What is the wavelength of the photons emitted by hydrogen at

What is the wavelength of the photons emitted by hydrogen atoms when they undergo n=4 to n=2 transitions in nm? What region of the eletromagnetic spectrum does this radiation occur?

Solution

1) The energy associated with the hydrogen atom with the electron revolving in the nth orbit is given by

E = - me^4 / 8*?o*h^2 ( 1 / n^2),
n = 1.2.3......correspond to fist, second , third,...... levels

n = 1 corresponds to ground state energy of the H atom. It is given by
E1 = - me^4 / 8*?o*h^2 = -21.76 x10^-19 J = - 13.6 eV ,
(using the known values of the constants and 1eV = 1.6 x10^-19 J)

In working out problems on hydrogen atom energy levels it is always convenient to remember this value of ground state energy.

Now in your problem,
E = E2 - E4 = (13.6 / 2^2 - 13.6 / 4^2) = 2.55 eV = 2.55 x1.6 x10^-19 J
E = 4.08 x10^-19 J

This energy difference appears as the photon energy which is given by
E = hf = h c / ?
? = h c / E = 6.6256 x 10^-34 x 3 x10^8 / 4.08 x10^-19 = 487 x10^-9 = 487 nm

2) Balmer series: Visible spectrum (~400 - 800 nm)