An electron in a hydrogen atom is excited to a higher energy state such that the maximum number of spectral lines it can emit while returning to the ground state is 6 . What is the wavenumber of the highest energy photon emitted in this entire process? ( R_H = Rydberg constant )

Step-by-step solution

The formula for the maximum number of spectral lines emitted when an electron returns from the n^ th state to the ground state is n(n-1) 2 . Given that the maximum number of lines is 6 : n(n-1) 2 = 6 n^2 - n - 12 = 0 Solving this quadratic equation, we get (n-4)(n+3) = 0 . Since n must be positive, n = 4 . The highest energy photon is emitted during the transition with the largest energy gap, which is from n = 4 directly to the ground state ( n = 1 ). Using the Rydberg equation for wavenumber : = R_H ( 1 n₁^2 - 1 n