshells, they create the atom's chemical properties. In determining energy, lower (n + l) values indicate lower energy, helping to prioritize pattern filling.

Electron Configuration

Electron configuration is the arrangement of electrons in an atom's orbitals. It shows how electrons fill the various energy levels and subshells. The Aufbau principle, Pauli exclusion principle, and Hund's rule dictate this arrangement. The Aufbau principle suggests electrons fill the lowest energy orbitals first. This aligns with the n+l rule's order. For instance, if you're arranging electrons in an atom, you would fill the 1s subshell before 2s, 2p, then 3s, and so on. The Pauli exclusion principle states no two electrons in an atom can have the same set of all four quantum numbers. This principle ensures electrons occupy different states. Hund's rule emphasizes maximum multiplicity, meaning electrons will first fill empty orbitals in a subshell singly, before pairing up. This reduces electron-electron repulsion and leads to a more stable configuration. Understanding electron configuration helps predict how an atom will interact chemically and informs the periodic table's structure.