Question

Identify the orbital that has (a) one radial node and one angular node; (b) no radial nodes and two angular nodes; (c) two radial nodes and three angular nodes.

Short answer

The orbitals are (a) \(3p\), (b) \(3d\), and (c) \(6f\).

Step-by-step solution

Understand the concepts

Orbital is a specific region within an atom where there is a high probability of finding an electron. Nodes are areas where the probability of finding an electron is zero. There are two types of nodes: radial and angular. A radial node occurs when the electron is never found at a certain distance from the nucleus. An angular node (also known as a nodal plane) is a plane where the electron cannot be found. The number of radial nodes is given by \( n - l - 1 \) and the number of angular nodes is equal to \( l \). where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number.

Solve for (a) with one radial node and one angular node

From the formula for the number of radial nodes, we have \( n - l - 1 = 1 \), and from the formula for the number of angular nodes, we just get \( l = 1 \). From the principle quantum number and azimuthal quantum number, it's found that \( n = 3 \) and \( l = 1 \), corresponding to the \( 3p \) orbitals.

Solve for (b) with no radial nodes and two angular nodes

Following a similar process, for no radial nodes, we get \( n = l + 1 \) and for the two angular nodes, we just get \( l = 2 \). From these equations, we get \( n = 3 \) and \( l = 2 \), corresponding to the \( 3d \) orbitals.

Solve for (c) with two radial nodes and three angular nodes

Again, for two radial nodes we get \( n = l +3 \) and for three angular nodes, we just get \( l = 3 \). From these equations, we get \( n = 6 \) and \( l = 3 \), corresponding to the \( 6f \) orbitals.

Key concepts

Radial Nodes

When we talk about radial nodes in the context of quantum chemistry, we're discussing specific points within an atomic orbital where the probability of finding an electron drops to zero. These nodes depend heavily on the principal quantum number, denoted as \( n \), and the azimuthal quantum number, or \( l \). To determine the number of radial nodes within an orbital, one can use the formula:\[ \text{Radial Nodes} = n - l - 1 \]

• As \( n \) increases, the size of the orbital expands, and the number of radial nodes also grows.
• The presence of more radial nodes suggests more complex wave functions for electron probability.

For instance, if you're examining a 3p orbital with \( n = 3 \) and \( l = 1 \), the orbital will have:\[ 3 - 1 - 1 = 1 \text{ radial node} \]This means there's one particular distance from the nucleus where electrons cannot be located, distinct from the smooth, increasing probability seen close to the atomic nucleus.

Angular Nodes

Angular nodes describe regions where there is a zero probability of finding an electron due to the orientation or shape of the orbital. These nodes depend entirely on the azimuthal quantum number \( l \). The number of angular nodes is equal to \( l \), hence providing clues about the orbital's shape:

• For \( l = 0 \), an \( s \)-orbital has no angular nodes and is spherical in shape.
• With \( l = 1 \), a \( p \)-orbital features one angular node, dividing space into lobes around a nucleus.
• At \( l = 2 \), d orbitals have two angular nodes, showcasing even more complex structures.

Understanding angular nodes helps predict how orbitals align and overlap in different molecules, influencing molecular geometry and chemical bonding patterns.

Quantum Numbers

Quantum numbers are vital indicators that specify the properties of atomic orbitals and the electrons within them. There are four primary quantum numbers:

• **Principal Quantum Number (\( n \))**: Determines the size and energy of the orbital. Larger \( n \) suggests a greater energy level and larger orbital.
• **Azimuthal Quantum Number (\( l \))**: Defines the shape of the orbital. Values of \( l \) range from 0 to \( n-1 \), designating different shapes (s, p, d, f).
• **Magnetic Quantum Number (\( m_l \))**: Provides the orbital's orientation in space, varying between \(-l\) and \(+l\).
• **Spin Quantum Number (\( m_s \))**: Indicates the spin of the electron, either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

These quantum numbers collectively give a complete address to an electron's identity in an atom, dictating its energy, shape, and the spatial orientation of its orbital. They are essential for understanding complex atomic behaviors and have widespread implications in predicting chemical interactions.