A gas of hydrogen-like ions is prepared in a specific excited state A . It subsequently emits photons that have a wavelength equal to the wavelength of the first line of the Lyman series, as well as photons of five other wavelengths. Identify the gas and determine the principal quantum number of the state A .

Step-by-step solution

Let the principal quantum number of the excited state A be n . The total number of different wavelengths emitted is 1 + 5 = 6 . The number of spectral lines emitted when a gas de-excites from state n to the ground state is given by n(n-1) 2 . Equating this to 6 : n(n-1) 2 = 6 n^2 - n - 12 = 0 (n-4)(n+3) = 0 Since n must be a positive integer, n = 4 . The wavelength of the first line of the Lyman series for hydrogen ( Z=1 ) corresponds to the transition from n=2 to n=1 . Its wavelength is given by: 1 = R (1)^2 ( 1 1