Question

According to Bohr's theory, the angular momentum of an electron in \(5^{\text {th }}\) orbit is (a) \(\frac{10 h}{\pi}\) (b) \(\frac{25 h}{\pi}\) (c) \(\frac{1.5 h}{\pi}\) (d) \(\frac{2.5 h}{\pi}\)

Short answer

\( \frac{10h}{\pi} \)

Step-by-step solution

Understanding Bohr's Quantization of Angular Momentum

According to Bohr's theory of the hydrogen atom, the angular momentum (L) of an electron in a given orbit is quantized and can be calculated using the formula: \( L = n\frac{h}{2\pi} \), where \( n \) is the principal quantum number of the orbit and \( h \) is Planck's constant.

Identify the Principal Quantum Number for the Given Orbit

For an electron in the 5th orbit, the principal quantum number is \( n = 5 \).

Calculate the Angular Momentum for the 5th Orbit

Using the formula for quantized angular momentum, we substitute \( n = 5 \) and Planck's constant \( h \) to get: \( L = 5\frac{h}{2\pi} \).

Simplify the Expression for Angular Momentum

Simplify the expression to represent the angular momentum in terms of \( h \) and \( \pi \): \( L = \frac{5h}{2\pi} = \frac{10h}{4\pi} = \frac{10h}{\pi} \), after multiplying the numerator and denominator by 2 to clear the fraction.

Key concepts

Bohr's Quantization of Angular Momentum

Bohr's theory of the atom is fundamental in the field of quantum mechanics, especially when it comes to understanding electron behavior. According to Niels Bohr, electrons orbit the nucleus in distinct paths or 'shells' without radiating energy, much like planets orbiting the sun. But, unlike the continuous orbits of planets, electrons can only inhabit certain allowed orbits.

These discrete orbits are a result of what Bohr termed the 'quantization of angular momentum'. So, what does this mean? In essence, the angular momentum of an electron is not arbitrary but comes in 'packets' of fixed size. The mathematical representation of this concept is given by the formula:
\[ L = n\frac{h}{2\pi} \]
Where:\

• \( L \) is the angular momentum of the electron in its orbit.
• \( n \) is the principal quantum number, which is an integer and determines the size and energy of the orbit. It can be 1,2,3, and so on.
• \( h \) is Planck's constant, a fundamental physical constant in quantum mechanics.

Thus, only orbits with angular momentum that is an integer multiple of \( \frac{h}{2\pi} \) are permitted. This means that an electron can only gain or lose energy in discrete amounts (or quanta) when it jumps from one allowed orbit to another, which leads to the emission or absorption of photons with specific energies.

Principal Quantum Number

Diving deeper into Bohr's theory brings us to the concept of the principal quantum number, symbolized as \( n \). This number is germane to the quantum mechanics of an atom because it designates the electron's orbit - its 'address' within the atom, if you will.

Each value of \( n \) corresponds to a particular electron shell, and this, in turn, correlates to the energy level of the electron. The rule of thumb here is: the greater the value of \( n \), the farther the electron shell from the nucleus and the higher the potential energy of the electron. In simple terms, as \( n \) increases, the electron's orbit becomes larger and its energy higher.

For the first orbit, \( n = 1 \); for the second orbit, \( n = 2 \), and so on. The principal quantum number plays a crucial role in the formula that quantifies angular momentum presented in Bohr's theory, linking the concept of quantization directly to the shell structure of the atom. In a classroom setting, understanding the principal quantum number is akin to learning the alphabet before spelling - it's an essential building block for grasping more complex quantum mechanical concepts.

Planck's Constant

Planck's constant, denoted as \( h \), is a cornerstone in the world of quantum physics, and it's a physical constant that carries profound implications. This value was first discovered by Max Planck, one of the pioneers of quantum theory, and it essentially dictates the scale at which quantum effects become significant.

Here's what you need to know about Planck's constant:\

• Its value is approximately \( 6.62607015 \times 10^{-34} \) joule seconds.
• Planck's constant is used to describe the size of quanta, the discrete units of energy transfer, prevalent in quantum mechanics.
• It appears in various fundamental formulas, the most well-known perhaps being the energy of a photon, \( E = hu \), where \( u \) is the frequency of the photon.

Planck's constant is so critical that it forms the keystone of Bohr's quantization of angular momentum and is integral to Heisenberg's uncertainty principle, another fundamental theory in quantum mechanics. Understanding Planck's constant and its applications not only helps in solving numerical problems in physics but also provides insight into the essential nature of the quantum realm, where the classical laws of physics as we know traditionally, take on a new, often non-intuitive, form.