Problem 72 - General Chemistry: Principles and Modern Applications

Question

Identify the orbital that has (a) two radial nodes and one angular node; (b) five radial nodes and zero angular nodes; (c) one radial node and four angular nodes.

Step-by-Step Solution

Answer

The orbitals are: (a) 3p; (b) 5s; (c) 5f

Step 1: Identify the Orbital in case (a) two radial nodes and one angular node.

For this case, since there are two radial nodes and one angular node, the total number of nodes or 'n', will be the sum of these two giving 'n' = 3. Also, for one angular node, the orbital must be of 'p type' which is denoted as 'l = 1'. Therefore, the orbital can be identified as '3p'.

Step 2: Identify the Orbital in case (b) five radial nodes and zero angular nodes.

For this case, since there are five radial nodes and zero angular nodes, the total number of nodes or 'n', will be the sum of these two giving 'n' = 5. And for zero angular nodes, the orbital must be of 's type' which is denoted as 'l = 0'. Therefore, the orbital can be identified as '5s'.

Step 3: Identify the Orbital in case (c) one radial node and four angular nodes.

For this case, since there is one radial node and four angular nodes, the total number of nodes or 'n', will be sum of these two giving 'n' = 5. And for four angular nodes, the orbital must be of 'f type' which is denoted as 'l = 3'. Therefore, the orbital can be identified as '5f'.

Key Concepts

Orbital Types

In chemistry, orbitals are regions around the nucleus of an atom where electrons are most likely to be found. Different types of orbitals exhibit unique shapes and orientations within the atom. They are typically categorized as s, p, d, and f orbitals.

• s-orbitals: These orbitals are spherical in shape and can be found in any energy level (n). Each energy level has one s-orbital.


• p-orbitals: Shaped like dumbbells, p-orbitals exist in sets of three and come in any energy level from n=2 and upwards.


• d-orbitals: These orbitals have a complex, cloverleaf shape and come in sets of five. They are present from energy level n=3 onwards.


• f-orbitals: With even more complex shapes than the d-orbitals, f-orbitals come in sets of seven and are found in energy levels starting from n=4.

Understanding these orbital types is vital in predicting how atoms bond and react with one another. You can visualize the shape of these orbitals as clouds representing the most likely locations where an electron can be found. This can help clarify why certain molecules have their specific shapes and properties.

Radial Nodes

Radial nodes are regions within atomic orbitals where the probability of finding an electron is exactly zero. These nodes appear as concentric spheres around the nucleus, where the electron density falls to zero. The number of radial nodes an orbital has is determined by subtracting the angular momentum quantum number, \( l \), from the principal quantum number, \( n \).

The formula is:

• Number of radial nodes = \( n - l - 1 \).

For example, a 3p orbital will have \( 3 - 1 - 1 = 1 \) radial node. Understanding radial nodes is necessary for predicting the behavior and reactivity of atoms, as they influence how electrons are distributed in space. They also play a role in quantum mechanical calculations involving atoms and molecules.

Angular Nodes

Angular nodes are planes or surfaces where the probability of finding an electron is zero. These are caused by the orbital's shape and the orientation of its lobes rather than its distance from the nucleus. For example, in p-orbitals, the angular nodes align with planes that bisect the dumbbell shapes, and they manifest as nodes along the axes in the Cartesian coordinate system.

• Number of angular nodes = \( l \), where \( l \) is the azimuthal quantum number.

This quantum number determines the overall shape of the orbital. More angular nodes typically mean more complex shapes, as seen with d and f orbitals. For instance, a d orbital has l = 2, thus having 2 angular nodes, while an f orbital has l = 3, resulting in 3 angular nodes. The angular nodes are significant in understanding which directions electrons may pair, affecting molecular geometry and bonding.