Question

The energies of the hydrogen atom are quantized by integer number n from 1 to infinity. This number if called the principal quantum number. Find:

a) principal quantum number

b) binding energy

c) orbital radius

d) total enery

e) excitation energy of the fifth state of hydrogen

Answer 1

According to Bohr's atomic model:

(i) radius of n_th orbit is given by,

  

or,    ...................................................................(1)

Where,

is radius of n_th orbit

n is the principal quantum number

e = 1.602 x 10^-19 C is charge on electron

m_e = 9.1 x10^-31 kg is mass of electron

h = 6.626 x 10^-34 J-s is Planck's constant

= 8.85 x 10^-12 F/m is permittivity of free space

and     

a₀ is known as Bohr radius.

(ii) Total energy is given by,

or, ...........................................(2)

Where,

  

is the ground state energy (with a negative sign).

(iii)  Binding Energy : The energy required to free the elecctron, or, the amount of energy required to separate the electron from the atom is known as binding energy.

For an electron in n_th state, the binding energy will be

  

or,   

or,    .......................................(3)

(iv) Excitation energy : The amount of energy required to excite the electron from one state to another state is known as excitation energy.

The excitation energy of an electron from the ground state to an excited state n is (E_n - E₁).

Answers :

For the fifth state of hydrogen,

(a) Principal quantum number n = 5 .

(b) From the equation(3), Binding energy of the fifth state of hydrogen is

  

(c) Using equation (1), orbital radius of fifth state of hydrogen will be

or,

or,

(d) From equation(2), total energy of the fifth state of hydrogen will be

or,

or,

(e) Excitation energy of the fifth state of hydrogen is given by,

Excitation Energy = E₅ - E₁ = (-0.544) - (-13.6) = 13.056 eV   

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