Problem 91 - Chemistry: An Atoms-Focused Approach

Question

Identify the subshells with the following combinations of quantum numbers and arrange them in order of increasing energy in a multielectron atom: a. \(n=3, \ell=2\) b. \(n=7, \ell=3\) c. \(n=3, \ell=0\) d. \(n=4, \ell=1\)

Step-by-Step Solution

Answer

Question: Arrange the following subshells in order of increasing energy in a multi-electron atom, given the principal quantum number (n) and azimuthal quantum number (ℓ) as follows: a) n=3, ℓ=2; b) n=7, ℓ=3; c) n=3, ℓ=0; d) n=4, ℓ=1. Answer: The subshells, arranged according to increasing energy in multi-electron atoms are: 3s, 3d, 4p, 7f.

Step 1: Identifying the subshells for given quantum numbers:

For each combination of principal and azimuthal quantum numbers, we will identify the subshell using the standard electron orbital notation (s, p, d, f, etc.). a. \(n=3, \ell=2\) - This corresponds to the 3d subshell. b. \(n=7, \ell=3\) - This corresponds to the 7f subshell. c. \(n=3, \ell=0\) - This corresponds to the 3s subshell. d. \(n=4, \ell=1\) - This corresponds to the 4p subshell.

Step 2: Applying the (n + ℓ) rule:

To arrange the subshells in order of increasing energy in a multielectron atom, we will apply the (n + ℓ) rule. This rule states that subshells with a lower value of (n + ℓ) have lower energy. If there is a tie between two subshells, then the subshell with the lower value of n has lower energy. a. 3d - (n + ℓ) = (3 + 2) = 5 b. 7f - (n + ℓ) = (7 + 3) = 10 c. 3s - (n + ℓ) = (3 + 0) = 3 d. 4p - (n + ℓ) = (4 + 1) = 5

Step 3: Arranging the subshells according to energy levels:

Using the (n + ℓ) values calculated above, we'll arrange the subshells in order of increasing energy in a multielectron atom. The order is: 3s (n + ℓ = 3) < 3d (n + ℓ = 5, n = 3) < 4p (n + ℓ = 5, n = 4) < 7f (n + ℓ = 10) The subshells, arranged according to increasing energy in multielectron atoms are: 3s, 3d, 4p, 7f.

Key Concepts

Electron Configuration

Electron configuration refers to the arrangement of electrons in atoms or molecules. In chemistry, this knowledge allows us to discern the stability, reactivity, and properties of elements.

• The electrons in an atom inhabit distinct energy levels or shells, represented by the principal quantum number, \( n \).
• The azimuthal quantum number, \( \ell \), designates the subshell or orbital type—such as \( s \), \( p \), \( d \), and \( f \)—within an energy level.
• Each orbital type can hold a certain number of electrons: \( s \) holds 2, \( p \) holds 6, \( d \) holds 10, and \( f \) holds 14.

By recognizing these configurations, scientists can predict how atoms will bond and interact with others, formative to the field of quantum chemistry.

Multielectron Atoms

In multielectron atoms, electrons not only interact with the nucleus but also with each other. This means the electron energies are more complex compared to hydrogenic atoms.

• One major difference is shielding, where inner electrons partially block the nuclear charge from outer electrons. This alters the actual energy levels experienced by the electrons.
• Repulsion between electrons in multielectron atoms adjusts the energy levels and results in splitting that turbine simple in hydrogen atoms.
• Energy levels in multielectron atoms are closer together, making the energy calculation more intricate than in single-electron systems.

Understanding how these forces interrelate gives insight into electron dynamics and consequently the chemical properties of elements.

n+l Rule

The \( n + \ell \) rule is an essential principle for organizing electron configurations in multielectron atoms.

• This rule suggests subshell energy can be predicted by the sum \( n + \ell \), where \( n \) is the principal quantum number and \( \ell \) is the azimuthal quantum number.
• A lower \( n + \ell \) value indicates a lower energy level, and between ties, the subshell with the lower \( n \) value is prioritized.

In practical terms, it means subshells are filled following a certain sequence based on their energy, beginning with the lowest \( n + \ell \) values. Understanding this rule is critical for predicting the stability and electron arrangement in a given atom.