Question

The values of \(n\) and \(\ell\) are useful to determine the order of filling (Aufbau principle). Use \(n\) and \(\ell\) to determine which orbital, \(4 f, 5 d,\) or \(6 s,\) will fill first.

Short answer

The 6s orbital fills first according to the Aufbau principle.

Step-by-step solution

Understanding quantum numbers

The quantum number \( n \) is the principal quantum number and \( \ell \) is the azimuthal quantum number, or angular momentum quantum number. They together help determine the energy level and shape of an atomic orbital respectively. \( n \) is a positive integer \( n = 1, 2, 3, ... \), and \( \ell \) can range from \( 0 \) to \( n-1 \). For example, \( \ell = 0 \) indicates an 's' orbital, \( \ell = 1 \) indicates a 'p' orbital, and so on.

Calculating \( n+\ell \) for each orbital

To determine which orbital fills first according to the Aufbau principle, calculate \( n+\ell \) for each orbital: \( 4f \), \( 5d \), and \( 6s \).- For \( 4f \), \( n = 4 \) and \( \ell = 3 \) (since \( f \) orbital corresponds to \( \ell = 3 \)). So, \( n+\ell = 4 + 3 = 7 \).- For \( 5d \), \( n = 5 \) and \( \ell = 2 \) (since \( d \) orbital corresponds to \( \ell = 2 \)). So, \( n+\ell = 5 + 2 = 7 \).- For \( 6s \), \( n = 6 \) and \( \ell = 0 \) (since \( s \) orbital corresponds to \( \ell = 0 \)). So, \( n+\ell = 6 + 0 = 6 \).

Comparing \( n+\ell \) values

The orbital with the lowest \( n+\ell \) value fills first according to the Aufbau principle. Comparing the values:- \( 4f \) has an \( n+\ell = 7 \)- \( 5d \) has an \( n+\ell = 7 \)- \( 6s \) has an \( n+\ell = 6 \)The \( 6s \) orbital has the lowest \( n+\ell \) value of 6, so it will fill before \( 4f \) and \( 5d \).

Conclusion: Orbital filling order

Based on the \( n+\ell \) rule and comparison, the \( 6s \) orbital fills first before both \( 4f \) and \( 5d \) because it has the lowest \( n+\ell \) value.

Key concepts

Quantum Numbers

Quantum numbers are fundamental to understanding how electrons are arranged in an atom. They describe different properties of the electrons in orbitals. There are four quantum numbers, but let's focus on the two crucial ones for our purpose here: the principal quantum number (") and the angular momentum quantum number (\( \ell \)).

These numbers help in determining the position and energy of electrons within an atom:

• The principal quantum number \( n \) indicates the main energy level or shell of an electron, essentially telling us how far the electron is from the nucleus.
• The angular momentum quantum number \( \ell \) describes the shape of the orbital and influences the angular distribution.

Understanding these two quantum numbers is vital for applying the Aufbau principle correctly.

n+l Rule

The \( n + \ell \) rule, also known as the Madelung energy ordering rule, is key to determining the order in which atomic orbitals fill.

According to this rule, the sum of the principal quantum number \( n \) and the angular momentum quantum number \( \ell \) determines the energy of an orbital:

• An orbital with a lower \( n + \ell \) value fills before an orbital with a higher \( n + \ell \) value.
• If two orbitals have the same \( n + \ell \) value, the orbital with the lower \( n \) will fill first.

This simple yet effective rule provides insight into the sequence of electron occupation in multi-electron atoms under the guidance of the Aufbau principle.

Orbital Filling Order

The orbital filling order is a critical process in atomic physics and chemistry, as it dictates how electrons populate the various available orbitals. This order adheres to the Aufbau principle, which states that electrons occupy the lowest energy orbitals available.

Here’s how it works:

• Start filling electrons from the lowest \( n + \ell \) value.
• Proceed to fill orbitals with increasing \( n + \ell \) until reaching the parameter needed for the element in question.
• For orbitals with the same \( n + \ell \) value, prioritize the one with the lower \( n \).

By following these steps, you ensure that electrons distribute themselves in a way that corresponds to their energies and minimizes the energy of the atom.

Principal Quantum Number

The principal quantum number, symbolized by \( n \), is a fundamental property in the quantum mechanical model of the atom. It represents the main electron shell or level and is a positive integer \( n = 1, 2, 3, \ldots \).

The importance of \( n \) includes:

• Indicating the relative overall energy of electrons and their distance from the nucleus.
• Determining the number of orbitals in each shell as they increase with increasing \( n \).

Higher \( n \) values denote higher energy levels and orbitals farther away from the nucleus, thus playing a significant role in electronic configurations and chemical properties.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted by \( \ell \), is integral to understanding the shape and structure of electron clouds in an atom. It specifies the subshell or sublevel of an electron within a given principal quantum number shell.

Key points about \( \ell \) include:

• It can take on integer values from 0 up to \( n-1 \), where \( n \) is the principal quantum number.
• The value of \( \ell \) corresponds to different subshells: \( \ell = 0 \) for 's', \( \ell = 1 \) for 'p', \( \ell = 2 \) for 'd', and \( \ell = 3 \) for 'f'.

This number is pivotal in dictating the electron configuration within an atom, influencing both the electronic structure and the atom's interaction with other atoms.