A sample of gaseous hydrogen at room temperature is bombarded by an electron beam. If the resulting emission spectrum contains exactly 6 distinct spectral lines, the minimum kinetic energy of the incident electrons must be:

Step-by-step solution

Let the maximum excited state reached by the hydrogen atoms be n . The number of distinct spectral lines emitted is given by n(n-1) 2 . Given that the number of lines is 6 , we have: n(n-1) 2 = 6 n(n-1) = 12 n = 4 The energy required to excite a hydrogen atom from the ground state ( n=1 ) to the n=4 state is the difference in their energy levels: E = E₄ - E₁ E = -13.6 ( 1 4^2 - 1 1^2 ) eV E = -13.6 ( 1 16 - 1 ) eV E = 13.6 15 16 eV = 12.75 eV Therefore, the incident electrons must have a minimum kinetic energy of 1