Problem 43 - Master Resource Book in JEE Main Physics

Question

Consider an electron in the \(n\)th orbit of a hydrogen atom in the Bohr model. The circumference of the 5 orbit can be expressed in terms of the de-Broglie wavelength of that electron as (a) \((0.529) n \lambda\) (b) \(\sqrt{n} \lambda\) (c) \((13.6) \lambda\) (d) \(n \lambda\)

Step-by-Step Solution

Answer

The correct answer is (d) \(n \lambda\).

Step 1: Understanding the Bohr Model

In Bohr's model, an electron orbits the nucleus in a circular path, which is called an orbit. Each orbit corresponds to a specific energy level designated by the quantum number \(n\). The circumference of the orbit relates to the de Broglie wavelength by the condition that the orbit's circumference should be an integer multiple of the wavelength.

Step 2: Using de Broglie's Relationship

According to de Broglie's hypothesis, the wavelength \( \lambda \) of a particle like an electron is given by \( \lambda = \frac{h}{mv} \), where \(h\) is Planck's constant, \(m\) is the electron's mass, and \(v\) is its velocity. For an electron in an orbit, this wavelength relationship ensures that only specific orbits meet the requirements for stable electron wave patterns.

Step 3: Determine the Circumference Condition

Bohr postulated that for an electron to have a stable orbit, the circumference \(2\pi r\) of its orbit must be an integer multiple of its de Broglie wavelength. In terms of \(n\) and \( \lambda \), this condition is written mathematically as \[2\pi r = n\lambda\] where \(r\) is the radius of the nth orbit.

Step 4: Determine the Correct Option

Plugging in the condition determined, \(2\pi r = n\lambda\). It shows the circumference is directly proportional to \(n\) multiplied by \(\lambda\). Therefore, the option that matches this condition is \(d)\; n \lambda\). The factor of proportionality is simply 1, considering the circumference equals \(n\) times the de Broglie wavelength.

Key Concepts

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept that connects the wave nature of particles like electrons to their motion characteristics. According to de Broglie's hypothesis, every moving particle or object has an associated wavelength, which is described by the formula:\[\lambda = \frac{h}{mv}\]where:

• \(\lambda\) is the wavelength.
• \(h\) is Planck's constant \( (6.626 \times 10^{-34} \, \text{m}^2 \text{kg/s}) \).
• \(m\) is the mass of the particle (in this case, the electron).
• \(v\) is the velocity of the particle.

This formula implies that the wavelength of a particle is inversely proportional to its momentum \((mv)\). This wavelength concept was crucial in developing early quantum theory because it suggested that particles like electrons have properties of both particles and waves, known as wave-particle duality.
In the Bohr model of the hydrogen atom, the de Broglie wavelength determines the dimensions of electron orbits, ensuring they form stable wave patterns, which is essential for understanding the quantized nature of electron orbits.

Quantum Number

In quantum mechanics, the quantum number \(n\) is an important identifier that represents the energy level of an electron in an atom. In the Bohr model, these quantum numbers are integer values starting from 1 and increase as the orbits or energy levels move outward from the nucleus.

• \(n = 1\) denotes the closest orbit to the nucleus, also known as the ground state.
• Higher values of \(n\) signify orbits further away from the nucleus, with increasing energy levels.

The quantum number is not only a measure of the electron's energy but also indicates the size of the orbit it occupies. The greater the quantum number, the more significant is the radius of the orbit. This correlates directly with de Broglie's concept, as larger orbits must encompass more wavelengths, tying back to the stability condition for electron wave patterns. Thus, understanding quantum numbers is crucial for predicting the properties and behaviors of electrons within atoms as described by the Bohr model and later refined by quantum mechanics.

Electron Orbit

An electron orbit in the context of the Bohr model refers to the specific path that electrons take as they revolve around the nucleus of an atom. These orbits, or shells, are quantized, meaning each orbit corresponds to a discrete energy level.The Bohr model simplistically likens these paths to those of planets orbiting a sun but with key distinctions:

• Electron orbits are defined by fixed energy levels, governed by the principal quantum number \(n\).
• Electrons can only inhabit these specific orbits, not in between them, which leads to the quantization that is central to Bohr's model.
• The circumference of these orbits must exactly accommodate a whole number of de Broglie wavelengths \( (n\lambda) \), enabling stable standing wave patterns.

These criteria ensure that electrons in these orbits help define the atom's stability and electron arrangement. When an electron moves between orbits, it absorbs or emits energy equal to the difference between the energy levels, a principle pivotal in explaining atomic emission and absorption spectra. Hence, understanding electron orbits provides a foundational concept essential for further studies in atomic physics and quantum mechanics.