Class 12 Physics NCERT Solutions Chapter 12: Atom Important Questions

Introduction

Class 12 Physics is a crucial stage in a student's academic journey, especially when preparing for higher education and competitive exams. Chapter 12 of the NCERT Class 12 Physics textbook, titled "Atom," covers fundamental concepts related to atomic structure, including models of the atom, atomic spectra, and the concept of quantization. Mastering this chapter is essential for students aiming to excel in their board exams and various entrance tests. This article delves into the important questions from Chapter 12, providing detailed solutions and explanations to help students grasp the core concepts effectively.

1. Atomic Models: Key Concepts

1.1. The Thomson Model

Question: Describe the Thomson model of the atom and explain its limitations.

Answer: The Thomson model, also known as the "plum pudding model," proposed by J.J. Thomson in 1904, depicts the atom as a positively charged sphere with negatively charged electrons embedded within it. According to this model, the positive charge is uniformly distributed throughout the atom, and the electrons are scattered within this positive charge cloud.

Limitations:

1. Electron Stability: The model could not explain the stability of electrons within the atom.
2. Spectral Lines: It failed to account for the discrete spectral lines observed in atomic spectra.
3. Nuclear Structure: The model did not account for the presence of a nucleus, as evidenced by later experiments.

1.2. The Rutherford Model

Question: Explain Rutherford's gold foil experiment and how it led to the Rutherford model of the atom.

Answer: Rutherford's gold foil experiment, conducted in 1909, involved directing alpha particles at a very thin gold foil. Most of the particles passed through with minimal deflection, but a small fraction was deflected at large angles.

Key Findings:

1. Atom's Structure: The atom consists of a very small, dense nucleus at the center, which contains most of the atom's mass and positive charge.
2. Electron Arrangement: Electrons orbit around this nucleus, and the majority of the atom is empty space.

Model: Rutherford proposed that the atom has a small, dense nucleus surrounded by orbiting electrons. This model replaced the Thomson model but had its limitations in explaining the stability of electron orbits and the discrete spectral lines.

1.3. The Bohr Model

Question: Discuss Bohr's model of the atom and its significance.

Answer: Niels Bohr, in 1913, proposed a new model to address the limitations of Rutherford's model. The Bohr model introduced the concept of quantized orbits for electrons.

Key Features:

1. Quantized Orbits: Electrons orbit the nucleus in fixed, quantized orbits or energy levels without radiating energy.
2. Energy Levels: Electrons can transition between these orbits by absorbing or emitting discrete amounts of energy, which explains the atomic spectra.
3. Stability: The model accounts for the stability of electrons in these orbits and the observed spectral lines.

Significance: The Bohr model successfully explained the Rydberg formula for the spectral lines of hydrogen and laid the foundation for quantum mechanics.

2. Quantum Mechanical Model

2.1. Wave-Particle Duality

Question: What is the wave-particle duality of electrons, and how does it impact the understanding of atomic structure?

Answer: Wave-particle duality is a fundamental concept in quantum mechanics, stating that electrons exhibit both wave-like and particle-like properties. This duality impacts the understanding of atomic structure in several ways:

1. Wave Nature: Electrons, when treated as waves, can be described by wavefunctions, which provide information about the probability distribution of an electron's position.
2. Particle Nature: When observed, electrons exhibit particle-like behavior, such as in electron impact experiments.
3. Quantum Mechanics: The wave-particle duality led to the development of quantum mechanics, which provides a more accurate description of atomic structure than classical models.

2.2. Heisenberg Uncertainty Principle

Question: Explain the Heisenberg Uncertainty Principle and its significance in atomic theory.

Answer: The Heisenberg Uncertainty Principle, formulated by Werner Heisenberg in 1927, states that it is impossible to simultaneously determine both the exact position and momentum of a particle. Mathematically, it is expressed as:

$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$

where $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum, and $h$ is Planck's constant.

Significance:

1. Quantum Mechanics: The principle highlights the limitations of classical mechanics and emphasizes the need for quantum mechanics.
2. Electron Orbits: It explains why electrons cannot be described as moving in precise orbits but rather exist in probabilistic regions called orbitals.

2.3. Schrödinger's Wave Equation

Question: Describe Schrödinger's wave equation and its role in quantum mechanics.

Answer: Schrödinger's wave equation, developed by Erwin Schrödinger in 1926, is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is given by:

$\hat{H} \Psi = E \Psi$

where $\hat{H}$ is the Hamiltonian operator, $\Psi$ is the wavefunction, and $E$ is the energy of the system.

Role:

1. Wavefunctions: The wave equation provides solutions called wavefunctions, which describe the probability distribution of particles.
2. Quantum States: It helps in determining the allowed energy levels of electrons in an atom and the shapes of atomic orbitals.

3. Atomic Spectra

3.1. Hydrogen Atom Spectrum

Question: Explain the Balmer series in the hydrogen atom spectrum.

Answer: The Balmer series refers to the set of spectral lines of hydrogen that are visible in the visible region of the electromagnetic spectrum. These lines are produced when an electron in a hydrogen atom falls from a higher energy level (n ≥ 3) to the second energy level (n = 2).

Balmer Formula: The wavelengths of the spectral lines in the Balmer series can be calculated using the formula:

$\frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{n^2} \right)$

where $\lambda$ is the wavelength, $R$ is the Rydberg constant, and $n$ is the principal quantum number of the higher energy level.

3.2. Rydberg Formula

Question: Derive the Rydberg formula for the wavelengths of spectral lines in hydrogen.

Answer: The Rydberg formula describes the wavelengths of spectral lines of many chemical elements. For hydrogen, it is given by:

$\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

where:

• $\lambda$ is the wavelength of the emitted light,
• $R$ is the Rydberg constant ($1.097 \times 10^7 \text{ m}^{-1}$),
• $n_1$ and $n_2$ are the principal quantum numbers of the lower and higher energy levels, respectively.

Derivation: The formula is derived from the Bohr model of the atom, where the energy difference between two quantized orbits corresponds to the energy of the emitted or absorbed photon. The wavelength $\lambda$ is related to the energy $E$ of the photon by:

$E = \frac{hc}{\lambda}$

where $h$ is Planck's constant and $c$ is the speed of light. By substituting the energy levels into this equation and solving, the Rydberg formula is obtained.

4. Quantum Numbers

4.1. Types of Quantum Numbers

Question: Describe the four quantum numbers and their significance in determining the state of an electron in an atom.

Answer: Quantum numbers are parameters used to describe the unique state of an electron in an atom. They include:

1. Principal Quantum Number ($n$): Defines the main energy level of an electron and its distance from the nucleus. Higher $n$ values indicate higher energy levels.

2. Azimuthal Quantum Number ($l$): Determines the shape of the electron's orbital. It can take integer values from 0 to $n-1$. Each value of $l$ corresponds to a different type of orbital (s, p, d, f).

3. Magnetic Quantum Number ($m_l$): Specifies the orientation of the orbital within a given energy level and shape. It can take integer values from $-l$ to $+l$.

4. Spin Quantum Number ($m_s$): Describes the intrinsic angular momentum (spin) of the electron. It can be $+\frac{1}{2}$ or $-\frac{1}{2}$.

Significance:

• These quantum numbers collectively describe the electron's position, energy, and angular momentum, providing a complete description of its state.

4.2. Pauli Exclusion Principle

Question: State the Pauli Exclusion Principle and its impact on the arrangement of electrons in an atom.

Answer: The Pauli Exclusion Principle, formulated by Wolfgang Pauli in 1925, states that no two electrons in an atom can have the same set of all four quantum numbers. This principle ensures that each electron in an atom has a unique set of quantum numbers, which impacts electron configuration and the structure of the periodic table.

Impact:

• Electron Configuration: Determines the maximum number of electrons that can occupy each energy level and orbital.
• Chemical Properties: Influences the chemical properties and bonding behavior of elements.

5. Important Derivations and Formulas

5.1. Derivation of the Rydberg Formula

Question: Derive the Rydberg formula using the Bohr model of the atom.

Answer: The Rydberg formula can be derived from the Bohr model by calculating the difference in energy between two quantized electron orbits. The energy of an electron in the $n$-th orbit is given by:

$E_n = - \frac{k e^2}{2a_0 n^2}$

where $a_0$ is the Bohr radius and $k$ is Coulomb's constant. The energy difference between two levels $n_1$ and $n_2$ is:

$\Delta E = E_{n_1} - E_{n_2} = \frac{k e^2}{2a_0} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

The wavelength $\lambda$ of the emitted photon is related to the energy by:

$E = \frac{hc}{\lambda}$

Combining these equations gives the Rydberg formula:

$\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

where $R$ is the Rydberg constant.

5.2. Derivation of the Energy Levels in the Hydrogen Atom

Question: Derive the expression for the energy levels of a hydrogen atom using the Bohr model.

Answer: According to the Bohr model, the centripetal force on the electron is provided by the Coulomb force between the electron and the nucleus.

The centripetal force is:

$F_c = \frac{m v^2}{r}$

where $m$ is the electron mass, $v$ is its velocity, and $r$ is the radius of the orbit.

The Coulomb force is:

$F_e = \frac{k e^2}{r^2}$

Setting these equal gives:

$\frac{m v^2}{r} = \frac{k e^2}{r^2}$

Solving for $v^2$ gives:

$v^2 = \frac{k e^2}{m r}$

The energy of the electron in orbit is the sum of its kinetic energy ($K$) and potential energy ($U$):

$K = \frac{1}{2} m v^2 = \frac{k e^2}{2r}$

$U = - \frac{k e^2}{r}$

The total energy $E$ is:

$E = K + U = - \frac{k e^2}{2r}$

Using the quantization condition $m v r = n \hbar$ (where $\hbar$ is the reduced Planck's constant), and solving for $r$ and $E$, gives the energy levels:

$E_n = - \frac{k e^2}{2a_0 n^2}$

where $a_0$ is the Bohr radius.

Conclusion

Mastering Chapter 12 of the Class 12 Physics NCERT textbook requires a thorough understanding of atomic models, quantum mechanics, and atomic spectra. The questions discussed in this article cover essential aspects of the chapter, providing a solid foundation for students preparing for exams. By reviewing these important questions and their solutions, students can enhance their comprehension of atomic structure and quantum theory, paving the way for success in their academic endeavors.