Problem 77 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

If the radius of first Bohr orbit is \(x\) unit, then de-Broglie wavelength of electron in the third orbit is (a) \(2 \pi x\) unit (b) \(6 \pi x\) unit (c) \(9 x\) unit (d) \(18 \pi x\) unit

Step-by-Step Solution

Answer

The de-Broglie wavelength of electron in the third orbit is \( 18\pi x \) units.

Step 1: Understanding the Bohr Model and de Broglie Wavelength

According to the Bohr model of the hydrogen atom, the radius of the nth orbit is directly proportional to the square of the quantum number, n. In this case, the radius of the first Bohr orbit is given as x. The de Broglie wavelength, \( \lambda \), for an electron in orbit is given by \( \lambda = \frac{nh}{mv} \), where \( \) is the principal quantum number, \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is the velocity of the electron. In a quantized orbit, an integral number of wavelengths must fit into the circumference of the orbit, which is \( 2\pi r \).

Step 2: Expressing the de Broglie Wavelength in Terms of the Bohr Radius

The circumference of the nth orbit is \( 2\pi n^2 x \), since the radius increases with n squared, and n=1 corresponds to x units. For the third orbit, where n=3, the circumference would be \( 2\pi \cdot 3^2 \cdot x = 18\pi x \). Since the electron's de Broglie wavelength must fit an integral number of times into the circumference of the orbit (as stated above), the wavelength in the third orbit would be equal to the circumference of the orbit, which leads to a wavelength of \( 18\pi x \) units.

Key Concepts

Bohr Model

The Bohr model is a pivotal concept in understanding atomic structure as posited by Niels Bohr in 1913. It was a tremendous leap forward in quantum physics, providing a theoretical framework for the hydrogen atom that integrated classical mechanics with early quantum theory.

The model showcases electrons in discrete orbits around the nucleus, each with a fixed energy level and not emitting radiation unless they change orbits. The key aspects of this model include the quantization of electron angular momentum and the idea that electron energy levels are fixed and can only change via quantum leaps between these orbits. Bohr's predictions about the emission spectrum of hydrogen were confirmed by experiment, making his model a cornerstone of quantum mechanics.

Bohr's model resolved issues that classical mechanics could not explain, such as why atoms exhibited only certain discrete spectral lines. This model helped pave the way for the complex and more accurate quantum mechanical models that would follow, like the Schrödinger equation.

Quantum Number

Quantum numbers are vital for understanding the Bohr model and beyond. Each electron in an atom is described by a set of quantum numbers that specify its distinct energy state including its orbital path. There are four kinds of quantum numbers: principal (n), angular momentum (l), magnetic (m_l), and spin (m_s).

The principal quantum number, represented by 'n', plays a central role in the Bohr model. It determines the energy level and size of an electron's orbit within the atom, which is further related to other properties such as the atom's ionization energy and the radius of each orbit. Higher values of 'n' denote greater distances from the nucleus and greater energy levels. Within the framework of the Bohr model, the principal quantum number can only take positive integer values starting from one, signifying the ground state, or lowest energy level, of the electron.

Bohr Orbit Radius

Concerning the Bohr model of the atom, the Bohr orbit radius is vital to understand how electrons inhabit distinct energy levels. Bohr postulated that the radius of these orbits increases with the square of the principal quantum number (n).

Mathematically, the radius of the n-th orbit (r_n) is proportional to n², which is shown in the formula r_n = n²r₁ where r₁ is the radius of the first Bohr orbit, often denoted as the Bohr radius. This squared relationship explains why orbit radii grow much larger at higher energy levels—specifically, when an electron transitions from a lower orbit to a higher one, it is effectively moving to an orbit with a radius that is n-squared times larger than the radius of the first orbit. As a result, the largest allowed jump in wavelength, as per the Bohr model, would be directly tied to this relationship. Thus, understanding this concept is essential for calculations involving electron transitions, energy level differences, and spectral lines.