Question

Radial nodes present in \(3 \mathrm{~s}\) and \(2 \mathrm{p}\) orbitals are respectively: (a) 0,2 (b) 2,0 (c) 2,1 (d) 1,2

Short answer

The answer is (b) 2,0.

Step-by-step solution

Understanding the Radial Node Formula

The formula for calculating radial nodes is given by: \( n-l-1 \), where \(n\) is the principal quantum number, and \(l\) is the azimuthal quantum number (0 for \(s\), 1 for \(p\), 2 for \(d\), etc.).

Calculate Radial Nodes for \(3s\) Orbital

For the \(3s\) orbital, \(n = 3\) and \(l = 0\). Using the formula:\[\text{Radial Nodes} = 3 - 0 - 1 = 2\]Thus, the \(3s\) orbital has 2 radial nodes.

Calculate Radial Nodes for \(2p\) Orbital

For the \(2p\) orbital, \(n = 2\) and \(l = 1\). Applying the formula:\[\text{Radial Nodes} = 2 - 1 - 1 = 0\]So, the \(2p\) orbital has 0 radial nodes.

Identify the Correct Option

Based on the calculations, the \(3s\) orbital has 2 radial nodes and the \(2p\) orbital has 0 radial nodes. Thus, the answer is (b) 2,0.

Key concepts

Radial Nodes

In quantum mechanics, radial nodes are specific points within an atomic orbital where the probability of finding an electron is exactly zero. These nodes are critical in understanding the shape and structure of orbitals. They provide insight into how electrons are organized around the nucleus.

The number of radial nodes can be calculated using the formula:

• Radial Nodes = \(n - l - 1\)

Here, \(n\) is the principal quantum number, and \(l\) is the azimuthal quantum number. This formula helps us find the points where electron density is absent.

For instance, in the case of the \(3s\) orbital, where \(n = 3\) and \(l = 0\), the number of radial nodes is \(3 - 0 - 1 = 2\). This means there are two radial nodes, indicating two spherical surfaces where finding an electron is not possible. Conversely, the \(2p\) orbital, with \(n = 2\) and \(l = 1\), yields \(2 - 1 - 1 = 0\) radial nodes, indicating no such spherical surfaces of zero electron probability.

Principal Quantum Number

The principal quantum number, denoted as \(n\), is a fundamental concept in quantum mechanics that determines the size and energy level of an electron's orbit within an atom. The larger the value of \(n\), the higher the energy level, and the more spread out the electron shell is from the nucleus.

The principal quantum number is integral in predicting the electron's behavior:

• \(n\) can be any positive integer (1, 2, 3, etc.).
• Higher \(n\) values correspond to electrons found in outer shells, further from the nucleus.

In our exercise example, the \(3s\) orbital has \(n = 3\), indicating it's in the third shell. Meanwhile, the \(2p\) orbital has \(n = 2\), meaning it resides in the second shell. Understanding \(n\) helps one comprehend not only the orbital's location within an atom but also its energy compared to other orbitals.

Azimuthal Quantum Number

The azimuthal quantum number, represented as \(l\), defines the shape and subshell or type of the atomic orbital. It provides deeper insight into the electron's angular momentum and helps distinguish between different orbital shapes within the same energy level.

Here are some key points to remember:

• \(l\) values range from 0 to \(n-1\) for each principal quantum number \(n\).
• Its value dictates the shape of the orbital:
  • \(l = 0\) for s orbitals (spherical shape)
  • \(l = 1\) for p orbitals (dumbbell shape)
  • \(l = 2\) for d orbitals, and so on

For the \(3s\) orbital, \(l=0\), indicating a spherical shape, while the \(2p\) orbital's \(l=1\) reflects a dumbbell shape. The azimuthal quantum number thus not only helps in classifying the orbital's shape but also plays a crucial role in determining other properties like angular momentum, essential for detailed atomic modeling.