4 Most SAQ’s of Atoms Chapter in Inter 2nd Year Physics (TS/AP)

4 Marks

SAQ-1 : What are the limitations of Bohr’s theory of hydrogen atom?

Introduction

Bohr’s theory, developed by Niels Bohr in 1913, marked a significant advancement in our understanding of the hydrogen atom. Although it successfully explained several aspects of hydrogen atom behavior, it encountered limitations that were later addressed by quantum mechanics. This analysis aims to elucidate these limitations for Class 12 Board Exam students and enhance online accessibility.

Limitations of Bohr’s Theory of the Hydrogen Atom

1. Applicability to Hydrogen Atom Only:
   • Bohr’s theory was specifically tailored for the hydrogen atom, characterized by an atomic number (z) of 1.
   • It fails for atoms with an atomic number greater than 1, thus limiting its application to more complex atoms.
2. Unclear Explanation of Electron Orbits:
   • The theory posits that electrons move in circular orbits.
   • However, it does not justify the exclusion of elliptical orbits, which could also be feasible.
3. Inability to Explain Spectral Line Intensities: Bohr’s theory falls short in explaining why spectral lines exhibit varying intensities, lacking clarity on why some lines appear brighter or darker.
4. Ignoring the Wave Properties of Electrons:
   • Electrons exhibit wave-particle duality, possessing both particle and wave-like properties.
   • The theory solely treats electrons as particles, overlooking their wave properties.

Summary

Bohr’s theory significantly contributed to the atomic structure’s comprehension, particularly that of the hydrogen atom. Nonetheless, its limitations—including its exclusive applicability to hydrogen, lack of explanation for electron orbit shapes, insufficient details on spectral line intensities, and negligence of electron wave properties—prompted the evolution of more sophisticated quantum mechanics models, offering a deeper insight into atomic structure.

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SAQ-2 : Explain the different types of spectral series of hydrogen atom.

Introduction

The spectrum of the hydrogen atom, consisting of diverse spectral lines, reveals critical insights into the atom’s structure and behavior. This exploration covers the various spectral series resulting from electron transitions between energy levels within the atom, an essential concept for students preparing for their Class 12 Board Exam and anyone interested in atomic physics.

Hydrogen Atom and Energy Transitions

A hydrogen atom features an electron orbiting a nucleus, capable of existing at different energy levels. Transitions from a higher energy level to a lower one emit light, forming the hydrogen spectrum.

Rydberg Formula

The wavelength of emitted light is calculated using the Rydberg formula:

$$1/\lambda = RZ^2(1/p^2 – 1/n^2)$$

Where

1. R is the Rydberg constant (1.09737 × 10⁷ m⁻¹)
2. Z is the atomic number (for hydrogen Z = 1)
3. P is the lower energy level
4. n is the higher energy level

Spectral Series in Hydrogen Spectrum

Different energy level transitions yield various spectral lines:

1. Lyman Series:
   • Transitions to the first energy level (p = 1) from higher levels (n=2,3,4,5,…).
   • Located in the ultraviolet region.
2. Balmer Series:
   • Transitions to the second energy level (p = 2) from higher levels (n=3,4,5,6,…).
   • Found in the visible region.
3. Paschen Series:
   • Transitions to the third energy level (p = 3) from higher levels (n=4,5,6,7,…).
   • Located in the infrared region.
4. Brackett Series:
   • Transitions to the fourth energy level (p = 4) from higher levels (n=5,6,7,8,…).
   • Also in the infrared region.
5. Pfund Series:
   • Transitions to the fifth energy level (p = 5) from higher levels (n=6,7,8,9,…).
   • Found in the infrared region.

Summary

The hydrogen atom’s spectrum and its spectral series, including Lyman, Balmer, Paschen, Brackett, and Pfund, delineate the electron transitions between energy levels, categorized by their electromagnetic spectrum position. These spectral lines offer profound insights into the atom’s structure, aiding in the analysis of unknown samples and understanding their electronic configurations.

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SAQ-3 : Derive an expression for potential and kinetic energy of an electron in any orbit of a hydrogen atom according to Bohr’s atomic model. How does P.E. change with increasing n?

Introduction

Bohr’s atomic model offers a fundamental explanation of how electrons orbit the nucleus in a hydrogen atom. It elucidates that electrons travel in specific paths or orbits, with each orbit corresponding to a distinct energy level. This article will delve into the calculations for the kinetic and potential energy of an electron within these orbits and examine the variation in potential energy with increasing principal quantum number (n).

Derivation of Expressions for Potential and Kinetic Energy

1. Understanding the Forces: Electrons revolve around the nucleus in circular paths due to the electrostatic force of attraction between the oppositely charged electron and nucleus.
2. Bohr’s Rule for Angular Momentum: Bohr postulated that the angular momentum of an electron is quantized and given by L = nℏ, where n is the principal quantum number, and ℏℏ is the reduced Planck’s constant.
3. Finding the Radius: The radius (r_n​) of the electron’s orbit is derived from Bohr’s quantization condition, indicating that specific, quantized orbits are possible.
4. Kinetic Energy: The kinetic energy (K.E.) of an electron in orbit is given by the expression: $$K.E. = \frac{1}{2}mv^2$$ Using Bohr’s model, the kinetic energy can also be expressed as: $$K.E. = \frac{k_e^2Z^2e^4m}{2\hbar^2n^2}$$ ​where k_e​ is Coulomb’s constant, Z is the atomic number (for hydrogen, Z = 1), e is the charge of an electron, m is the mass of an electron, and n is the principal quantum number.
5. Potential Energy: The potential energy (P.E.) is associated with the electrostatic attraction between the electron and the nucleus and is given by: $$P.E. = -\frac{k_eZ^2e^4m}{\hbar^2n^2}$$ ​Notably, the potential energy is twice the kinetic energy but negative, indicating that the total energy is negative, signifying bound states.

Summary

Bohr’s atomic model provides the framework for calculating the kinetic and potential energy of an electron in a hydrogen atom’s orbit. The kinetic energy reflects the electron’s motion, whereas the potential energy is indicative of its position relative to the nucleus. With increasing n, the potential energy becomes less negative, illustrating the electron’s increasing distance from the nucleus and a decrease in the absolute value of its binding energy.

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SAQ-4 : Describe Rutherford atom model. What are the drawbacks of this model?

Introduction

The Rutherford atomic model plays a crucial role in the atomic structure’s understanding. Proposed by Ernest Rutherford in 1911, this model introduced the concept of a small, dense, positively charged center known as the nucleus, around which electrons circulate, akin to planets orbiting the Sun. Despite its significant contributions, this model encountered notable shortcomings.

The Rutherford Atomic Model

1. Concept:
   • The atom possesses a central nucleus that contains most of the atom’s mass and positive charge.
   • Electrons, which are light and negatively charged, orbit the nucleus.
2. Experiment
   • Rutherford, alongside Ernest Marsden and Hans Geiger, executed an experiment involving alpha particles and gold foil.
   • A radioactive source emitted alpha particles, which were directed through gold foil. The deflection of these particles by the foil was observed using a fluorescent screen.
   • This experiment led to the conclusion that an atom’s positive charge is concentrated in a compact, dense nucleus.

Drawbacks of Rutherford’s Atomic Model

1. Stability of Atoms: The model could not elucidate why atoms are stable, failing to address why electrons orbiting the nucleus do not spiral inward and collide with it.
2. Electron Behavior: It did not offer a sustainable explanation for electron orbits around a positively charged nucleus, overlooking the fact that circulating charged particles should emit radiation and gradually lose energy.
3. Hydrogen Spectrum: Rutherford’s model was unable to account for the distinct lines observed in the hydrogen spectrum, lacking insight into electrons occupying specific energy levels, a phenomenon later clarified by quantum mechanics.

Summary

While the Rutherford atomic model significantly advanced our comprehension of the atom’s structure by highlighting the nucleus and electron orbits, it had pivotal drawbacks. These limitations, particularly regarding atomic stability, electron behavior, and the hydrogen spectrum, prompted the development of more advanced atomic models.