A sample of He ⁺ ions is excited to a specific higher energy state n . When the ions return to the ground state, a maximum of 6 different spectral lines are observed in the emission spectrum. If the difference between the maximum and minimum wave numbers of these emitted lines is expressed as x R_ H 36 (where R_ H is the Rydberg constant), the value of x is _______.

Step-by-step solution

The maximum number of spectral lines emitted when an electron drops from the n^ th state to the ground state is given by n(n-1) 2 . Given that the number of lines is 6 : n(n-1) 2 = 6 n^2 - n - 12 = 0 n = 4 The maximum wave number (highest energy) corresponds to the transition from n = 4 to n = 1 . For He ⁺ ( Z=2 ): _ max = R_ H (2)^2 [ 1 1^2 - 1 4^2 ] = 4R_ H ( 1 - 1 16 ) = 15R_ H 4 = 135R_ H 36 The minimum wave number (lowest energy) corresponds to the transition from n = 4 to n = 3 . _ min = R_ H (2)^2 [ 1 3^2 -