Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? (b) from an orbit of radius 0.846 nm to one of radius 0.212 nm

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Identify the initial and final energy levels (n_i and n_f) for the electron transition. Use the formula for the radius of the nth orbit in the Bohr model of the hydrogen atom, r_n = n^2 \(\times\) 0.053 nm, to find n_i and n_f.

Calculate the energy associated with the initial and final orbits using the energy level formula for hydrogen, E_n = -\(\frac{13.6 \text{ eV}\)}{n^2}.

Compare the energies of the initial and final orbits. If the energy of the final orbit (E_f) is higher than the initial orbit (E_i), energy is absorbed. If E_f is lower than E_i, energy is emitted.

Determine the change in energy (\(\Delta\) E) by subtracting the energy of the initial orbit from the energy of the final orbit, \(\Delta\) E = E_f - E_i.

Conclude whether energy is emitted or absorbed based on the sign of \(\Delta\) E. If \(\Delta\) E is negative, energy is emitted; if positive, energy is absorbed.

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Energy Levels in Hydrogen

In a hydrogen atom, electrons occupy specific energy levels, which correspond to distinct orbits around the nucleus. The energy of these levels is quantized, meaning electrons can only exist in certain states. The radius of these orbits is directly related to the energy level; as the radius decreases, the energy of the electron increases.

Photon Emission and Absorption

When an electron transitions between energy levels, it either emits or absorbs a photon, which is a particle of light. If an electron moves to a lower energy level, it emits energy in the form of a photon; conversely, if it moves to a higher energy level, it absorbs energy. The energy of the photon corresponds to the difference in energy between the two levels.

Rydberg Formula

The Rydberg formula allows for the calculation of the wavelengths of light emitted or absorbed during electronic transitions in hydrogen. It relates the wavelengths to the principal quantum numbers of the initial and final energy levels. This formula is essential for predicting the energy changes associated with electron transitions and understanding the spectral lines of hydrogen.