Alpha Particle Scattering And Rutherford S Nuclear Model Of Atom

Bohr’s Postulates for Hydrogen Atom:

(A) The electron in the atom revolves in specific orbits without radiating energy.
(B) The electron can only occupy discrete energy levels or orbits. The energy of the electron in the n-th orbit is given by:

where k is Coulomb's constant, e is the electron charge, and r_n is the radius of the n-th orbit.

(C) The transition of the electron between orbits results in the emission or absorption of energy corresponding to the difference in energy levels.

Emission and Absorption Spectra:

For transition between n = 1 and n = 4, the emission and absorption spectra can be derived using the following formulas:

1. Emission:

When the electron transitions from a higher to a lower energy state, the frequency of the emitted radiation is:

2. Absorption:

The electron absorbs energy when it moves to a higher energy state.

(A) Emission Spectrum Lines:

The number of lines in the emission spectrum is given by the number of possible transitions from n=4 to n=1, including intermediate states. The transitions are as follows:

There are a total of 6 emission lines.

(B) Absorption Spectrum Lines:

Similarly, for absorption, the number of lines will be the same, as absorption also happens when the electron moves from a lower to a higher energy state. The number of lines is 6.

Step 1: Understanding the Concept:
The proof is based on the principle of conservation of energy. An -particle (charge ) is accelerated by a potential difference V, gaining kinetic energy. As it approaches a positively charged nucleus (charge ), the electrostatic repulsive force slows it down. The distance of closest approach ( ) is the point where the -particle momentarily stops and turns back. At this point, all of its initial kinetic energy has been converted into electrostatic potential energy of the system.

Step 2: Key Formula or Approach:
1. Kinetic Energy (KE) gained by the -particle:
2. Electrostatic Potential Energy (PE) at separation :
By conservation of energy:

Step 3: Detailed Explanation (Derivation):
Equating the initial kinetic energy with the final potential energy: We can cancel from both sides: Now, we solve for the distance of closest approach, : To get the numerical value, we substitute the known constants: \begin{itemize} \item Coulomb's constant, N·m /C . \item Elementary charge, C. \end{itemize} Since 1 Angstrom ( ) = meters, we can express the distance in Angstroms: This proves the required relation.

Step 1: Understanding the Concept:
Rutherford's gold foil experiment (or -particle scattering experiment) involved bombarding a thin sheet of gold foil with positively charged alpha particles. The scattering pattern of these particles led to the discovery of the atomic nucleus. The question asks for the fundamental force responsible for this scattering.

Step 2: Detailed Explanation:
Let's analyze the forces involved:
\begin{itemize} \item Nature of Particles: The -particle is a helium nucleus ( ), which is positively charged. The target, the nucleus of a gold atom, is also positively charged (as it contains protons).
\item Coulomb's force: This is the electrostatic force of repulsion between two like charges (positive -particle and positive nucleus). This force is long-range and follows an inverse-square law ( ), which perfectly explains why most particles passed straight through (far from a nucleus) while a few were deflected at large angles (passed close to a nucleus).
\item Gravitational force: This is the force of attraction between masses. While it exists, it is extremely weak compared to the electrostatic force at the atomic scale and is completely negligible in this context.
\item Nuclear force: This is a very strong, short-range attractive force that holds protons and neutrons together within the nucleus. It only acts over distances of about m. The -particles are repelled by the Coulomb force long before they get close enough for the nuclear force to be significant, except in very high-energy head-on collisions. The scattering itself is governed by the electrostatic repulsion.
\item Magnetic force: This force acts on moving charges within a magnetic field. There is no external magnetic field applied in this experiment, so this force is not the cause of the scattering.
\end{itemize} The primary interaction causing the -particles to deviate from their straight paths is the electrostatic repulsion between the positively charged -particles and the positively charged gold nuclei. This force is known as Coulomb's force.

Step 3: Final Answer:
The force responsible for the scattering is Coulomb's force. Therefore, option (B) is correct.