Question

Bohr’s model of the hydrogen atom. a) State the three postulates underlying Bohr’s theory of the hydrogen atom. [3 marks] The radius of an electron orbiting the nucleus is given by r = 0h 2 πme e 2 n 2 , where me is the electron mass, e the electron charge and 0 the vacuum permittivity. b) Calculate the Bohr radius of a hydrogen atom in an excited state of n = 2. [1 mark] c) By considering the sum of kinetic energy Ekin and potential energy Epot of the electron at a distance r from the nucleus, derive an expression for the total energy E of the electron. Use this expression to calculate the wavelength (in nm) of a single photon emitted if the electron makes a transition from n = 4 to n = 2.

Answer 1

a) Postulates of Bohr's theory:

1. The electron in Hydrogen atom can't revolve around the nucleus in all possible orbits but it revolves only in certain fixed orbits in which it doesn't radiate its energy.

2. The atom radiates energy only when the electron jumps from a higher energy level (orbit) to lower energy level. If the electron jumps from E2 energy level to E1 energy level then the radiation energy will be E= E2 - E1.

3. The electron revolves only in those orbits for which the angular momentum of the electron is an integral multiple of h/2. These are known as stationary orbits.

b) From the equation of radius r=kn² where k contains all other values of the r.h.s. k is also known as Bohr radius and its value is calculated to be equal to 0.053 nm.

Now, for n = 2,

r = 0.053 * 2²

= 0.053 * 4 =0.212 nm.

c) From Coulomb's law,

Epot =

Similarly one can also find the value of kinetic energy of the revolving electron

Ekin =

The total energy of the electron

E = Ekin + Epot

=

Substituting the value of radius r, the energy of the electron in nth orbit

En = C/n² , where C is a constant factor.

If an electron jumps from n2 otbit to n1 orbit, the radiation emitted E = En2 - En1

Since E = hc/l where h is the Planck's constant and l is the wavelength of the emitted radiation, gives 1/l = E/hc = R where R is known as the Rydberg constant=1.097*10⁷ /m.

Substituting the value of R, n1 = 2 and n2 = 4, we can easily calculate the value of 1/l. After inverting this value we will get the value of wavelength of the emitted photon.

1/l =