This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Determine the initial energy level of the hydrogen atom. Hydrogen atoms are in the ground state, which corresponds to the principal quantum number n_i = 1. The energy of an electron in the n-th orbit of a hydrogen atom is given by: E_n = -13.6 (1)/(n^2) eV For the ground state (n_i = 1): E_1 = -13.6 (1)/(1^2) = -13.6 eV Step 2: Calculate the final energy level after excitation. The atoms are excited by monochromatic radiation with a photon energy of 12.1 eV. The final energy E_f of the electron will be the initial energy plus the absorbed photon energy: E_f = E_1 + Photon Energy E_f = -13.6 eV + 12.1 eV E_f = -1.5 eV Step 3: Determine the principal quantum number n_f corresponding to the final energy level. Using the energy formula for E_f: E_f = -13.6 (1)/(n_f^2) -1.5 = -13.6 (1)/(n_f^2) n_f^2 = (-13.6)/(-1.5) n_f^2 = (13.6)/(1.5) n_f^2 ≈ 9.066... n_f = sqrt(9.066...) ≈ 3.01 Since the principal quantum number must be an integer, the electron is excited to the n_f = 3 energy level. Step 4: Calculate the number of possible spectral lines emitted during de-excitation. When an electron de-excites from an excited state n to lower energy levels, the total number of possible spectral lines N is given by the formula: N = (n(n-1))/(2) In this case, the electron is excited to n=3. N = (3(3-1))/(2) N = (3 × 2)/(2) N = (6)/(2) N = 3 The possible transitions are 3 2, 3 1, and 2 1. The final answer is 3. 3 done, 2 left today. You're making progress.