Problem 57 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following List-I contains some quantities for the \(n^{\text {w }}\) orbit of the atom and List-II contains options showing how they depend on \(n .\) List-I (I) Radius of the \(n^{\text {th }}\) orbit (II) Angular momentum of the electron in the \(n^{\text {th }}\) orbit (III) Kinetic energy of the electron in the \(n^{\text {t }}\) orbit (IV) Potential energy of the electron in the \(n^{\text {t }}\) orbit List-II (P) \(\propto n^{-2}\) (Q) \(\propto n^{-1}\) (R) \(\propto n^{0}\) \((\mathrm{S}) \propto n^{1}\) (T) \(\propto n^{2}\) \((\mathrm{U}) \propto n^{1 / 2}\) Which of the following options has the correct combination considering List-I and List-II? (a) (III), (S) (b) (IV), (Q) (c) (III), (P) (d) (IV), (U)

Step-by-Step Solution

Answer

(c) (III), (P)

Step 1: Understanding Radius in Bohr's Model

In Bohr's model, the radius of the \( n^{\text{th}} \) orbit is given by \( r_n = n^2 \times a_0 \), where \( a_0 \) is the Bohr radius. Since the radius is proportional to the square of \( n \), the dependence here is \( n^2 \). Therefore, it matches with option \( (T) \propto n^2 \).

Step 2: Analyzing Angular Momentum

The angular momentum of an electron in the \( n^{\text{th}} \) orbit is quantized and follows \( L_n = n \times \hbar \). Thus, the dependence is proportional to \( n \), which aligns with option \( (S) \propto n^1 \).

Step 3: Kinetic Energy Dependence

The kinetic energy for the \( n^{\text{th}} \) orbit in a hydrogen-like atom is \( K_n = \frac{E_n}{2} = -\frac{1}{2} \left( \frac{kZe^2}{r_n} \right) \), and using the expression for radius \( r_n = n^2 a_0 \), it results in \( \propto n^{-2} \). Thus, option \( (P) \propto n^{-2} \) is correct for kinetic energy.

Step 4: Potential Energy Dependence

Potential energy in the \( n^{\text{th}} \) orbit is given by \( U_n = -\frac{kZe^2}{r_n} \), implying \( \propto -\frac{1}{n^2} \). Hence, it is also proportional to \( n^{-2} \), similarly like the kinetic energy corresponds with option \( (P) \propto n^{-2} \).

Key Concepts

orbit radius

In Bohr's Model, the radius of an electron's orbit is one of the fundamental aspects. The radius for the nth orbit, denoted by \( r_n \), is calculated as \( r_n = n^2 \times a_0 \), where \( a_0 \) is known as the Bohr radius. This expression tells us that the orbit radius is proportional to the square of the principal quantum number \( n \).

• The Bohr radius \( a_0 \) represents the radius of the smallest orbit in the model, which is the ground state for a hydrogen atom.
• As the electron moves to higher energy levels (larger \( n \)), the orbit radius increases rapidly, following a quadratic relationship.

This quadratic dependence implies that if you double the principal quantum number \( n \), the radius of the orbit increases by a factor of four. Understanding this dependency helps explain the spatial scaling of different elements in hydrogen-like atoms.

angular momentum

Angular momentum in Bohr's model is a significant concept used to describe the motion of electrons in orbits. It is quantized, meaning it can only take on specific discrete values. The angular momentum \( L_n \) of an electron in the nth orbit is given by the formula \( L_n = n \times \hbar \), where \( \hbar \) is the reduced Planck's constant.

• This implies that angular momentum is directly proportional to \( n \), the quantum number.
• For example, if \( n = 1 \), the angular momentum is \( \hbar \); if \( n = 2 \), it becomes \( 2\hbar \).

This quantization of angular momentum was a revolutionary concept introduced by Bohr, and it helped in explaining why electrons don’t spiral into the nucleus, providing stability to the atom by maintaining electrons in fixed energy levels.

kinetic energy

The kinetic energy of an electron in the Bohr model reveals how the electron's energy changes with its position in an orbit. For the nth orbit, the kinetic energy \( K_n \) can be derived from the total energy \( E_n \), using the relation \( K_n = -\frac{1}{2}E_n \). For hydrogen-like atoms, substituting the expression for energy and radius gives \( K_n \propto n^{-2} \).

• This means as \( n \) increases, the kinetic energy decreases rapidly, by a factor of the square of \( n \).
• Kinetic energy inversely depends on the square of \( n \), implying that it becomes much smaller as the electron transitions to higher energy levels.

This relationship indicates that higher orbits have lower kinetic energy, reflecting a more relaxed electron motion farther from the nucleus, closely tied to the associated potential energy changes.

potential energy

Potential energy in Bohr's model refers to the energy due to the electron’s position in relation to the nucleus. This energy is given by \( U_n = -\frac{kZe^2}{r_n} \), where \( k \) is Coulomb's constant, \( Z \) is the atomic number, and \( e \) is the electron charge. By substituting the expression for the radius, the potential energy \( U_n \propto -n^{-2} \), showing an inverse square relationship.

• The potential energy is negative, indicating an attractive force between the negatively charged electron and the positively charged nucleus.
• As \( n \) increases, the magnitude of potential energy becomes less negative, reflecting the lower energy attraction in higher orbits.

Understanding potential energy and its dependency on the orbit number is crucial. It helps explain phenomena like ionization energy, which is the energy required to remove an electron from an atom, and how atoms interact with other electrons and nuclei in different chemical scenarios.