Problem 14 - A Complete Resource Book in Chemistry for JEE Main

Question

Ratio of radii of second and first Bohr orbits of \(\mathrm{H}\) atom is (a) 2 (b) 4 (c) 3 (d) 4

Step-by-Step Solution

Answer

The ratio is 4; correct option is (d).

Step 1: Understanding the Formula for Bohr Radius

The radius of the n-th Bohr orbit for a hydrogen atom is given by the formula \( r_n = n^2 a_0 \), where \( a_0 \) is the Bohr radius, a constant. Each orbit's radius depends on \( n^2 \), where \( n \) is the principal quantum number.

Step 2: Calculate the Radius of the First Bohr Orbit

For the first Bohr orbit, the principal quantum number \( n = 1 \). Using the formula, the radius is \( r_1 = 1^2 \cdot a_0 = a_0 \).

Step 3: Calculate the Radius of the Second Bohr Orbit

For the second Bohr orbit, the principal quantum number \( n = 2 \). Applying the formula gives the radius \( r_2 = 2^2 \cdot a_0 = 4a_0 \).

Step 4: Find the Ratio of the Radii

The ratio of the radii of the second to the first Bohr orbit is \( \frac{r_2}{r_1} = \frac{4a_0}{a_0} = 4 \).

Step 5: Verify and Choose the Correct Option

Therefore, the ratio of the radii of the second and first Bohr orbits of the hydrogen atom is 4. This corresponds to option (d).

Key Concepts

Bohr radius

The Bohr radius is a fundamental component in atomic physics, especially when studying the hydrogen atom. It represents the smallest possible distance between the proton and the electron in a hydrogen atom when in its ground state. This fundamental constant is denoted as \( a_0 \).

• Defined mathematically, the Bohr radius is given as \( a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_e e^2} \), where \( \varepsilon_0 \) is the vacuum permittivity, \( \hbar \) is the reduced Planck's constant, \( m_e \) is the electron mass, and \( e \) is the electric charge of the electron.
• In simpler terms, it sets the scale for the size of atoms and is approximately equal to 0.529 angstroms or \( 0.529 \times 10^{-10} \text{m} \).
• The Bohr radius is pivotal in calculating the sizes of other orbits in the atom, influencing how we understand atomic structure and behavior.

Understanding this concept is crucial for anyone diving into atomic structure and quantum mechanics, as it's foundational for calculating atomic orbitals in the Bohr model.

Principal quantum number

The principal quantum number, denoted by \( n \), is a primary factor in the Bohr model, which quantifies an electron's energy level within an atom. Each positive integer value of \( n \) corresponds to a distinct energy level or shell.

• \( n = 1 \) is the smallest and represents the ground state, the lowest energy level of an atom.
• Higher numbers, such as \( n = 2, 3, 4, \ldots \), indicate excited states which are farther from the nucleus and associated with higher energy.
• The formula relating the radius of an orbit to \( n \) is \( r_n = n^2 a_0 \).

The principal quantum number not only aids in calculating the radius of electron orbits but also helps predict the emission spectrum of hydrogen. As electrons move between these levels, energy is absorbed or emitted, leading to spectral lines specific to each element.

Hydrogen atom

The hydrogen atom is the simplest atom and serves as a fundamental model in atomic physics due to its single electron and proton. Its study is crucial for understanding atomic behavior and quantum mechanics.

• The hydrogen atom was the first to be described accurately using quantum theory by Niels Bohr in 1913.
• The Bohr model of the hydrogen atom simplified complex atomic behaviors using quantum theory concepts. It introduced quantized orbits for electrons, where they can only exist in fixed, discrete energy levels.
• In the Bohr model, electron transitions between orbits correspond to the absorption or emission of photons, leading to the unique spectral lines of hydrogen.

Studying the hydrogen atom allowed scientists to develop critical insights into atomic structure and energize the development of quantum mechanics, paving the way for more complex atomic theories.