Problem 45 - Chemistry and Chemical Reactivity

Question

How many nodal surfaces through the nucleus (planar nodes) are associated with each of the following orbitals? (a) \(2 s\) (b) \(5 d\) (c) \(5 f\)

Step-by-Step Solution

Answer

(a) 0 planar nodes, (b) 2 planar nodes, (c) 3 planar nodes.

Step 1: Understand Nodal Surfaces

Nodal surfaces are areas where the probability of finding an electron is zero. For orbitals, planar nodes (angular nodes) are determined by the quantum number \(\ell\). Each angular node corresponds to one planar or nodal surface.

Step 2: Determine the Angular Momentum Quantum Number \(\ell\)

For each orbital, identify the angular momentum quantum number \(\ell\) based on the type of orbital. - \(s\) orbitals have \(\ell = 0\)- \(p\) orbitals have \(\ell = 1\)- \(d\) orbitals have \(\ell = 2\)- \(f\) orbitals have \(\ell = 3\).

Step 3: Apply the Planar Node Formula

The number of planar nodes is equal to \(\ell\). - For each orbital, use the formula. - (a) Since the \(2s\) orbital has \(\ell = 0\), it has 0 planar nodes. - (b) The \(5d\) orbital has \(\ell = 2\), resulting in 2 planar nodes.- (c) The \(5f\) orbital has \(\ell = 3\), resulting in 3 planar nodes.

Key Concepts

Nodal Surfaces

Nodal surfaces are fascinating areas in atomic orbitals where the likelihood of locating an electron is exactly zero. This means electrons simply cannot be found at these surfaces. These surfaces are critical in understanding how electrons behave and distribute themselves within an atom. In general, there are two types of nodal surfaces: radial and angular. Radial nodes occur at certain distances from the nucleus, while angular nodes relate to the shapes of orbitals. The presence of these surfaces is determined by the quantum numbers of an electron in a specific orbital. Angular nodes, often referred to as planar nodes, are connected to the orbital's geometry and are defined by the angular momentum quantum number \( \ell \). Understanding nodal surfaces helps in visualizing the complex shapes of electron clouds around a nucleus, making it easier to predict chemical behavior.

Angular Momentum Quantum Number

Central to the behavior of electrons in orbitals is the angular momentum quantum number, denoted by the symbol \( \ell \). This quantum number essentially gives an idea about the shape and number of angular nodes in an orbital. It is connected directly with the orbital type. Each kind of atomic orbital (s, p, d, f, etc.) has a specific \( \ell \) value:

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