Atomic Physics

Step 1: Lyman series ends at . Transition from is the series limit.
Step 2: Balmer series ends at . 1st member: , 2nd member: .
Step 3: Paschen series ends at . 1st member: , 2nd member: .
*Note: Based on standard textbook diagrams for A, B, C.*

The time period of a simple pendulum is determined by the formula:

where is the length of the pendulum and is the effective acceleration due to gravity. Initially, with the metallic bob in air, the time period is given as 10 seconds. Upon immersing the bob in water, its apparent weight changes due to the buoyant force, thus changing .

The relative density (specific gravity) of the metallic bob is 5, meaning the bob is 5 times as dense as water. When immersed, the effective acceleration due to gravity becomes:

Given that the relative density is 5, or , then:

The new time period when the bob is immersed in water can be calculated as:

Comparing with the original time period in air, for the immersed time period we can set:

Since was initially 10 s, we equate:

Solving for :

Thus, the value of is 1.25. The task specifies a range of 5 to 5, which appears to be an oversight since the value of logically derives as 1.25 from the calculations.

To solve the problem, we first need to find the time at which the ratio of the number of nuclei of B to A is . Begin with the decay formula for any radioactive material: , where is the initial number of nuclei and is the decay constant.

Since both materials initially have the same number of nuclei, , the number of nuclei at a given time will be:

For material A:

For material B:

The given condition is that the ratio . Substitute the expressions:

This simplifies to:

Taking the natural logarithm on both sides gives:

Thus,

We need to identify the value of in such that the solution fits within the expected range.

Comparing with , we find .

Confirming the range: The value of we calculated is , which coincides with the given range of 9 to 9.

Therefore, the value of is 9.

The change in energy during the transition is given by:

The relationship between energy and wavelength is given by:

Rearranging for :

To find the effect of recoil, consider the conservation of momentum:

Solving for the velocity of the recoiling atom:

The total energy change considering kinetic energy and emitted photon energy is given by:

Substituting for and simplifying:

Using the energy conservation equation:

Solving this quadratic equation for :

Approximating for small changes:

Substituting the given values:

The percentage change in wavelength is approximately .

To solve this problem, we use the Rydberg formula for hydrogen-like ions to calculate the frequency of emitted radiation during electron transitions: where is the frequency, is the Rydberg constant ( ), is the atomic number, and and are the principal quantum numbers of the two levels.

Given , we find:

Using this as a basis, let's calculate :

From ,

Solving for ,

Substitute this into :

Comparing this to the requested format , . The value .

Step 1: Calculate the energy of the emitted photon.

Energy difference between two energy levels of hydrogen: Convert to joules:

Step 2: Relate photon energy to its momentum.

For a photon:

Step 3: Momentum conservation for recoil.

The atom recoils with momentum equal and opposite to the photon’s: So, recoil speed:

Step 4: Compare with given expression.

Final Answer:

The question provides an assertion and a reason regarding the Bohr model.

• Assertion (A): The Bohr model is applicable to hydrogen and hydrogen-like atoms only.
• Reason (R): The formulation of Bohr model does not include repulsive force between electrons.

Let's evaluate both statements:

1. Understanding the Assertion (A):
   • The Bohr model, introduced by Niels Bohr in 1913, describes the hydrogen atom or hydrogen-like ions (such as He⁺, Li²⁺, etc.) which contain only one electron. It accounts for the quantization of angular momentum and provides expressions for the energy levels of these one-electron systems. The model works effectively only for single-electron systems because it does not consider electron-electron interactions, which are significant in multi-electron atoms.
2. Understanding the Reason (R):
   • The Bohr model assumes that electrons orbit the nucleus in fixed paths without accounting for the repulsive forces between multiple electrons, as it focuses solely on the electrostatic force of attraction between a single electron and the positively charged nucleus. This is why it is not applicable to multi-electron atoms where inter-electronic repulsions play an important role.

Now, let's match these with the given options:

• Option 1: Both A and R are true but R is NOT the correct explanation of A.
• Option 2: A is false but R is true.
• Option 3: Both A and R are true and R is the correct explanation of A. (Correct Answer)
• Option 4: A is true but R is false.

Conclusion: Both the assertion and reason are true, and importantly, the reason correctly explains why the Bohr model is limited to hydrogen and hydrogen-like atoms. Therefore, the correct answer is: Both A and R are true and R is the correct explanation of A.

This problem requires finding the final principal quantum number ( ) of an electron in a hydrogen atom after it transitions from a higher energy state by emitting a photon of a specific energy.

Concept Used:

The energy of an electron in the energy level (or orbit) of a hydrogen atom is given by the formula:

The "ground state" corresponds to . The "first excited state" corresponds to , the "second excited state" to , and so on. Therefore, the excited state corresponds to the principal quantum number .

When an electron makes a transition from a higher initial energy state (with quantum number ) to a lower final energy state (with quantum number ), it emits a photon. The energy of this photon ( ) is equal to the difference in the energy levels.

Step-by-Step Solution:

Step 1: Determine the initial principal quantum number ( ) of the electron.

The problem states that the electron is initially in the "fourth excited state". Using the relationship that the excited state corresponds to , for the fourth excited state ( ), the principal quantum number is:

Step 2: Calculate the energy of the electron in this initial state ( ).

Using the energy formula with :

Step 3: Use the energy of the emitted photon to find the energy of the final state ( ).

The energy of the emitted photon is given as eV. The final state is the energy state, so its energy is . The relationship is:

Rearranging to solve for :

Substituting the known values:

Step 4: Determine the principal quantum number ( ) of the final energy state.

We use the energy formula again, this time with the final energy :

Substituting the value of we just found:

Now, we solve for :

Since the principal quantum number must be an integer, we can round the value of to the nearest integer.

Thus, the integer value of n for the final energy state is 2.

Concept:
When an -particle approaches a heavy nucleus head-on, it is slowed down by the electrostatic (Coulomb) repulsion. At the distance of closest approach, the initial kinetic energy of the -particle is completely converted into electrostatic potential energy. For a head-on collision: The Coulomb potential energy between two charges is:
Step 1: Identify the charges.
Charge of -particle:
Charge of gold nucleus:
Step 2: Write the energy balance equation. Given kinetic energy: At closest approach : Using the standard nuclear physics constant:
Step 3: Substitute numerical values.
Step 4: Find the diameter of the nucleus. Since the -particle just touches the nuclear boundary: