Question

Write down the three sets of quantum numbers that define the three \(3 p\) atomic orbitals.

Short answer

The three sets of quantum numbers for the \(3p\) orbitals are \((3, 1, -1)\), \((3, 1, 0)\), and \((3, 1, 1)\).

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers are used to describe the properties of atomic orbitals and the electrons within them. There are four quantum numbers: the principal quantum number \(n\), azimuthal quantum number \(l\), magnetic quantum number \(m_l\), and spin quantum number \(m_s\). For the \(3p\) orbitals, \(n = 3\) and \(l = 1\).

Determine the Azimuthal Quantum Number \(l\)

For a \(p\) orbital, the azimuthal quantum number \(l\) is always 1, indicating the shape of the orbital.

Setting the Principal Quantum Number \(n\)

Given that the atomic orbitals are \(3p\) orbitals, the principal quantum number \(n\) is 3, which defines the energy level and size of the orbital.

Find Possible Values of Magnetic Quantum Number \(m_l\)

The magnetic quantum number \(m_l\) can take integer values from \(-l\) to \(+l\), inclusive. For our \(3p\) orbitals: \(l = 1\), so \(m_l\) can be \(-1, 0,\) or \(+1\). Each corresponds to a different \(3p\) orbital.

Assign Spin Quantum Number \(m_s\)

Although problem didn't ask for spins specifically, each orbital can hold two electrons with different spins. The spin quantum number \(m_s\) can be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Formulate the Sets of Quantum Numbers

The three sets of quantum numbers for the \(3p\) orbitals are: \((3, 1, -1)\), \((3, 1, 0)\), and \((3, 1, 1)\). These describe the three different orientations of the \(3p\) orbitals.

Key concepts

Atomic Orbitals

Atomic orbitals are regions of space around the nucleus where the probability of finding an electron is high. They are the fundamental building blocks of quantum mechanics for atoms. Each orbital defines a specific "zone" where electrons can be found, described by unique sets of quantum numbers.

Atomic orbitals come in various shapes, determined by their energy levels and subshells:

• s orbitals: Spherical shapes.
• p orbitals: Dumbbell-shaped, oriented along different axes.
• d orbitals: More complex shapes, often clover-like.
• f orbitals: Even more complex shapes.

Each type of orbital is associated with specific quantum numbers which define its properties like size, shape, and orientation.

Principal Quantum Number

The principal quantum number, denoted as \( n \), is crucial in defining the size and energy of the atomic orbitals. It is an integer that specifies the main energy level, or shell, an electron occupies in an atom.

The higher the value of \( n \), the farther the electron is from the nucleus, and consequently, the higher the energy level:

• \( n = 1 \): Closest to the nucleus, lowest energy level.
• \( n = 2 \): Higher energy, larger distance.
• Higher values follow this trend.

For the 3p atomic orbitals, \( n = 3 \), placing these electrons in the third shell, which is at a significant distance from the nucleus with higher potential energy.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized as \( l \), describes the shape of the atomic orbital and is also known as the angular momentum quantum number.

This quantum number can take values ranging from zero to \( n-1 \) for any given principal quantum number \( n \). It helps define the subshells within a principal shell:

• \( l = 0 \): s subshell.
• \( l = 1 \): p subshell.
• \( l = 2 \): d subshell.
• \( l = 3 \): f subshell.

In the case of the 3p orbitals, \( l = 1 \), linking it directly to the p subshell known for its distinct dumbbell shape.

Magnetic Quantum Number

The magnetic quantum number, \( m_l \), determines the orientation of the atomic orbital in three-dimensional space. It can take on values ranging from \(-l\) to \(+l\), including zero. This quantum number shows how the different orbitals in a subshell are oriented in relation to an external magnetic field.

For the 3p subshell:

• \( m_l = -1, 0, +1 \): Reflects three possible orientations in space.

These possibilities determine the different 3p orbitals, each uniquely oriented relative to the others. They indicate how electrons can be distributed in various planes within the p subshell.

Spin Quantum Number

The spin quantum number, represented as \( m_s \), is essential for defining the intrinsic spin of an electron within an atomic orbital. This quantum number can take one of two possible values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Each atomic orbital can host two electrons, provided they have opposite spins:

• \(+\frac{1}{2}\): Spin up.
• \(-\frac{1}{2}\): Spin down.

The opposite spins help in minimizing repulsion between electrons in the same orbital, adhering to the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of all four quantum numbers.