Electronic Transitions in the Hydrogen Atom Using Figure, predict which of these electronic transitions produces the spectral line having the longest wavelength: n = 2 to n=1,n = 3 to n = 2, or n = 4 to n = 3. ? Figure Energy levels in the hydrogen atom from the Bohr model. The arrows refer to the transitions of the electron from one allowed energy state to another. The states shown are those for which n = 1 through n = 6 and the state for n = °° for which the energy, E, equals zero. In the top part of Figure, the four lines in the H atom spectrum are due to transitions from a level for which ni > 2 to the nf = 2 level. What is the value of ni for the blue-green line in the spectrum? (a) 3 (b) 4 (c) 5 (d) 6 (e) 7 ? Figure Line spectra of hydrogen and neon. The colored lines occur at wavelengths present in the emission. The black regions are wavelengths for which no light is produced in the emission.

Solution: Here, we are going to determine the value of ni for the blue-green light in the spectrum. Step1: The spectrum of hydrogen atom has played a very important role in the development of atomic structure. It can be obtained by passing an electric discharge through the gas taken in the discharge tube under low pressure. The emitted light is analysed with the help of spectroscope. The spectrum consists of a large number of lines appearing in different regions of wavelengths. In order to calculate the wavelength of all the spectral lines of hydrogen, a general expression was forwarded by Rydberg, known as Rydberg equation. The equation is: --------(1) Where, = wavelength of a spectral line 7 - 1 RH Rydberg constant and its value is 1.096776 x 10 m n and n are positive integers, n < n 1 2 1 2 Step2: The wavelength corresponding to the blue-green line = 486 nm -9 = 486 x 10 m = 4.86 x 10 m -7 The value of n = 2 1 Thus, substituting the values in equation (1), we get, -7 7 2 2 1/(4.86 x 10 ) = (1.096776 x 10 )[(1/2 ) - (1/n )] 2 7 7 2 2 0.206 x 10 = (1.096776 x 10 )[(1/2 ) - (1/n )] 2 2 7 7 (1/4) - (1/n ) = 2.206 x 10 ) / (1.096776 x 10 ) (n - 4) / 4n = 0.192 2 2 n2 4 = 0.19 x 4n 2 2 2 n2 4 = 0.76n 2 2 2 -4 = 0.76n - n 2 2 2 -4 = -0.24n 2 n = 4/0.24 2 n = 16.67 2 n2 16.67 n2 4 (approx.) Thus the correct option is (b). ------------------------