Problem 56 - 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 \(n^{\text {h }}\) orbit of the atom and List-II contains options showing how they depend on \(n\) List-I List-II (I) Radius of the \(n^{\text {h }}\) orbit \(\quad\) (P) \(\propto n^{-2}\) (II) Angular momentum of the electron in the \(n^{\text {th }}\) orbit \(\quad(\mathrm{Q}) \propto n^{-1}\) (III) Kinetic energy of the electron in the \(n^{\text {th }}\) orbit \((R) \propto n^{0}\) (IV) Potential energy of the electron in the \(n^{\text {th }}\) orbit \(\quad(\mathrm{S}) \propto n^{1}\) (T) \(\propto n^{2}\) \((\mathrm{U}) \propto n^{1 / 2}\) Which of the following options has the correct ombination considering List-I and List-II? (a) (II), (R) (b) \((\mathrm{II}),(\mathrm{Q})\) (c) (I), (P) (d) (I), (T)

Step-by-Step Solution

Answer

Correct option is (I), (T) due to radius proportional to \( n^2 \).

Step 1: Understanding Bohr's Model

In Bohr's model, an electron revolves around the nucleus in permitted orbits. Each orbit corresponds to a certain quantized value of angular momentum, radius, and energy.

Step 2: Analyzing Quantity I – Radius of the n-th Orbit

According to Bohr's model, the radius of the n-th orbit is given by the formula \( r_n = n^2 a_0 \), where \( a_0 \) is the Bohr radius. Here, radius is proportional to \( n^2 \). Hence, the corresponding option in List-II is (T).

Step 3: Analyzing Quantity II – Angular Momentum

In Bohr's model, the angular momentum \( L \) of an electron in the n-th orbit is quantized and given by \( L = n\hbar \), where \( \hbar \) is the reduced Planck's constant. Therefore, angular momentum is proportional to \( n \) or \( n^1 \). Hence, the corresponding option in List-II is (S).

Step 4: Analyzing Quantity III – Kinetic Energy

The kinetic energy \( KE \) of an electron in the n-th orbit decreases with increasing n and is inversely proportional to \( n^2 \). Since kinetic energy is proportional to \( n^{-2} \), the corresponding option in List-II is (P).

Step 5: Analyzing Quantity IV – Potential Energy

The potential energy \( PE \) of an electron in an orbit is proportional to \(-\frac{1}{n^2}\). Since it is inversely proportional to \( n^2 \), the corresponding option from List-II is also (P).

Step 6: Conclusion on Correct Combinations

Based on the options given in the question, none of the provided options match with our derived dependencies for List-I quantities. However, examining the possible experimental mix-ups, only option (I), (T) matches possibly due to interpretation errors in the defined dependencies in this context.

Key Concepts

Quantized Orbits

In Bohr's model, electrons revolve around the nucleus in fixed paths known as quantized orbits. Unlike earlier models, Bohr introduced the concept that these orbits have specific allowed radii and energy levels. This means that electrons can only exist in these defined paths and nowhere else.

• Each orbit is characterized by an integer, called the principal quantum number, denoted by \(n\).
• As \(n\) increases, the orbit radius increases, indicating that the electron is further from the nucleus.
• The concept of quantized orbits explains why electrons don't spiral into the nucleus despite attraction from protons.

This model helped in explaining the stability of atoms and the emission spectra of elements.

Angular Momentum

Angular momentum in Bohr's model is quantized, meaning that it can only take on certain discrete values. This is a significant departure from classical mechanics.

• The angular momentum \(L\) of an electron in an orbit is given by \(L = n\hbar\), where \(\hbar\) is the reduced Planck's constant.
• It is directly proportional to the principal quantum number \(n\).
• This quantization condition helps in determining the possible energy levels an electron can have within an atom.

Bohr's introduction of quantized angular momentum reconciled atomic stability and spectral lines not explained by the earlier models.

n-th Orbit Energy Dependencies

In the n-th orbit of an atom, several energy-related quantities depend on \(n\).

• The energy of the electron in the n-th orbit is given by \(E_n = -\left(\frac{1}{n^2}\right) E_0\), indicating it is inversely proportional to \(n^2\).
• The energy levels become less negative as \(n\) increases, meaning higher energy orbits are less tightly bound to the nucleus.
• This dependency explains why excited electrons can move to higher energy levels by absorbing specific amounts of energy.

Thus, these dependencies are crucial in explaining electronic transitions and related electromagnetic spectra.

Kinetic Energy Proportionality

The kinetic energy of an electron in a Bohr orbit also shows a dependency on \(n\).

• It can be expressed as \(KE \propto \frac{1}{n^2}\), indicating an inverse square relationship.
• As the orbit number \(n\) increases, the kinetic energy decreases.
• This reduction in kinetic energy at higher orbits aligns with electrons being further from the nucleus and experiencing less force.

This relationship helps in understanding why electrons in higher orbits require lesser energy to maintain their orbit compared to those closer to the nucleus.

Potential Energy in Bohr Orbits

In Bohr's model, the potential energy of an electron within an orbit similarly depends on its principal quantum number \(n\).

• The potential energy \(PE\) for an electron in an orbit is given by \(PE \propto -\frac{1}{n^2}\).
• This negative sign indicates an attractive force between the electron and nucleus.
• As \(n\) increases, the negative potential energy value becomes less negative, which means it requires less energy to remove the electron from the atom.

Understanding potential energy in Bohr orbits is essential for explaining ionization and electron affinity.