Problem 53 - Master Resource Book in JEE Main Physics

Question

An ionised H-molecule consists of an electron and two protons. The protons are separated by a small distance of the order of angstrom. In the ground state, \(\quad\) [NCERT Exemplar] (a) the electron would not move in circular orbits. (b) the energy would be \((2)^{4}\) times that of a h-atom. (c) the electrons, orbit would go around the protons. (d) the molecule will soon decay in a proton and a H-atom.

Step-by-Step Solution

Answer

(a) The electron would not move in circular orbits.

Step 1: Understanding the System

An ionized hydrogen molecule is composed of two protons and one electron. Unlike a hydrogen atom, which has one electron orbiting one proton, this system involves two protons that can affect the electron's orbit due to their mutual electromagnetic interactions.

Step 2: Analyzing Electron Motion

In this molecular ion, the electron is under the influence of the electric fields of both protons. Unlike a classical scenario with a single nucleus (as in an atom), this configuration does not favor simple circular orbits for the electron. Hence, statement (a) that the electron would not move in circular orbits is likely true.

Step 3: Energy Consideration

The energy of a hydrogen atom is determined by the electron's motion around a single proton. The presence of a second proton changes the potential landscape significantly. The statement claiming the energy would be \((2)^4\) times that of a hydrogen atom isn’t straightforward to check without additional calculations. Doubling charge might imply squared influence, but it does not typically translate multiplicatively like this, listing (b) as improbable.

Step 4: Electron Orbit Analysis

Considering (c), in a typical hydrogen atom, the electron orbits a single proton. When two protons are close together, the electron tends to be shared in a more complex, often linear or oval orbit rather than entirely encircling them, making this statement unlikely.

Step 5: Stability Evaluation

Molecules are energetically more stable than individual atoms unless disturbed. The two-proton, one-electron molecule, despite possessing a peculiar electron configuration, is typically stable regarding dissociation into a proton and hydrogen atom shortly. Therefore, any such decay into a proton and hydrogen atom, as per statement (d), would not occur soon.

Key Concepts

Electron Motion

In the ionized hydrogen molecule, the electron's motion is influenced by both protons. This is different from a standard hydrogen atom where the electron orbits one proton. Here, the electric fields from both protons create a unique environment. This means that electron paths won't be simple circles.

In classical terms, electrons might be expected to move in neat circles around a nucleus. But in this molecular ion, there are two nuclei (the protons) causing a tug-of-war on the electron.

• Instead of circular paths, more complex paths such as elliptical or even linear alignments could occur.
• The electron does not simply choose one proton to encircle; its motion is influenced by both.

This overlap and dual influence lead to paths that cannot precisely be described as circular, affirming that assumption (a) is reasonable.

Energy Consideration

Energy in the context of the ionized hydrogen molecule differs greatly from a simple hydrogen atom. The electron in a hydrogen atom moves around a single proton, but here, it is attracted by two protons, leading to additional interactions that modify the energy landscape.

The statement suggesting that energy would be \( (2)^4 \) times that of a hydrogen atom appears exaggerated.

• The energy calculations do not follow such a simple multiplicative rule. The physical system's energy depends on potential and kinetic interactions between all charged particles.
• Including another proton changes the potential energy significantly but not in a way that multiplies simply.

The real energy of this system needs careful quantum mechanical computation, which often can be approximated but isn't merely \( (2)^4 \) times that of a simple hydrogen atom.

Stability Evaluation

The stability of any molecular formation is an essential consideration. For our ionized hydrogen molecule, despite having a non-traditional electron arrangement, it generally maintains stability over decay into a proton and a hydrogen atom.

Stability is determined by the energy states of the molecule: lower energy configurations are more stable.

• This molecule configuration doesn’t favor immediate breakdown into a singular hydrogen atom and a free proton because the energy required to separate them is high.
• The molecular bond and the electron's shared involvement between the two protons offer several energy states that further avoid rapid decay.

Hence, the molecule is likely to remain energetically favorable for a significant period, negating the assumption that it will quickly decay.