Problem 2 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The correct statement about probability density (except at infinite distance from nucleus) is: [Main Sep. \(\mathbf{0 5}, \mathbf{2 0 2 0}\) (II)] (a) It can be zero for \(1 s\) orbital (b) It can be negative for \(2 p\) orbital (c) It can be zero for \(3 p\) orbital (d) It can never be zero for \(2 s\) orbital

Step-by-Step Solution

Answer

The correct statement is (c) It can be zero for 3p orbital.

Step 1: Understanding Probability Density

Probability density is a measure used in quantum mechanics to find the likelihood of an electron being at a specific point in space. It is a non-negative function and cannot be negative.

Step 2: Identifying the Possibilities for 1s Orbital

For the 1s orbital, the probability density is highest at the nucleus and decreases as you move away. However, the probability density does not become zero unless at infinite distance. Nevertheless, there are radial nodes or points where density could theoretically be zero, but these do not occur for 1s.

Step 3: Checking the 2p Orbital

The 2p orbital has a dumbbell shape with a nodal plane where the probability density is zero. It cannot, however, be negative.

Step 4: Evaluating the 3p Orbital

The 3p orbital has radial nodes, leading to points where the probability density is zero. This is due to the wave function having zero crossings (nodes) within the 3p orbital.

Step 5: Verifying the 2s Orbital

The 2s orbital has a spherical shape and also contains one radial node. This means that within this orbital, there is a spherical surface at a certain distance from the nucleus where the probability density is zero.

Step 6: Conclusion Based on Step Analysis

From the analysis, probability density can indeed be zero for the 3p orbital where radial nodes exist. Hence, statement (c) is correct.

Key Concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. Unlike classical mechanics, which uses deterministic equations to predict the behavior of matter, quantum mechanics introduces probability as a principal feature of the universe. This means that, rather than predicting an exact outcome, quantum mechanics provides probabilities of different outcomes. This is where the concept of probability density comes into play, giving us insight into the likelihood of finding an electron at a particular location in an atomic orbital. Quantum mechanics is essential for understanding how atoms bond in chemistry and how complex molecules interact.

Atomic Orbitals

Atomic orbitals are regions in an atom where there is a high probability of finding electrons. These orbitals can be thought of as clouds around the nucleus within which electrons are likely to be found. Each orbital corresponds to a different energy level and shape, determined by quantum numbers. There are several types of atomic orbitals, including s, p, d, and f, each with its characteristic shape:

• s orbitals are spherical in shape.
• p orbitals are dumbbell shaped.
• d and f orbitals have more complex shapes.

The shape and structure of these orbitals help determine the chemical and physical properties of an element, including its reactivity and the type of bonds it can form.

Radial Nodes

Radial nodes are specific points or spherical surfaces within atomic orbitals where the probability density of finding an electron is zero. These nodes occur as a result of the wave nature of electrons. For each increase in the principal quantum number (n), there are certain radial points where the wave function that describes the electron’s behavior has zero amplitude.

• In the 2s orbital, there is one radial node.
• For the 3p orbital, two radial nodes are present.

Radial nodes are crucial in determining the geometry of orbitals and help explain why, for instance, electrons might be found in certain regions more frequently than in others.

Nodal Planes

Nodal planes are flat (two-dimensional) regions in atomic orbitals where the probability density of finding an electron is exactly zero. Unlike radial nodes that are spherical, nodal planes divide the orbital into distinct lobes, often with differing phases of the wave function.

• The 2p orbitals have a single nodal plane perpendicular to the axis along which the orbital is oriented.
• Each orbital type (such as 3p or 3d) may contain various numbers of nodal planes based on their shape and orientation.

Understanding nodal planes helps clarify the symmetry and shape of molecular orbitals, which is key in predicting the behavior of molecules during chemical reactions.