Problem 4 - Chemistry in 30 days

Question

According to Bohr's theory, the angular momentum for an electron of \(5^{\text {th }}\) orbit is : (a) \(5 h / \pi\) (b) \(2.5 h / \pi\) (c) \(5 \pi h\) (d) \(25 h / \pi\)

Step-by-Step Solution

Answer

The correct answer is (b) \(2.5 h / \pi\).

Step 1: Recall Bohr's Quantization Condition

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

Step 2: Substitute for Principal Quantum Number

Given that the electron is in the 5th orbit, we have \( n = 5 \). Substituting this value into Bohr's formula yields: \( L = 5 \cdot \frac{h}{2\pi} \).

Step 3: Simplify the Expression

Simplifying the expression from Step 2, we obtain: \( L = \frac{5h}{2\pi} \).

Step 4: Relate to Given Options

Checking the options, none match \( \frac{5h}{2\pi} \) directly, but can be rewritten as \( 2.5 \frac{h}{\pi} \). This matches option (b).

Key Concepts

Angular Momentum

Angular momentum plays a fundamental role in Bohr’s atomic model. In Bohr's theory, an electron revolving around the nucleus is not just moving randomly. Its behavior is carefully described. Angular momentum in this context refers to the motion of an electron as it orbits the nucleus of an atom.
A crucial feature of Bohr's model is that the angular momentum of an electron is quantized. This means it can only take on specific, discrete values based on certain conditions.
For electrons in Bohr's orbits, the angular momentum is calculated with the formula:\[ L = n \hbar \]Here, \( L \) represents angular momentum, \( n \) is the principal quantum number, and \( \hbar \) is the reduced Planck's constant.
Bohr's insight into quantization brought a massive leap in understanding atomic structure, as it established that electrons are not in arbitrary motion but confined to specified orbits.

Quantization Condition

The concept of quantization was a groundbreaking addition to atomic theory introduced by Bohr. It simplifies the understanding of how electrons behave around an atomic nucleus. This concept states that certain physical quantities, like angular momentum, can only assume discrete values rather than any value within a range.
For electrons, the quantization condition specifies that their angular momentum is an integer multiple of the reduced Planck's constant. That condition is given by:\[ L = n \hbar \]where \( n \) signifies the principal quantum number. Quantization ensures that the electron's path is stable and predictable. It defines the allowed energy levels of electrons, fundamentally altering the way atoms were viewed and setting the foundation for quantum mechanics.

Principal Quantum Number

The principal quantum number, denoted by \( n \), is essential in Bohr's model, determining key properties of electron orbits. It describes the size and energy of the electron's orbit and is a positive integer (1, 2, 3,...). The further an electron is from the nucleus, the larger its principal quantum number.
In Bohr’s theory, each principal quantum number corresponds to a specific orbit that an electron can occupy. This means an electron in the 1st orbit has \( n = 1 \), in the 2nd orbit \( n = 2 \), and so on. The principal quantum number also affects the angular momentum, as described by the relationship:\[ L = n \hbar \]Here, the electron in the 5th orbit would have \( n = 5 \), showing why quantization ties directly with these discrete energy levels.

Reduced Planck's Constant

The reduced Planck's constant, denoted by \( \hbar \), is central to quantum theory. It modifies Planck's constant \( h \). It appears in the formula for angular momentum in Bohr’s models:\[ \hbar = \frac{h}{2\pi} \]This factor arises naturally when working with systems involving rotational motion and wavefunctions. In Bohr's quantization condition, the reduced Planck's constant helps define the allowed physical state or orbit of an electron.
By understanding \( \hbar \), students see how these constants unify broader concepts of wave and particle physics. It provides a bridge between macroscopic classical mechanics and the microscopic world of quantum mechanics.