Structure Of Atom

Step 1: Analyze Statement-I (Rutherford Model and Stability).
The Rutherford model (the planetary model) proposes that electrons revolve around a positively charged nucleus.
According to classical electromagnetic theory (Maxwell's equations), an accelerating charged particle (like an electron in orbit) must continuously radiate energy.
The continuous radiation of energy would cause the electron's orbit to shrink rapidly, eventually spiraling into the nucleus.
This process would take about seconds, meaning the atom would be highly unstable. The Rutherford model offers no explanation for this inherent instability.
Therefore, Statement-I is correct.

Step 2: Analyze Statement-II (Wavelength of X-rays and Microwaves).
Electromagnetic waves are ordered by their frequency ( ) and wavelength ( ) in the electromagnetic spectrum. The relation is .
The order of increasing wavelength (decreasing frequency) is: Gamma rays Microwaves have a much longer wavelength than X-rays. to , while to .
The statement claims , which is false.
Therefore, Statement-II is not correct.

Step 3: Conclude the correct option.
Statement I is correct, but Statement II is not correct.

We use the Rydberg formula for the wavelength of spectral lines in the hydrogen atom:
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Case (i): Transition from to . The wavelength is .
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So, .
Case (ii): Transition from to . The wavelength is .
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So, .
Now, we calculate the difference .
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To subtract the fractions, we find a common denominator, which is 24.
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The wavelengths and are given in Angstroms (\AA{}), but the Rydberg formula with in cm gives in cm. The question asks for the difference in cm. Since the calculation is done in terms of , the unit is automatically handled.

Step 1: Understanding the Concept:
This question is about the Bohr model of the hydrogen atom. To excite an electron from a lower energy orbit to a higher energy orbit, a photon of light must be absorbed. The energy of this photon must be exactly equal to the energy difference between the two orbits. The energy of a photon is related to its frequency by the Planck-Einstein relation, .
Step 2: Key Formula or Approach:
1. The energy of an electron in the n-th orbit of a hydrogen-like atom is given by , where is the energy of the first orbit.
2. The energy of the absorbed photon ( ) is the difference between the final and initial energy levels: .
3. The frequency of the light ( ) is related to the photon energy by .
Step 3: Detailed Explanation:
Given values:
- Energy of the first orbit, J.
- Initial orbit, .
- Final orbit, .
- Planck's constant, J·s.
First, calculate the energy of the electron in the fifth orbit ( ):
Next, calculate the energy difference ( ) required for the excitation:
An alternative way to calculate is:
Finally, calculate the frequency ( ) of the light:
Wait, let me re-evaluate the calculation with the provided option's value. The correct option is C: Hz. Let's check. This does not match my calculated . Let me re-read the question. Ah, the provided correct answer in the document seems to be (D), Hz. Let's assume there was a typo in my transcription. The calculation gives Hz which matches option (D). Let's assume (D) is the correct answer.
However, the solution key in the image marks option 3, which is . Let me check my calculation again.
Energy difference: J.
Frequency: Hz.
My calculation consistently gives Hz, which is option (D). The provided key marking option (C) appears to be incorrect. I will proceed with my calculated answer. Re-checking the source, the checkmark is on option 4. So my initial assessment was correct.
Step 4: Final Answer:
The required frequency is Hz. Therefore, option (D) is correct.