The Hydrogen Atom -- Bohr Model?

"The wavelength of light emitted when the electron drops from n = 3 to n = 2 is shorter than the wavelength of light emitted if the electron falls from n = 3 to n = 1."

False.

The energy released when the electrons drops from n = 3 to n = 2 is less than the energy released when the electron falls from n = 3 to n = 1 (from n = 3 to n = 2 and then from n = 2 to n = 1).
The less the energy released, the longer the wavelength emitted is.
Therefore, t he wavelength of light emitted when the electron drops from n = 3 to n = 2 is "shorter" than the wavelength of light emitted if the electron falls from n = 3 to n = 1.


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"The wavelength of the light emitted when the electron returns to the ground state from n=3 is larger than the wavelength of light absorbed to go from n=1 to n=3".

False.

For ground state, n = 1
The wavelength of the light emitted when the electron returns from n=3 to n = 1 is "equal to" the wavelength of light absorbed to go from n=1 to n=3


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The first excited state corresponds to n=2.

True.


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"It takes more energy to ionize the electron from n=3 than it does from the ground state."

False.

In ionization, the electron is migrate to the n = ∞.
The energy different between n = 3 and n = ∞ is smaller than the energy different between n = 1 (the ground state) and n = ∞.
Therefore, it takes "less" energy to ionize the electron from n=3 than it does from the ground state.


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"The electron is closer to the nucleus on average in the n=3 state than in the ground state."

False.

The smaller the number n, the closer on average that the electron to the nucleus.
Therefore, the electron is "farther apart" to the nucleus on average in the n=3 state than in the ground state (n = 1).