An unknown hydrogen-like ion undergoes a transition from n=8 to n=4 . If the wavelength of the photon emitted in this process is identical to that of the first line of the Lyman series of the hydrogen atom, identify the unknown ion.

Step-by-step solution

The wavenumber for the first line of the Lyman series of hydrogen ( n=2 to n=1 ) is given by: 1 _H = R_H (1)^2 ( 1 1^2 - 1 2^2 ) = R_H ( 1 - 1 4 ) = 3R_H 4 For the unknown hydrogen-like ion with atomic number Z , the transition is from n=8 to n=4 . The wavenumber is: 1 _ ion = R_H Z^2 ( 1 4^2 - 1 8^2 ) = R_H Z^2 ( 1 16 - 1 64 ) 1 _ ion = R_H Z^2 ( 4 - 1 64 ) = 3R_H Z^2 64 Since the wavelengths are identical, we equate the two expressions: 3R_H 4 = 3R_H Z^2 64 1 4 = Z^2 64 Z^2 = 16 Z = 4 The atomic number Z=4 corres