Consider Rydberg (Hydrogen-like) atoms in a highly excited state with n around 300. The wavelength of radiation coming out of these atoms for transitions to the adjacent states lies in the range:a)Gamma rays (λ~ pm)b)UV (λ~ nm)c)Infrared (λ~ mm)d)RF (λ~ m)Correct answer is option 'D'. Can you explain this answer?

Question

Consider Rydberg (Hydrogen-like) atoms in a highly excited state with n around 300. The wavelength of radiation coming out of these atoms for transitions to the adjacent states lies in the range:a)Gamma rays (&#955;~ pm)b)UV (&#955;~ nm)c)Infrared (&#955;~ mm)d)RF (&#955;~ m)Correct answer is option 'D'. Can you explain this answer?

Answer

Rydberg atoms are hydrogen-like atoms that have highly excited states with large principal quantum numbers (n). In this case, we are considering atoms with n around 300. These highly excited states have very tightly bound electrons, which results in large energies and correspondingly short wavelengths of radiation emitted during transitions to adjacent states.

To determine the range of wavelengths of the radiation emitted, we can use the Rydberg formula:

1/λ = R_H * (1/n_1^2 - 1/n_2^2)

where λ is the wavelength of the emitted radiation, R_H is the Rydberg constant for hydrogen, and n_1 and n_2 are the principal quantum numbers of the initial and final states, respectively.

Since we are interested in transitions to adjacent states, we can consider n_2 = n_1 + 1. Plugging this into the formula, we get:

1/λ = R_H * (1/n_1^2 - 1/(n_1 + 1)^2)

Simplifying this expression, we find:

1/λ = R_H * (2n_1 + 1)/(n_1^2 * (n_1 + 1)^2)

Now, let's consider the limit as n_1 approaches infinity (which is the case for highly excited states). In this limit, the expression simplifies to:

1/λ ≈ R_H/(n_1^3)

Since n_1 is large, we can approximate the value of the Rydberg constant R_H as approximately 1.097 × 10^7 m^-1.

Now, let's consider the range of wavelengths for large values of n_1:

- For n_1 ≈ 300, the approximate wavelength is:

λ ≈ 1/(1.097 × 10^7 * (300^3)) ≈ 3.18 × 10^-4 m

- For n_1 ≈ 301, the approximate wavelength is:

λ ≈ 1/(1.097 × 10^7 * (301^3)) ≈ 3.16 × 10^-4 m

As we can see, the wavelength of radiation emitted for transitions to adjacent states in highly excited Rydberg atoms is in the range of approximately 3.16 × 10^-4 m to 3.18 × 10^-4 m, which corresponds to radiofrequency (RF) radiation. Therefore, the correct answer is option 'D'.

Answer

B

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