Problem 17 - A Complete Resource Book in Chemistry for JEE Main

Question

Which one of the following expressions represent the electron probability function D? (a) \(4 \pi \mathrm{rr} \mathrm{dr} \psi^{2}\) (b) \(4 \pi \mathrm{r}^{2} \mathrm{dr} \psi\) (c) \(4 \pi \mathrm{rr}^{2} \mathrm{dr} \psi^{2}\) (d) \(4 \pi \mathrm{dr} \psi\)

Step-by-Step Solution

Answer

The correct answer is (c) \(4 \pi \mathrm{rr}^{2} \mathrm{dr} \psi^{2}\).

Step 1: Understanding the Electron Probability Function

The electron probability function, also known as the radial probability distribution, describes the likelihood of finding an electron within a thin spherical shell at a radial distance from the nucleus. It is mathematically expressed as the probability density function, which is the squared wave function, \(\psi^2\), multiplied by an element of volume in spherical coordinates.

Step 2: Identifying the Correct Formula Structure

The volume element in spherical coordinates for the radial probability distribution involves a shell of thickness \dr\ and a spherical surface area, so the expression needs \(4\pi r^2\), representing the surface area of the sphere at radius \(r\), and the probability density, \(\psi^2\). Correctly forming this gives \(4\pi r^2 dr \psi^2\).

Step 3: Analyzing Given Options

Review each option to find the match:- Option (a) is \(4\pi rr dr \psi^2\).- Option (b) is \(4\pi r^2 dr \psi\).- Option (c) is \(4\pi rr^2 dr \psi^2\).- Option (d) is \(4\pi dr \psi\).The expression that matches the required form is option (c) with the corrected arrangement as \(4\pi r^2 dr \psi^2\).

Step 4: Conclusion of the Correct Expression

The electron probability function should include both the squared wave function \(\psi^2\) and the term \(4\pi r^2 dr\) for the radial shell's volume. Among the given choices, option (c) best fits the criteria needed for the radial probability distribution.

Key Concepts

Radial Probability Distribution

The radial probability distribution is invaluable when trying to determine the likelihood of locating an electron at a particular distance from the nucleus. Imagine concentric spherical shells, spaced incrementally outward from the nucleus. The radial probability helps us visualize where the electron is most likely to linger. This concept revolves around the probability density function multiplied by a spherical shell's volume segment, which involves integrating over spherical coordinates.
To put it simply:

• It represents probability over thin spherical layers.
• Utilizes spherical coordinates for calculations.
• Aids in understanding electron positioning within an atom.

Once you combine the squared value of the wave function, \( \psi^2 \), with the volume element of a shell, \( 4\pi r^2 dr \), you achieve the radial probability distribution. This elegantly helps unravel the probable electron positioning mystery.

Wave Function

The wave function, often denoted as \( \psi \), is central to quantum mechanics. It's a mathematical description that provides insight into the behavior of an electron. Though \( \psi \) itself doesn't yield a probability directly, its square \( \psi^2 \) gives us the probability density.

• Wave function explains the wavelike nature of particles.
• The absolute square \( |\psi|^2 \) is the probability density.
• Utilized in Schrödinger's equation for atomic models.

The wave function helps in drawing predictions about particle locations and states. In atomic structures, especially, it is pivotal in anticipating electron arrangements. By solving Schrödinger's equation for an atom, predictions about energy levels and electron configurations become accessible.

Probability Density Function

In quantum mechanics, the probability density function derived from a wave function is a cornerstone. Given the wave-like nature of electrons, this function evaluates the electron's presence in a particular position.
This stands distinct from a general probability function by providing a density—meaning it doesn't just consider presence, but presence per unit volume.

• Crucial for understanding electron spatial distribution.
• Results from squaring the wave function: \( \psi^2 \).
• Integrating this over a volume gives the actual probability.

The density description is particularly useful because it allows physicists to map out the electron cloud of an atom, giving insight into aspects like chemical bonding and molecular shape based on electron position likelihood.

Spherical Coordinates

Spherical coordinates work splendidly when dealing with problems that showcase inherent symmetry about a point. In atomic physics, this is incredibly beneficial when assessing electron positioning around a nucleus, which is innately spherical.

• Defined by three values: radial distance, polar angle, and azimuthal angle.
• A powerful tool for problems with central symmetry.
• Streamlines the integration process in quantum mechanics calculations.

This system aids particularly in formulating the radial probability distribution, where the integration process becomes far more straightforward than it would using Cartesian coordinates. Understanding this system is key when exploring atomic models and electron densities.

Radial Distance

Radial distance is a measure from a central point, useful in describing how far an electron is from the nucleus in an atom. Within the context of spherical coordinates, it highlights one primary component of the system.

• Acts as a critical element in spherical coordinates.
• Helps determine electron positions in terms of distance from nucleus.
• Essential for calculating radial probability distribution.

In quantum mechanics, radial distance is utilized to help visualize and calculate the likelihood of electron presence in different shells around the nucleus. By focusing on radial distance, it’s easier to predict aspects like electron cloud size and probable electron locations throughout the atom's various energy levels.