Problem 76 - A Complete Resource Book in Chemistry for JEE Main

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

Step-by-Step Solution

Answer

The correct answer is (b) 2,0.

Step 1: Understanding the Concept of Radial Nodes

Radial nodes can be calculated using the formula: \( n - l - 1 \), where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number. Radial nodes represent the points where the probability density of finding an electron is zero in an atom.

Step 2: Calculating Radial Nodes for 3s Orbital

For a \( 3s \) orbital, the principal quantum number \( n = 3 \) and \( l = 0 \) (since it's an 's' orbital). Plug these into the formula for radial nodes: \( 3 - 0 - 1 = 2 \). Thus, the radial node count for the \( 3s \) orbital is 2.

Step 3: Calculating Radial Nodes for 2p Orbital

For a \( 2p \) orbital, \( n = 2 \) and \( l = 1 \) (since it's a 'p' orbital). Using the formula, we have \( 2 - 1 - 1 = 0 \). Therefore, the radial node count for the \( 2p \) orbital is 0.

Step 4: Matching the Calculated Values with the Options

Comparing our calculated results with the provided options: \( 3s \) has 2 radial nodes, and \( 2p \) has 0 radial nodes. The correct option is (b) 2,0.

Key Concepts

Principal Quantum Number

In quantum mechanics, the principal quantum number, denoted by \( n \), is a crucial element for defining the energy level of an electron in an atom. It primarily determines the energy and size of an orbital.
The principal quantum number is always a positive integer (such as 1, 2, 3, etc.) and tells us how far the electron is most likely to be found from the nucleus.

• An electron with a lower principal quantum number is closer to the nucleus, hence more tightly bound due to a stronger electrostatic pull.
• Conversely, a higher \( n \) indicates that the electron is more spread out, residing farther from the nucleus, and contributing to larger atomic size.

In the context of our exercise, the \( 3s \) orbital has a principal quantum number of 3, while the \( 2p \) orbital has a principal quantum number of 2. This helps us understand the spacing and energy levels that affect radial nodes and electron dynamics within these orbitals.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( l \), plays a vital role in quantum chemistry as it defines the shape of the electron cloud, known as an orbital. It essentially adds to the refinement of the electron's position by influencing its angular momentum.

• The azimuthal quantum number can take any integer values from 0 to \( n-1 \), where \( n \) is the principal quantum number.
• These values correspond to the type of orbital: \( l = 0 \) for an \( s \)-orbital, \( l = 1 \) for a \( p \)-orbital, \( l = 2 \) for a \( d \)-orbital, and \( l = 3 \) for an \( f \)-orbital.

In our example, the \( 3s \) orbital has an azimuthal quantum number of 0, while the \( 2p \) orbital has \( l = 1 \). These values crucially impact the number of radial nodes, as seen in the formula \( n - l - 1 \), guiding us to the precise number of points where the probability of finding an electron is zero.

Electron Probability Density

Electron probability density is an essential concept in understanding how electrons are distributed in an atom. It refers to the probability of finding an electron in a particular region of space around the nucleus.
In quantum mechanics, unlike classical electrons that follow fixed paths, electrons exhibit wave-like behavior. This means their position is not defined precisely; instead, we talk in terms of probabilities.

• The probability density function allows chemists and physicists to predict where an electron is likely to be found.
• Regions of high electron probability density mean a higher chance of finding an electron, while nodal points (where the electron probability is zero) mean the electron cannot be found there.

In relation to radial nodes, the concept is crucial. Radial nodes, calculated using the formula \( n - l - 1 \), are the points in the radial distribution where the electron probability density drops to zero. Understanding this helps us comprehend the internal structure of the atom and how electron energy levels are organized.