Problem 62 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The ratio of circumference of third and second orbits of \(\mathrm{He}^{+}\) ion is (a) \(3: 2\) (b) \(2: 3\) (c) \(9: 4\) (d) \(4: 9\)

Step-by-Step Solution

Answer

(c) 9: 4

Step 1: Understand the Bohr Model

In the Bohr model of the atom, electrons orbit the nucleus in specific circular paths called orbits, with quantized energy levels. The circumference of each orbit is related to its radius and energy level.

Step 2: Determine the Circumference of an Orbit

The circumference of an orbit in the Bohr model is given by the formula \(C_{n} = 2\pi r_{n}\), where \(r_{n}\) is the radius of the nth orbit. According to Bohr's postulates, the radius of the nth orbit is given by \(r_{n} = n^2 \times r_1\), where \(r_1\) is the radius of the first orbit.

Step 3: Calculate the Ratio of Circumferences

To find the ratio of the circumferences of the third and second orbits, use the expression for the circumference with the radii \(r_3 = 3^2 \times r_1\) and \(r_2 = 2^2 \times r_1\). Therefore, the ratio of the circumferences is \(\frac{C_{3}}{C_{2}} = \frac{2\pi (3^2 r_1)}{2\pi (2^2 r_1)} = \frac{3^2}{2^2} = \frac{9}{4}\).

Key Concepts

Circumference of Atomic Orbits

The Bohr model provides us with a visual picture of the atom, one that portrays electrons circling the nucleus in fixed orbits. Each of these orbits has a circumference that's crucial to understanding how atoms absorb and emit energy. The circumference, which is the distance around the orbit, is directly related to the size of the orbit; the larger the orbit, the greater the circumference.

For any given orbit labeled by an integer 'n', known as the quantum number, the circumference is determined by the formula \( C_n = 2\pi r_n \), where \( r_n \) is the radius of that particular orbit. Bohr's model asserts that the radius of these orbits increases as the square of the quantum number; thus as 'n' gets bigger, the orbits get considerably larger, following the expression \( r_n = n^2 \times r_1 \). An interesting consequence of this relationship is how it affects the behavior of electrons, and subsequently, how it governs the elemental signatures we see in emission and absorption spectra.

Bohr's Postulates

Niels Bohr proposed his model of the atom in 1913, fundamental postulates shaped this model and the understanding of atomic structure. One of these is that electrons can only orbit the nucleus in certain allowable paths—each with a fixed energy—without emitting radiation. Another crucial point is that electrons can transition between these defined orbits, or 'energy levels', by either absorbing or emitting a photon that corresponds to the energy difference between the levels.

This concept is deeply tied to the quantization of energy levels, where the energy of an orbit is related to its circumference and radius. For example, as an electron moves to a higher energy level, it must jump to a higher orbit with a larger circumference. Bohr's theory was groundbreaking because it introduced the idea that atomic energy levels are quantized, a major shift from the classical physics view of continuous energy variation.

Quantized Energy Levels

If you imagine an atom as a mini solar system, the Bohr model wonderfully illustrates how electrons can only 'live' at specific distances from the nucleus. What does 'quantized' mean in this context? Simply put, it means that the energy levels at which electrons exist are discrete, much like the individual steps of a staircase. These steps prevent an electron from existing in between levels, just as you can't stand between the steps of a staircase.

In mathematical terms, the energy associated with an orbit is given by \( E_n = \frac{-R_H}{n^2} \), where \( E_n \) is the energy of the nth orbit and \( R_H \) is the Rydberg constant for hydrogen. When an electron jumps from a lower orbit (n1) to a higher one (n2), it absorbs energy equal to the difference in energy levels \( \Delta E = E_{n2} - E_{n1} \). The concept of quantization helps explain the distinct lines observed in atomic emission spectra, as each jump corresponds to a specific energy, and hence a specific wavelength of light, that is emitted. As a fundamental principle, quantization helps us understand the stability of atoms and the exchange of energy in the form of light.