Problem 59 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Consider the hydrogen atom to be a proton embedded in a cavity of radius \(a_{0}\) (Bohr radius) whose charge is neutralised by the addition of an electron to the cavity in vacuum, infinitely slowly. Estimate the average total energy of an electron in its ground state in a hydrogen atom as the work done in the above neutralisation process. Also, if the magnitude of the average kinetic energy is half the magnitude of the average potential energy, find the average potential energy.

Step-by-Step Solution

Answer

The average potential energy is \(-27.2 \, \text{eV}\).

Step 1: Identify the Energy Concept

The average total energy of an electron in a hydrogen atom's ground state can be computed by considering the energy required to bring an electron from infinity to the Bohr radius, \(a_0\). This process is analogous to calculating the work done by an electric field on the electron, which is initially bound to a system with its potential energy defined by the electron-proton interaction.

Step 2: Calculate Potential Energy

The potential energy \(V\) of an electron in a hydrogen atom at the Bohr radius is given by the formula: \[ V = -\frac{k e^2}{a_0} \]where \(k\) is Coulomb's constant \(8.99 \times 10^{9} \, \text{N m}^2/\text{C}^2\), \(e\) is the elementary charge \(1.6 \times 10^{-19} \, \text{C}\), and \(a_0\) is the Bohr radius \(5.29 \times 10^{-11} \, \text{m}\). Substitute these values to compute \(V\).

Step 3: Compute Kinetic Energy

According to the problem statement, the magnitude of the average kinetic energy \(T\) is half the magnitude of the average potential energy \(V\). Since potential energy in a bound system like a hydrogen atom is negative, \[ |T| = -\frac{1}{2} V \].

Step 4: Calculate Total Energy

The total energy \(E\) of the system can be given by the sum of kinetic and potential energy: \[ E = T + V \].Substituting the relationship \(T = -\frac{1}{2}V\) into this equation, we find \[ E = -\frac{1}{2}V + V = \frac{1}{2}V \].

Step 5: Find Average Potential Energy

Since \(E = \frac{1}{2}V\), and we calculated \(V\) in Step 2, we can find \(V\) as \[ V = 2E \].Use the fact that the total energy of an electron in the ground state of hydrogen is \(-13.6 \, \text{eV}\), we can find the average potential energy in electron volts.

Key Concepts

Bohr Radius

The Bohr radius is a fundamental concept when dealing with the hydrogen atom. It's named after Niels Bohr, who proposed a model of the atom that helps us understand atomic structure. The Bohr radius ( a_0 e) represents the most probable distance between the proton and electron in a hydrogen atom in its ground state.
At a_0 = 5.29 imes 10^{-11} ext{m} e), this distance indicates how closely the electron orbits its proton under the influence of electrostatic forces. This radius is derived by balancing the attractive force between the negatively charged electron and the positively charged proton with the centrifugal force due to the electron's motion around the nucleus.
By understanding the Bohr radius, we can dive deeper into various atomic interactions and energy calculations within atoms.

Potential Energy

In the context of the hydrogen atom, potential energy (Ve) plays a critical role as it describes the energy due to the position of the electron relative to the proton at the Bohr radius.
The potential energy of an electron in a hydrogen atom is calculated using the formula: \[ V = -\frac{k e^2}{a_0} \]where k = 8.99 \times 10^{9} \, \text{N m}^2/\text{C}^2e) is Coulomb's constant, e = 1.6 \times 10^{-19} \, \text{C}e) is the elementary charge, and a_0e) is the Bohr radius.
This equation essentially quantifies the attractive electrostatic interaction between the proton and the electron. The negative sign indicates that energy is released when the electron is brought from infinity to the Bohr radius; hence, the system's potential energy decreases.

Kinetic Energy

The kinetic energy of an electron in a hydrogen atom, denoted by Te), reflects the energy due to its motion around the nucleus. In the quantum mechanical view, the electron orbits the nucleus under the influence of the attractive force from the proton.
According to the principle outlined in the exercise, the magnitude of the electron's average kinetic energy is half that of its potential energy but positive:
\[ |T| = -\frac{1}{2}V \]This relationship comes from the balance between forces acting on the electron in a stable orbit. The kinetic energy ensures that the electron neither crashes into the proton nor escapes it entirely.
On calculating, we find that if the potential energy is pegged at a specific negative value, kinetic energy will automatically adjust to be half that value in magnitude, but positive.

Electron Proton Interaction

The interaction between an electron and a proton is fundamental to understanding the behavior of hydrogen atoms. This interaction is primarily governed by electrostatic forces, where the negatively charged electron is attracted to the positively charged proton.
This attractive force results from Coulomb's law, which states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In the hydrogen atom, this interaction is what holds the electron in a "cloud" surrounding the nucleus.
Additionally, the interaction defines the potential and kinetic energy distribution in the atom. The precise balance of these energies results in the stable configuration of the electron orbit, maintaining it at the calculated Bohr radius.
Understanding this interaction is pivotal in calculating the total energy of an electron in the ground state, representing how much work must be done to move an electron into a stable orbit from a point of neutrality.