Problem 30 - Master Resource Book in JEE Main Physics

Question

The binding energy of a H-atom, considering an electron moving around a fixed nuclei (proton), is \(B=\frac{m e^{4}}{8 n^{2}{\underline{\phantom{xx}}}_{0}^{2} h^{2}} \cdot(m=\) proton mass \()\) If one decides to work in a frame of reference where the electron is at rest, the proton would be moving around it. By similar arguments, the binding energy would be \(B=-\frac{M e^{4}}{8 n^{2} \varepsilon_{0}^{2} h^{2}}(M=\) proton mass \() .\) This last expression is not correct because (a) \(n\) would not be integral (b) Bohr-quantisation applies only to electron (c) the frame in which the electron is at rest is not inertial. (d) the motion of the proton would not be in circular orbits, even approximately.

Step-by-Step Solution

Answer

The correct answer is (c): the frame in which the electron is at rest is not inertial.

Step 1: Understand the Initial Binding Energy Formula

The original binding energy formula for an electron moving around a proton is given as \( B = \frac{m e^{4}}{8 n^{2} \varepsilon_{0}^{2} h^{2}} \), where \(m\) is the electron mass, \(e\) is the charge of the electron, \(\varepsilon_0\) is the permittivity of free space, \(n\) is the principal quantum number, and \(h\) is Planck's constant.

Step 2: Consider the Proposed Frame Change

In the question, it proposes changing the reference frame such that the proton is moving around a stationary electron. By analogy, one would replace the electron mass \(m\) in the original formula with the proton mass \(M\). Then the binding energy expression becomes \( B = -\frac{M e^{4}}{8 n^{2} \varepsilon_{0}^{2} h^{2}} \).

Step 3: Evaluate Each Option to Determine the Error

Examine each given option to identify the reason this expression is incorrect:(a) \(n\) would not be integral: This explanation implies that changing the frame affects the quantization of energy levels, which might not hold as quantization generally depends only on conditions like boundary conditions or potential form.(b) Bohr-quantization applies only to electron: Bohr's model was specifically designed for electrons in hydrogen-like systems, not for scenarios with protons moving which have different characteristics.(c) The frame in which the electron is at rest is not inertial: This seems plausible as electrons and protons are not equal in mass and treating the electron stationary makes the frame accelerating due to the proton's influence.(d) The motion of the proton would not be in circular orbits, even approximately: Protons have much larger mass and would not fit neatly into the simpler assumptions of circular orbits required in Bohr's model.

Step 4: Conclude the Correct Answer

The most convincing explanation is option (c). Since the mass of the proton is significantly larger than that of the electron, placing the electron at rest results in a non-inertial frame. The non-inertial nature of this frame makes the derived binding energy expression invalid.

Key Concepts

Binding Energy

Binding energy is a crucial concept in understanding atomic structure, especially within the Bohr model. In simple terms, binding energy is the energy required to remove an electron from an atom and move it to an infinite distance away. This measure tells us how strongly an electron is bound to the nucleus.

For the hydrogen atom, the binding energy can be calculated using the formula:

• \[ B = \frac{m e^{4}}{8 n^{2} \varepsilon_{0}^{2} h^{2}} \] , where:
  • \(m\) is the electron mass.
  
  
  • \(e\) is the charge of the electron.
  
  
  • \(\varepsilon_0\) is the permittivity of free space.
  
  
  • \(n\) is the principal quantum number.
  
  
  • \(h\) is Planck's constant.

This formula provides insight into how energy levels are quantized. As the principal quantum number \(n\) increases, the binding energy decreases, meaning the electron is less tightly bound to the nucleus the further it is from the center of the atom.

Overall, understanding binding energy allows us to realize the stability and reactivity of atoms at the quantum level.

Principal Quantum Number

The principal quantum number, represented by \(n\), is a fundamental concept in quantum mechanics and is critical to the Bohr model of the atom. This quantum number primarily describes the size of the electron's orbit and its energy level.

• \(n\) can take positive integer values such as 1, 2, 3, etc.


• The energy level of an electron is higher when \(n\) increases.


• In the hydrogen atom, each \(n\) corresponds to a specific electron orbit.

As \(n\) increases, electrons find themselves farther from the nucleus. Hence, larger values of \(n\) indicate larger orbits and greater potential energy.

The principal quantum number also plays a pivotal role in quantization, a vital principle in quantum mechanics, stating that not all energy levels are accessible, only those with discrete values of \(n\). This quantization causes the formation of distinct energy levels or shells within an atom.

So, \(n\) is not just a number; it influences the atom's energetic and structural properties substantially.

Inertial Frame

An inertial frame of reference is a foundational concept in both classical physics and quantum mechanics. It refers to a frame of reference where objects tend to remain at rest or move at a constant velocity unless acted upon by external forces.

In the context of the Bohr model, understanding whether a frame is inertial is essential, as it can significantly affect calculations and perceptions of atomic behavior.

• In an inertial frame, Newton's laws of motion are applicable, and objects feel no 'fictitious' forces.


• Any frame moving with constant velocity relative to an inertial frame is also considered inertial.

However, when we attempt to consider a frame where the electron is at rest and the proton is moving around it, complications arise.

• This setup leads to a non-inertial frame because the forces acting would cause the electron to accelerate.


• In such frames, calculations often result in inaccurate interpretations, as seen with the invalid binding energy calculation.

Recognizing whether a frame is inertial is key to correctly applying the Bohr model and other laws of physics. Thus, for accurate atomic and subatomic analysis, it is crucial to work within an inertial frame of reference.