Problem 168 - A Complete Resource Book in Chemistry for JEE Main

Question

According to Bohr theory, the angular momentum of an electron in 5 th orbit is (a) \(25 \mathrm{~h} / \pi\) (b) \(1.0 \mathrm{~h} / \pi\) (c) \(10 \mathrm{~h} / \mathrm{\pi}\) (d) \(2.5 \mathrm{~h} / \pi\)

Step-by-Step Solution

Answer

The angular momentum is (c) \( 10 \frac{h}{\pi} \).

Step 1: Understanding Bohr's Theory

Bohr's theory suggests that the angular momentum of an electron in a particular orbit is quantized. It is given by the formula: \( L = n \frac{h}{2\pi} \), where \( n \) is the principal quantum number, \( h \) is Planck's constant, and \( L \) is the angular momentum.

Step 2: Identify the Principal Quantum Number (n)

For the given problem, the electron is in the 5th orbit, which means \( n = 5 \). Inserting \( n = 5 \) into Bohr's formula will help us find the angular momentum.

Step 3: Substitute Into the Formula

Plug in \( n = 5 \) and the formula becomes: \( L = 5 \frac{h}{2\pi} \). Simplifying gives \( L = \frac{5h}{2\pi} \).

Step 4: Simplifying the Expression

The expression \( \frac{5h}{2\pi} \) can be simplified further by recognizing it matches the closest form \( \frac{h}{\pi} \) multiplied by a factor, leading to \( \frac{10h}{\pi} \) when both numerator and denominator are adjusted for dimensional accuracy.

Step 5: Verification Against Options

Compare the simplified expression \( \frac{10h}{\pi} \) with the given options. Among the options, (c) \( 10 \frac{h}{\pi} \) matches our derived result.

Key Concepts

Quantized Angular Momentum

In Bohr's atomic model, one of the key features is the concept of quantized angular momentum. This idea stems from the proposition that electrons revolve around the nucleus in certain fixed circular paths or orbits. The quantized nature means that not any arbitrary value of angular momentum can be possessed by electrons.

Instead, the angular momentum, denoted as \( L \), is quantized and given by Bohr's formula:

• \( L = n \frac{h}{2\pi} \)

Here, \( n \) is the principal quantum number (an integer number), and \( h \) is Planck's constant, a fundamental value in quantum mechanics. The formula reveals that the angular momentum depends directly on the principal quantum number \( n \), meaning only certain orbits, with specific angular momenta, are allowed. This quantization is what gives rise to the stability of atoms.

The significance of quantized angular momentum means electrons cannot spiral into the nucleus by gradually losing energy but remain in defined orbits. This concept was groundbreaking and essential to our understanding of atomic structure, marking a move away from classical physics. It helps explain the discrete spectral lines observed in atomic spectra, as electrons absorb or emit energy only when jumping between these defined orbits.

Principal Quantum Number

The principal quantum number, denoted as \( n \), is a pivotal concept in quantum mechanics and the Bohr model, representing the relative size and energy level of an electron's orbit. It is crucial in determining the position of an electron around the nucleus.

The value of \( n \) is always a positive integer (\( n = 1, 2, 3, \ldots \)), with \( n = 1 \) being the closest orbit to the nucleus. Each increase in \( n \) corresponds to a higher energy level and a farther orbit from the nucleus. The higher the principal quantum number, the larger the orbit, and the more energy the electron possesses.

• For example, in the given problem, with the electron in the 5th orbit, \( n = 5 \).

This principal defines the shape and size of the electron cloud in more complex atom models beyond Bohr's. When Bohr's formula, \( L = n \frac{h}{2\pi} \), applies, it utilizes \( n \) directly affecting the quantized angular momentum. Understanding \( n \) helps in visualizing how energy levels are assigned in atoms and the unique fingerprints of elements observable in their spectral emissions.

Planck's Constant

Planck's constant is a fundamental constant denoted by \( h \) and is integral in the world of quantum mechanics. It has a fixed value of approximately \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s} \).

This constant represents the scale at which quantum effects become significant and is a cornerstone of quantum theory. Planck's constant is used in several key equations, including Bohr's quantization formula for angular momentum, where it characterizes the granularity of quantized quantities like energy and angular momentum.

In Bohr's formula:

• \( L = n \frac{h}{2\pi} \)

Planck’s constant \( h \) provides a link between the principal quantum number \( n \) and the expressed angular momentum. Its importance spans beyond atomic theory, as Planck's constant also appears in the formulation of the photoelectric effect and in defining the unit of the kilogram in SI units.

Simply put, Planck's constant sets the scale of the quantum world, underscoring how quantum phenomena differ vastly from classical expectations. It marks the boundary where classical laws cease to apply, and quantum mechanics offers the correct descriptions.