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) \(10 \mathrm{~h} / \pi\) (d) \(2.5 \mathrm{~h} / \pi\)

Step-by-Step Solution

Answer

The angular momentum of an electron in the 5th orbit is \(2.5 \mathrm{~h} / \pi\) (option d).

Step 1: Understanding Bohr's Angular Momentum Formula

According to Bohr's model, the angular momentum of an electron in an orbit is quantized and is given by the formula \(L = n \cdot \frac{h}{2\pi}\), where \(n\) is the principal quantum number (orbit number), and \(h\) is Planck's constant. For the 5th orbit, \(n = 5\).

Step 2: Apply the Formula for the 5th Orbit

Substitute \(n = 5\) into the formula to find the angular momentum: \(L = 5 \cdot \frac{h}{2\pi} = \frac{5h}{2\pi}\).

Step 3: Simplify the Expression

Simplify the expression \(\frac{5h}{2\pi}\). This can also be expressed as \(2.5 \cdot \frac{h}{\pi}\) by factoring out the 2 in the denominator.

Step 4: Identify the Correct Option

The simplified angular momentum is \(2.5 \cdot \frac{h}{\pi}\), which corresponds to option (d) \(2.5 \mathrm{~h} / \pi\).

Key Concepts

Angular Momentum

Angular momentum is a crucial aspect of Bohr's model of the atom. This model describes how electrons move around the nucleus in specific paths, called orbits. The concept of angular momentum in these orbits is essential because it is considered to be quantized. In simpler terms, this means that electrons can only have certain, fixed amounts of angular momentum as they orbit the nucleus. In Bohr's model, the angular momentum (L) of an electron is described by the formula: \[L = n \cdot \frac{h}{2\pi}\]where \(n\) is the principal quantum number representing the orbit number, and \(h\) is Planck's constant, which is a fundamental constant in physics.For example, if an electron is in the 5th orbit (\(n = 5\)), its angular momentum is specific and quantized, calculated as \(\frac{5h}{2\pi}\). This indicates that the electron cannot just have any arbitrary value of angular momentum; instead, it must stick to these predefined, quantized values.

Quantized Orbits

Bohr introduced the idea of quantized orbits to explain the stability of electrons around the nucleus. Quantized orbits are those defined paths where electrons can revolve around the nucleus without losing energy. This concept is a major departure from classical physics, where such restrictions would not typically apply. In the Bohr model, quantization means that only certain allowable energy levels exist for electrons. Electrons cannot reside in between these defined energy levels but can jump from one level to another. This jump appears as an electron moves between orbits, absorbing or emitting energy in the form of photons. Quantization arises because of the constraints on the angular momentum, meaning electrons can only have specific values permitted by the equation: \[L = n \cdot \frac{h}{2\pi}\] This quantization ensures that each orbit has a specific energy level, and an electron moving from one orbit to another explains phenomena such as atomic emission spectra, where electrons change orbits and emit light at certain wavelengths specific to each element.

Principal Quantum Number

The principal quantum number (\(n\)) is another fundamental aspect of understanding Bohr's model. It effectively labels each orbit around the nucleus, with increasing values of \(n\) representing orbits that are farther from the nucleus. The principal quantum number starts from 1 and can assume positive integer values (1, 2, 3, ...), where each number corresponds to a specific energy level and orbit in the atom. Larger values of \(n\) indicate higher energy levels, where the electron would be farther from the nucleus.This number plays a critical role in defining both the energy and angular momentum of electrons in their orbits. The energy of the electron in a particular orbit is directly related to the principal quantum number, with higher \(n\) values corresponding to larger, more energetically demanding orbits. In terms of angular momentum, the principal quantum number directly determines the quantized angular momentum of the electron:\[L = n \cdot \frac{h}{2\pi}\]Understanding the principal quantum number is crucial for predicting the position and energy of electrons within the atom, providing a simple but profound insight into atomic structure.