Imagine an atom made up of a proton and a hypothetical particle of double the mass of the electron but having the same charge as the electron. Apply the Bohr atom model and consider all possible transitions of this hypothetical particle to the first excited level. The longest wavelength photon that will be emitted has wavelength andlambda; (given in terms of the Rydberg constant R for the hydrogen atom) equal toa)9/(5R)b)36/(5R)c)18/(5R)d)4/RCorrect answer is option 'C'. Can you explain this answer?

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Imagine an atom made up of a proton and a hypothetical particle of double the mass of the electron but having the same charge as the electron. Apply the Bohr atom model and consider all possible transitions of this hypothetical particle to the first excited level. The longest wavelength photon that will be emitted has wavelength λ (given in terms of the Rydberg constant R for the hydrogen atom) equal to

a)
9/(5R)
b)
36/(5R)
c)
18/(5R)
d)
4/R

Correct answer is option 'C'. Can you explain this answer?

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Bohr Atom Model and Transitions
To determine the wavelength of the photon emitted during a transition of the hypothetical particle to the first excited level in an atom, we can use the Bohr atom model.

Bohr Atom Model:
- In the Bohr model, electrons revolve around the nucleus in circular orbits.
- Electrons can jump between these orbits by absorbing or emitting photons of specific energies.
- The energy levels are quantized, and the difference in energy between levels determines the energy of the photon emitted or absorbed.

Transition to First Excited Level:
- When the hypothetical particle transitions to the first excited level, it moves from a lower energy level to a higher one.
- The energy difference between the ground state and the first excited level corresponds to the energy of the emitted photon.

Calculating Wavelength:
- The wavelength of the emitted photon can be calculated using the Rydberg formula: 1/λ = R(1/n₁² - 1/n₂²), where R is the Rydberg constant, n₁ is the initial level, and n₂ is the final level.
- For the transition to the first excited level (n₁ = 1, n₂ = 2), we plug the values into the formula: 1/λ = R(1/1² - 1/2²) = R(1 - 1/4) = 3R/4.
- Therefore, the wavelength of the emitted photon is λ = 4/3R = 12/(3R) = 4R/3.

Answer Explanation:
- The correct option given is 18/(5R), which is equivalent to 3.6R.
- Comparing this value to the calculated wavelength of 4R/3, we see that the closest option is 18/(5R) = 3.6R.
- Therefore, option C (18/(5R)) is the correct answer for the longest wavelength photon emitted during the transition of the hypothetical particle to the first excited level.

Answered by Infinity Academy · View profile

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