$E_n = \frac{-13.6 eV}{n^2}$ , $E = hf = \frac{hc}{\lambda}$
$h = 4.14 \times 10^{-15} eV \cdot s$ , $h = 6.63 \times 10^{-34} J \cdot s$ , $c = 3.0 \times 10^8 m/s$ , $1 eV = 1.60 \times 10^{-19} J$
Problem 1: In a Hydrogen atom, what is the energy of the photon emitted when an electron transitions from
(a) the $n = 3$ state to the $n = 1$ state. Answer: 12.09 eV.
(b) the $n = 6$ state to the $n = 2$ state. Answer: 3.02 eV.
(c) the $n = 5$ state to the $n = 4$ state. Answer: 0.31 eV.
Problem 2: In a Hydrogen atom, what is the wavelength of the photon emitted when an electron transitions from
(a) the $n = 4$ state to the $n = 2$ state. Answer: 487.1 nm.
(b) the $n = 4$ state to the $n = 1$ state. Answer: 97.4 nm.
(c) the $n = 5$ state to the $n = 2$ state. Answer: 434.3 nm.
Problem 3: How much energy is needed to ionize a Hydrogen atom in the $n = 2$ state? Answer: 3.4 eV.
Problem 4: How much energy is needed to ionize a Hydrogen atom in the $n = 5$ state? Answer: 0.54 eV.
Problem 5: In a Hydrogen atom, determine the energies of all the possible photons that can be emitted when an electron transitions from the $n = 7$ state to the $n = 5$ state.
Answer: $n = 7$ to $n = 5$: 0.27 eV, $n = 7$ to $n = 6$: 0.10 eV, $n = 6$ to $n = 5$: 0.17 eV.
Problem 6: In a Hydrogen atom, determine the energies of all the possible photons that can be emitted when an electron transitions from the $n = 4$ state to the $n = 1$ state.
Answer: $n = 4$ to $n = 1$: 12.75 eV, $n = 4$ to $n = 2$: 2.55 eV, $n = 2$ to $n = 1$: 10.2 eV,
$n = 4$ to $n = 3$: 0.66 eV, $n = 3$ to $n = 2$: 1.89 eV, $n = 3$ to $n = 1$: 12.09 eV.
Problem 7: In a Hydrogen atom, an electron in the initial state $n_i$ transitions to $n_f = 1$ by emitting a photon of energy 13.056 eV. What is $n_i$? Answer: $n_i = 5$.
Problem 8: In a Hydrogen atom, an electron in the initial state $n_i$ transitions to $n_f = 2$ by emitting a photon of energy 3.1875 eV. What is $n_i$? Answer: $n_i = 8$.
Problem 9: In a Hydrogen atom, an electron ends up in the final state $n_f = 3$ by emitting a photon.
