Summary of "Radial and Angular wave functions | Atomic structure - Bsc 1st year inorganic chemistry"

Summary of "Radial and Angular wave functions | Atomic Structure - Bsc 1st year inorganic chemistry"

Main Ideas and Concepts

Introduction to Radial and Angular Wave Functions
- The lecture covers the definitions, representations, and features of radial and angular wave functions in Atomic Structure.
- Emphasis on understanding electron distribution around the nucleus.

Radial Wave Function

Definition: The Radial Wave Function describes the distribution (probability) of an electron as a function of the distance \( r \) from the nucleus. It depends on the principal quantum number \( n \) and the azimuthal quantum number \( l \).

Representation: Denoted as \( R_{nl}(r) \), where \( n \) and \( l \) are quantum numbers and \( r \) is the distance from the nucleus.

Features of Radial Wave Functions:

1s Orbital:
- Radial part depends mainly on an exponential term \( e^{-Zr/a_0} \), where \( Z \) is atomic number and \( a_0 \) is the Bohr Radius (~0.529 Å for hydrogen).
- The radial function decreases exponentially with increasing distance \( r \) from the nucleus.
2s Orbital:
- Radial function includes the term \( (2 - Zr/a_0) e^{-Zr/2a_0} \).
- The function decreases with \( r \) but becomes zero at a certain radius (node), specifically at \( r = 2a_0/Z \).
- The point where the radial function equals zero is called a node.
3s Orbital:
- The Radial Wave Function has two nodes (two points where it becomes zero).
- The function shows positive and negative values separated by nodes, with maxima and minima in between.
2p Orbital:
- Radial function depends on \( (r/a_0) e^{-r/2a_0} \).
- Zero only at \( r=0 \), so no nodes exist at a positive radius.
- The function remains positive and reaches a maximum before decreasing.
3p Orbital:
- Radial function shows an initial increase, reaches a maximum, then decreases, crosses zero (node), becomes negative, reaches a minimum, and finally approaches zero asymptotically.
Size of Orbital:
- The Radial Wave Function indicates the size of the orbital by showing where electron probability density is significant.
Similarity Among Orbitals:
- The first orbital of each type (1s, 2p, 3d, 4f) has no nodes and maintains a positive sign throughout.
- These orbitals have similar radial function shapes and exhibit maxima before decreasing asymptotically.

Angular Wave Function

Context: Electron position in 3D space is described by spherical coordinates: radius \( r \), polar angle \( \theta \), and azimuthal angle \( \phi \). Radial Wave Function depends on \( r \), while Angular Wave Function depends on \( \theta \) and \( \phi \).

Definition: The Angular Wave Function is the product of functions depending on angles \( \theta \) and \( \phi \), often denoted as \( Y(\theta, \phi) \).

Dependence: It depends on the azimuthal quantum number \( l \) and magnetic quantum number \( m \).

Features:

s Orbitals:
- Angular Wave Function is independent of \( \theta \) and \( \phi \), meaning it is constant in all directions.
- This results in spherical symmetry (electron density is uniform in all directions).
p and d Orbitals:
- Angular Wave Function depends on \( \theta \) and \( \phi \), making these orbitals directional with specific orientations in space (along x, y, z axes).
- Changes in \( \theta \) and \( \phi \) alter the directionality and shape of the orbital.

Complete Wave Function: The total wave function of an electron in an atom is the product of the Radial Wave Function and the Angular Wave Function: \[ \Psi(r, \theta, \phi) = R_{nl}(r) \times Y_l^m(\theta, \phi) \] Radial part determines the size of the orbital. Angular part determines the shape of the orbital.