Atoms

In the Bohr model of the hydrogen atom, the total energy ( ), potential energy ( ), and kinetic energy ( ) in the n-th orbit are related as follows:
.
We are given the potential energy eV.
We can find the total energy of the electron in this orbit:
eV.
The total energy in the n-th orbit of a hydrogen atom is also given by the formula:
eV.
Equating the two expressions for :
.
So, the electron is in the second orbit, .
According to Bohr's second postulate, the angular momentum ( ) of the electron is quantized:
.
The de Broglie wavelength ( ) of the electron is given by .
From this, we can write .
Substitute this into the angular momentum equation:
.
.
This is the de Broglie wavelength in terms of the radius of the n-th orbit.
For the second orbit ( ), the wavelength is .
The radius of the n-th orbit is given by , where is the Bohr radius.
For , the radius is .
Substituting this into the expression for the wavelength:
.

Step 1: Understanding the Concept:
This question requires the use of the Rydberg formula, which describes the wavelengths of spectral lines emitted by a hydrogen atom when an electron transitions between different energy levels. We need to identify the specific transitions corresponding to the first Lyman line and the second Balmer line.
Step 2: Key Formula or Approach:
The Rydberg formula for the inverse of the wavelength ( ) is:
where is the Rydberg constant, is the principal quantum number of the final energy level, and is the principal quantum number of the initial energy level ( ).
- For the Lyman series, the final state is .
- For the Balmer series, the final state is .
Step 3: Detailed Explanation:
First line of the Lyman series ( ):
This corresponds to the transition from the next higher level to the ground state.
- Final level: .
- Initial level for the first line: .
Second line of the Balmer series ( ):
This corresponds to the transition from the second higher level to the first excited state.
- Final level: .
- The first line is from . The second line is from .
Calculate the ratio :
To find the ratio , we can divide equation (1) by equation (2):
The terms cancel out.
So, the ratio is 4:1.
Step 4: Final Answer:
The ratio of the wavelengths is 4:1. Therefore, option (C) is correct.

• Frequency of spectral line: .
• Lyman series: , second line: , .
• Balmer series: , third line: , .
• Ratio: .
• Hence correct answer = 27:5.

1. The distance of closest approach in Rutherford scattering occurs when kinetic energy is fully converted to electrostatic potential energy.
2. The formula is , where 2e is alpha charge, Ze nucleus charge.
3. Thus, , meaning the distance is inversely proportional to the kinetic energy.
4. For new energy , the new distance .
5. This assumes non-relativistic speeds and point-like particles.
6. Therefore, the correct option is (3) 2.5 x.