Question

How many radial nodes does each of the following orbitals (a) \(2 s ;(b) 4 s ;(c) 3 p ;(d) 5 d\) possess: \(l ;\) (e) \(1 s ;(f) 4 p ?\)

Short answer

2s: 1 node, 4s: 3 nodes, 3p: 1 node, 5d: 2 nodes, 1s: 0 nodes, 4p: 2 nodes.

Step-by-step solution

Understand the Concept of Radial Nodes

Radial nodes are regions in an atom where the probability of finding an electron is zero. The number of radial nodes, denoted as n_radial, for an atomic orbital can be calculated using the formula: \[ n_{\text{radial}} = n - l - 1 \]where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number.

Calculate Radial Nodes for Each Orbital

Let's apply the formula to each orbital provided:- **(a) 2s orbital:** - n = 2, l = 0 (s orbitals have l = 0) - \( n_{\text{radial}} = 2 - 0 - 1 = 1 \)- **(b) 4s orbital:** - n = 4, l = 0 - \( n_{\text{radial}} = 4 - 0 - 1 = 3 \)- **(c) 3p orbital:** - n = 3, l = 1 (p orbitals have l = 1) - \( n_{\text{radial}} = 3 - 1 - 1 = 1 \)- **(d) 5d orbital:** - n = 5, l = 2 (d orbitals have l = 2) - \( n_{\text{radial}} = 5 - 2 - 1 = 2 \)- **(e) 1s orbital:** - n = 1, l = 0 - \( n_{\text{radial}} = 1 - 0 - 1 = 0 \)- **(f) 4p orbital:** - n = 4, l = 1 - \( n_{\text{radial}} = 4 - 1 - 1 = 2 \)

Key concepts

Understanding Quantum Numbers

Quantum numbers are essential in determining the properties and behaviors of electrons in atoms. Each electron in an atom is described by a set of four quantum numbers, which are:

• Principal Quantum Number (): This number describes the energy level of an electron in an atom and is always a positive integer (1, 2, 3,...). The greater the value of the principal quantum number, the further away the electron is from the nucleus and the higher its energy.
• Azimuthal Quantum Number (\( l \)): Also known as the angular momentum quantum number, \( l \) determines the shape of the atomic orbital. It takes on values from 0 to \( n-1 \). For example, when \( l = 0 \), it represents an "s" orbital, \( l = 1 \) is a "p" orbital, \( l = 2 \) a "d" orbital, and so on.
• Magnetic Quantum Number (\( m_l \)): This quantum number indicates the orientation of an orbital in space and can have values ranging from \( -l \) to \( l \).
• Spin Quantum Number (\( m_s \)): Reflects the spin direction of an electron, which can either be +1/2 or -1/2.

By understanding quantum numbers, especially the principal and azimuthal ones, we can determine the structure and behavior of different atomic orbitals.

Exploring Atomic Orbitals

Atomic orbitals are regions around the nucleus of an atom where electrons are likely to be found. These orbitals have distinct shapes and sizes depending on their quantum numbers. The three common types of orbitals are s, p, and d orbitals:

• s orbitals: These are spherical in shape. Since the azimuthal quantum number \( l \) for s orbitals is 0, they only differ in size based on the principal quantum number. For example, a 2s orbital is larger than a 1s orbital but shares the same spherical shape.
• p orbitals: Characterized by a dumbbell shape, p orbitals have \( l = 1 \). They exist in sets of three at right angles to each other, termed as \( p_x \), \( p_y \), and \( p_z \).
• d orbitals: With more complex shapes, d orbitals have \( l = 2 \) and appear in sets of five. These orbitals have unique, cloverleaf patterns and some even have donut shapes around the nucleus.

Understanding these shapes and configurations is crucial for grasping how electrons occupy orbitals and how atoms bond with one another.

Probability Distribution and Radial Nodes

The concept of probability distribution in atomic orbitals explains where an electron is likely to be found around the nucleus. The electron density is not uniform, and there are specific regions, known as nodes, where the probability of finding an electron is zero.

Radial nodes are a type of node and are represented by the formula:\[ n_{\text{radial}} = n - l - 1\]where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number.

• For example, in a 2s orbital, the presence of one radial node means there's a spherical shell around the nucleus where there's no electron density.
• In larger orbitals like the 4s or 5d, multiple radial nodes indicate more such regions of zero probability between energy levels.

Understanding this concept helps in visualizing how electrons are arranged and the nature of chemical bonding in different elements.