How many radial and angular nodes does a 5 fxyz orbital?
The correct answer and explanation is:
Correct Answer:
- Radial nodes: 1
- Angular nodes: 3
Explanation:
The number of nodes in an atomic orbital can be divided into two categories: radial nodes and angular nodes. Understanding how to find these requires knowing the quantum numbers associated with the orbital.
For the 5f orbital labeled as 5fxyz, the principal quantum number (n) is 5, and the azimuthal quantum number (l) for any f orbital is 3 (since s = 0, p = 1, d = 2, f = 3).
The total number of nodes in any orbital is given by the formula n – 1. Therefore, the 5f orbital has a total of 5 – 1 = 4 nodes.
Next, the total number of angular nodes depends on the value of l. For f orbitals, l = 3, so there are 3 angular nodes. Angular nodes are regions where the probability of finding an electron is zero due to the angular part of the wavefunction. These nodes come from the shape and orientation of the orbital in three-dimensional space. For example, the 5fxyz orbital has complex angular shapes that include nodal planes and cones.
The remaining nodes are radial nodes. Radial nodes arise from the radial part of the wavefunction and are spherical surfaces at specific distances from the nucleus where the probability of finding an electron is zero. To find the number of radial nodes, subtract the number of angular nodes from the total nodes:
Radial nodes = (n – 1) – l = 5 – 1 – 3 = 1
So, the 5fxyz orbital has 1 radial node and 3 angular nodes. This means an electron in this orbital has one spherical surface within the orbital where it cannot be found and three angular regions (planes or cones) that define its directional shape and probability distribution. This pattern helps explain the shapes and energies of orbitals in atoms.