Problem 101 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

For an electron in a hydrogen atom, the wave function is given by \(\psi_{1 \mathrm{~s}}\) \(=(\pi / \sqrt{2}) e^{-r / a_{0}}\), where \(a_{0}\) is the radius of first Bohr's orbit and \(r\) is the distance from the nucleus with which probability of finding electron varies. What will be the ratio of probabilities of finding electrons at the nucleus to first Bohr's orbit \(a_{\mathrm{o}} ?\) (a) 0 (b) \(e\) (c) \(e^{2}\) (d) \(\frac{1}{e^{2}}\)

Step-by-Step Solution

Answer

\(e^{2}\)

Step 1: Write down the probability density

The probability of finding an electron at a particular location is proportional to the square of the magnitude of its wave function, i.e., the probability density is given by \(P(r) = |\psi(r)|^2\). In this case, the wave function is given by \(\psi = \frac{\pi}{\sqrt{2}} e^{-r/a_{0}}\), so the probability density becomes \(P(r) = |\frac{\pi}{\sqrt{2}} e^{-r/a_{0}}|^2 = \frac{\pi^2}{2} e^{-2r/a_{0}}\).

Step 2: Calculate probability at the nucleus (r = 0)

Substitute \(r = 0\) into the probability density formula to find the probability at the nucleus: \(P(0) = \frac{\pi^2}{2} e^{-2\cdot0/a_{0}} = \frac{\pi^2}{2}\).

Step 3: Calculate probability at the first Bohr orbit (r = a₀)

Substitute \(r = a_0\) into the probability density formula to find the probability at first Bohr's orbit: \(P(a_{0}) = \frac{\pi^2}{2} e^{-2\cdot a_{0}/a_{0}} = \frac{\pi^2}{2} e^{-2}\).

Step 4: Calculate the ratio of probabilities

Divide the probability at the nucleus by the probability at first Bohr's orbit: \(\text{Ratio} = \frac{P(0)}{P(a_{0})} = \frac{\frac{\pi^2}{2}}{\frac{\pi^2}{2} e^{-2}} = e^{2}\). Thus, the correct answer is that the ratio of probabilities is \(e^2\).

Key Concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces the concept that energy, typically thought of as continuous, is actually quantized into discrete amounts called quanta.

Unlike classical mechanics, quantum mechanics does not adhere to the principles of determinism. Rather, it deals with probabilities and uncertainty, which is where the wave function comes into play. The wave function, a central concept in quantum mechanics, is a mathematical description of the quantum state of a system. Its square is related to the probability of finding a particle in various locations, a cornerstone in solving problems like the one given for the hydrogen atom.

Bohr's Model

Bohr's model of the atom was a revolutionary step in understanding atomic structure. Proposed by Niels Bohr in 1913, this model introduced the idea that electrons orbit the nucleus in specific paths or shells without emitting radiation, contrary to the predictions of classical electromagnetism.

Each orbit in Bohr's model represents a certain energy level, and an electron can jump from one level to another by absorbing or emitting a photon with energy equal to the difference between these levels. The first orbit, known as the first Bohr's orbit, holds particular significance as it represents the ground state, the lowest energy state of an electron in the hydrogen atom. Bohr's model is critical for understanding the probability density of electrons because it sets important constraints for their positions and energies.

Wave Function of Electron

The wave function of an electron in quantum mechanics is a complex function whose absolute square gives the probability density for finding the electron at a given point in space. Specifically, the textbook problem provides a wave function \(\psi_{1s}\) for the ground state, or '1s' state, of a hydrogen atom, revealing the nuance of electron behavior at a quantum level.

The electron's wave function is a key element in determining its behavior, as it encompasses all possible information about an electron's position and momentum. In simple systems like the hydrogen atom, these wave functions can be calculated precisely and have distinct, characteristic shapes, often referred to as orbitals. Understanding the wave function is essential for interpreting the behavior of electrons within atoms and for calculating the probability of their locations.

Probability Distribution

Probability distribution in quantum mechanics involves the likelihood of finding a quantum particle, such as an electron, in a particular region of space. It's determined by the squared magnitude of the wave function, which acts as a mathematical description of the electron's location and movement within an atom.

In the provided exercise, the probability distribution tells us how likely we are to find an electron at various distances from the nucleus of a hydrogen atom. The calculation of probability density at specific points, such as the nucleus or the first Bohr orbit, highlights the quantum mechanical nature of electrons—that they are not just particles but also exhibit wave-like properties. By comparing the probability densities at these two points, we can gain insight into the electron's spatial distribution around the nucleus.