Problem 5 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The radius of the second Bohr orbit, in terms of the Bohr radius, \(a_{0}\), in \(\mathrm{Li}^{2+}\) is: (a) \(\frac{2 a_{0}}{3}\) (b) \(\frac{4 a_{0}}{9}\) (c) \(\frac{4 a_{0}}{3}\) (d) \(\frac{2 a_{0}}{9}\)

Step-by-Step Solution

Answer

(c) \( \frac{4a_0}{3} \)

Step 1: Identify the Formula

The radius of the nth Bohr orbit is given by the formula \( r_n = n^2 \frac{a_0}{Z} \), where \( n \) is the orbit number, \( a_0 \) is the Bohr radius, and \( Z \) is the atomic number of the nucleus.

Step 2: Substitute Values

For \( \text{Li}^{2+} \), the atomic number \( Z \) is 3 because lithium has 3 protons. For the second Bohr orbit, \( n = 2 \). Substitute these values into the formula: \( r_2 = 2^2 \frac{a_0}{3} \).

Step 3: Simplify the Expression

Calculate \( 2^2 \) which gives 4, and then substitute it into the expression: \( r_2 = \frac{4a_0}{3} \).

Step 4: Compare with Options

Compare the simplified expression \( \frac{4a_0}{3} \) with the given options. The correct option is (c) \( \frac{4a_0}{3} \).

Key Concepts

Bohr Radius

The Bohr radius is a fundamental physical constant that represents the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is denoted as \( a_0 \) and forms the basis for calculating the sizes of quantum orbits in the Bohr Model. This concept is crucial because:

• It provides a specific reference point for determining the electron's orbital paths around the nucleus.
• Its value is approximately \( 5.29 \times 10^{-11} \) meters, and it sets the scale for atomic dimensions.

In the exercise, the Bohr radius \( a_0 \) is used to calculate the radius of electron orbits in multi-electron atoms, such as the lithium ion \( \mathrm{Li}^{2+} \). By multiplying \( a_0 \) by factors determined by the orbit's quantum numbers and the atomic number, it allows us to find the size of specific orbits in different elements."

Atomic Number

The atomic number denoted as \( Z \) is a crucial integer variable that defines the number of protons in the nucleus of an atom. Each element has a unique atomic number which determines its position in the periodic table and its chemical properties:

• It equals the number of electrons in a neutral atom, balancing the positive charge of the protons.
• In a charged ion, such as \( \mathrm{Li}^{2+} \), the atomic number remains the same, but electron count varies.

For lithium, \( Z = 3 \) since it has three protons. This atomic number is essential in determining the radii of Bohr orbits when using the formula \( r_n = n^2 \frac{a_0}{Z} \). Understanding the atomic number is vital for visualizing electron distribution and size of atomic orbits in multi-electron systems."

Quantum Orbits

Quantum orbits in the Bohr Model describe the specific paths that electrons take as they travel around an atomic nucleus. These orbits are characterized by discrete, quantized energy levels, which means that electrons can only reside at certain distances from the nucleus.

• The principal quantum number \( n \) describes the orbit's size and energy level; larger \( n \) values indicate higher energy and larger orbits.
• The second Bohr orbit for an electron in a system like \( \mathrm{Li}^{2+} \) corresponds to \( n = 2 \).

In the solved exercise, the second orbit radius was calculated using \( n = 2 \) and the corresponding adjustments for the atomic number, resulting in the formula \( r_2 = \frac{4a_0}{3} \). Quantum orbits help visualize how electrons are arranged in atoms and play a significant role in determining the chemical properties of elements."