Question

The angular momentum of electron revolving in the second orbit of \(\mathrm{H}\) -atom is ' \(x^{\prime} \mathrm{J} \cdot \mathrm{s}\). The angular momentum of electron in the second orbit of \(\mathrm{He}^{+}\) ion should be (a) \(x \mathrm{~J} \cdot \mathrm{s}\) (b) \(2 x \mathrm{~J} \cdot \mathrm{s}\) (c) \(0.5 x \mathrm{~J} \cdot \mathrm{s}\) (d) \(4 x \mathrm{~J} \cdot \mathrm{s}\)

Short answer

The angular momentum of the electron in the second orbit of the \( \mathrm{He}^{+} \) ion should be \( x \text{ J} \cdot \text{s} \).

Step-by-step solution

Understanding Bohr's quantization of angular momentum

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

Determining the angular momentum for a hydrogen atom and helium ion

Since the angular momentum is quantized, it only depends on the principal quantum number \( n \) and Planck's constant \( h \), and not on the type of atom or ion. Therefore, both the hydrogen atom and helium ion have the same angular momentum for the second orbit (\( n = 2 \)) if they have a single electron.

Comparing the angular momentum for different atoms or ions

Given the angular momentum for a hydrogen atom in the second orbit is \( x' \text{ J} \cdot \text{s} \), and knowing that the number of protons does not affect the quantized angular momentum for a single electron, the angular momentum for the electron in the second orbit of a helium ion \( \mathrm{He}^{+} \) will be the same because it too has only one electron in the orbit.

Key concepts

Bohr's Model of the Atom

Niels Bohr's model of the atom is a fundamental concept in quantum mechanics that provided a new way of understanding atomic structure. Introduced in 1913, Bohr's model describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus, similar to the way planets orbit the sun.

The key innovation of Bohr's model was the introduction of quantization to explain the stability of these orbits. According to Bohr, electrons can only exist in certain allowed orbits where the angular momentum of the electron is an integral multiple of the reduced Planck's constant (\(\frac{h}{2\pi}\)). This means that not any orbit is possible, but only those where the angular momentum satisfies the condition
\[L = n\frac{h}{2\pi}\].

In this equation, \(L\) represents the angular momentum, \(n\) is the principal quantum number, which can take any positive integer value, and \(h\) is Planck's constant. The restriction of energy levels due to quantization resulted in the explanation for the discrete spectral lines of elements.

Although Bohr's model was later superseded by the more complex and accurate quantum mechanical model, it was an essential step in the development of atomic theory and remains a valuable tool for introducing the concepts of quantized energy levels and electron orbits.

Principal Quantum Number

The principal quantum number, denoted as \(n\), is an integer that determines the major electronic shell or energy level in which an electron resides in an atom. It is one of the four quantum numbers that describe the unique quantum state of an electron, with the others being the orbital angular momentum, magnetic, and spin quantum numbers. The principal quantum number takes on positive integer values: 1, 2, 3, and so forth.

Each value of \(n\) corresponds to a specific electron shell, with increasing energy as \(n\) increases. In the context of Bohr's model, the principal quantum number is directly related to the size of the electron's orbit and its energy, with higher \(n\) values signifying orbits that are further from the nucleus and contain higher energy.

For example, when an electron resides in the second shell, it has a principal quantum number of 2. This is significant in the explanation of angular momentum quantization in Bohr's model, as the angular momentum of an electron orbit is quantized and directly proportional to \(n\), as shown by the formula:
\[L = n\frac{h}{2\pi}\].

The concept of the principal quantum number remains crucial in modern quantum mechanics, where it influences the overall energy of an electron along with the shape and size of atomic orbitals.

Planck's Constant

Planck's constant is a fundamental physical constant denoted by \(h\), central to quantum mechanics, and has a value of approximately \(6.626 \times 10^{-34}\) joule-seconds. Discovered by Max Planck in 1900, it represents the quantization of energy, being the proportionality factor between the frequency of a photon and its energy.

The expression for Planck's relationship is:
\[E = hu\],
where \(E\) is the energy of the photon and \(u\) (nu) is its frequency. This constant sets the scale for the quantum of action and reflects the inherent granularity of the processes at the microscopic scale, contradicting the classical view of smooth, continuous changes.

In Bohr's model, Planck's constant is critical as it appears in the quantization rule for the angular momentum of orbiting electrons. The implications of Planck's constant extend beyond atomic physics to define the smallest possible energy change in quantum systems, influencing the Heisenberg uncertainty principle and shaping the very foundation of quantum mechanics.

Understanding the significance of Planck's constant allows one to appreciate the discrete nature of the energy phenomenon at the quantum level and its profound impact across the field of physics. With its revolutionary introduction, we recognize that energy, much like matter, is quantized, a concept that shapes our modern understanding of the atomic and subatomic worlds.