A large number of ground-state hydrogen atoms are bombarded by electrons having kinetic energy exactly 12.75 eV. Assume a successful inelastic collision raises an atom to the highest bound state accessible at this energy. The excited atoms are then allowed to de-excite through all possible paths. Find the maximum number of distinct emission wavelengths that can be observed.
Physics · Atomic Structure and X-rays · atomic structure and bohr atomic model
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Question

A large number of ground-state hydrogen atoms are bombarded by electrons having kinetic energy exactly 12.75 eV. Assume a successful inelastic collision raises an atom to the highest bound state accessible at this energy. The excited atoms are then allowed to de-excite through all possible paths. Find the maximum number of distinct emission wavelengths that can be observed.

Physics · Atomic Structure and X-rays · atomic structure and bohr atomic model

G1 Physics · Accepted Answer

The energy 12.75 eV is exactly the excitation energy from the ground state to the fourth Bohr orbit: ΔE₄ = 13.6[1−1 ÷ 16] = 12.75 eV. Thus the highest populated state is n = 4. For a large sample, every possible transition among the four levels can occur in different atoms. N(lines) = n(n−1) ÷ 2 = 4×3 ÷ 2 = 6. Hence N = 6.