Chapter 7, Problem 7.49QP

Interpretation Introduction

Interpretation:

A boundary surface diagram useful in representing an atomic orbital should be explained.

Concept Introduction:

The most important characters of atomic orbitals are their overall shapes and relative sizes which are sufficiently represented by boundary surface diagrams.  It is resultant of the solution of Schrodinger wave equation.  The orbitals are drawn in 3 dimensional which is difficult to draw.  In boundary surface diagrams, cross sections are used to get the general idea of their shapes.  For example, the boundary surface diagrams of ‘p’ orbitals show that each ‘p’ orbital has two lobes on opposite sides of the nucleus.

The exact position and momentum of an electron cannot be determined (according to Heisenberg uncertainty principle).  The probability density of finding an electron in a particular region is calculated.  Boundary surface diagram is a boundary surface or a contour surface drawn in a space for an orbital on which the value of probability density |ψ|² is constant.  The boundary surface diagram of constant probability density is considered as a good and acceptable approximation of shape of orbital if the boundary surface encloses the region or volume with probability density of more than 90%.  This means that the boundary surface enclosing a constant probability density of let’s say 50% won’t be considered good.

The boundary surface diagram is not taken with constant probability density of 100%.  The reason is at any distance from the nucleus, the probability density of finding an electron is never zero.  It will always have some finite value, so it is not possible to draw a boundary surface diagram which encloses region with 100% probability density.

Feature of boundary surface diagram

Shape of the surface diagram: The boundary surface diagram of an orbital is independent of principle quantum number.  For example: The boundary surface diagram of s orbital is spherical in shape, so it will be spherical for 1s, 2s, 3s and 4s or for any general ‘ns’.  The shape doesn’t depend on the principle quantum number (n).

Size of the surface diagram: The boundary surface diagram of an orbital increase in size or volume with increase in the value of principle quantum number (n).

Nodes in the surface diagram: Nodes are the region with very low probability density typically which goes to zero.  There are (n-1) nodes in the boundary surface diagram of s-orbital with ‘n’ principle quantum number.  This kind of nodes is also observed in the surface diagram of p, d and f orbitals.