Problem 4 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

If the de Broglie wavelength of the electron in \(\mathrm{n}^{\text {th }}\) Bohr orbit in a hydrogenic atom is equal to \(1.5 \pi \mathrm{a}_{0}\left(\mathrm{a}_{0}\right.\) is Bohr radius), then the value of \(\mathrm{n} / \mathrm{z}\) is : \(\quad\) [Main Jan. \(12,2019(\mathrm{II})]\) (a) \(0.40\) (b) \(1.50\) (c) \(1.0\) (d) \(0.75\)

Step-by-Step Solution

Answer

The value of \( \frac{n}{Z} \) is 0.75, so the answer is (d) 0.75.

Step 1: Understanding De Broglie Wavelength

The de Broglie wavelength associated with an orbiting electron is given by \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant, \( m \) is the electron mass, and \( v \) is the electron's velocity. For electrons in a Bohr orbit, this value should harmonize with the circumference of the orbit, given by \( 2\pi r \). This illustrates the wave nature of electrons in quantized orbits.

Step 2: Connecting De Broglie Wavelength to Bohr’s Model

For a given \( n^{th} \) orbit in a hydrogenic atom, the de Broglie wavelength formula can be equated to get \( 2 \pi r_n = n \lambda \). Given that \( \lambda = 1.5 \pi a_0 \), we can substitute to find \( 2 \pi r_n = n \times 1.5 \pi a_0 \). Thus, the condition for the de Broglie wavelength is satisfied when the wavelength fits evenly into the orbit's circumference.

Step 3: Applying Bohr’s Radius Formula

For the \( n^{th} \) Bohr orbit, the radius \( r_n \) can be expressed as \( r_n = \frac{n^2 a_0}{Z} \), where \( Z \) is the atomic number of the hydrogenic atom. Substituting \( r_n \) into the previous equation, we get \( 2 \pi \frac{n^2 a_0}{Z} = n \times 1.5 \pi a_0 \). Simplify this equation to isolate \( \frac{n}{Z} \).

Step 4: Solving for \( \frac{n}{Z} \)

Simplify the equation from the previous step: \( \frac{2 \pi n^2 a_0}{Z} = 1.5 n \pi a_0 \). Cancel out common terms (\( \pi \) and \( a_0 \)), and rearrange to find \( 2 n^2 = 1.5 n Z \). Solving for \( \frac{n}{Z} \) gives \( n = 0.75 Z \). Thus, \( \frac{n}{Z} = 0.75 \).

Step 5: Final Answer

From the simplification, the value of \( \frac{n}{Z} \) is \( 0.75 \). Thus, the correct answer to the problem is option (d) \( 0.75 \).

Key Concepts

de Broglie Wavelength

The de Broglie wavelength is a concept that illustrates the wave nature of particles like electrons. According to de Broglie, particles such as electrons not only exhibit particle characteristics but also wave-like qualities. This is depicted by associating a wavelength to a moving particle. The de Broglie wavelength is given by the formula: \( \lambda = \frac{h}{mv} \) where \( h \) is Planck’s constant, \( m \) is the mass of the particle, and \( v \) is its velocity.
To understand it better:

• The smaller the particle's velocity, the larger the de Broglie wavelength.
• This concept is crucial in explaining the behavior of electrons in an atom's orbit.
• The quantized orbits in Bohr’s model accommodate whole numbers of these wavelengths around the nucleus. This is why the wavelength must fit neatly into the orbit’s circumference, \( 2\pi r \).

The notion of wave-particle duality is fundamental in quantum mechanics and helps delineate the behavior of subatomic particles like electrons.

Hydrogenic Atom

A hydrogenic atom is a term used for any atom that resembles hydrogen but may have different atomic numbers. In essence, these are single-electron systems similar to hydrogen.
A hydrogenic atom is characterized by:

• Having one electron orbiting the nucleus, similar to hydrogen.
• Exhibiting energy levels that depend on the atomic number \( Z \).
• Using the atomic number \( Z \), energy levels and radii of orbits can be calculated. This makes them easier to model and understand using Bohr's theories.

The understanding of hydrogenic atoms is central in quantum mechanics and spectroscopy, providing insights into the behavior of atoms under various conditions.

Bohr Radius

The Bohr radius is a fundamental constant in physics that describes the most probable distance between the nucleus and the electron in a hydrogen atom. Denoted as \( a_0 \), it is approximately equal to \( 0.529 \times 10^{-10} m \).
Here's what you need to know about the Bohr radius:

• It's derived from Bohr’s model of the hydrogen atom and provides a scale for atomic structures.
• This constant is critical for illustrating the size of atoms and helps calculate orbit radii for hydrogenic atoms.
• In Bohr’s formula for the radius of an orbit \( r_n \), the radius is related to \( a_0 \) and is expressed as \( r_n = \frac{n^2 a_0}{Z} \), where \( n \) is the principal quantum number and \( Z \) is the atomic number.

The Bohr radius plays an integral role in understanding atomic structure and quantum physics by providing a fundamental physical quantity.