De Broglie Equation

De Broglie equation

source :http://switkes.chemistry.ucsc.edu

According to De Broglie, a moving small particle like electron, proton, neutron, dust particle, a small ball etc. has the properties of a wave (Dual nature of matter).

The wave length of a moving particle may be calculated as

λ=h/mv =h/p

De Broglie equation

λ =wave length

h= planck’s constt.

m=mass of particle

v=velocity of particle

p=momentum of the particle

Dual nature of electrons – Moving electrons in an orbit have the properties of both particle & wave. The following facts confirm the dual nature of electron.

1. Cathode rays (beam of electrons) rotate a pin whell placed in their path. This property shows the particle nature of electron.
2. A beam of electrons (Cathode rays) show diffraction & interference phenomenon (wave nature). This confirms the wave nature of moving electrons.

Dual nature of Radiations

Radiations consist of photons (Packets of energy). These photons possess the property of the wave & particle energy of photon,

According to Planck,

E= hv               (1)

According to Einstein’s equation,

E=mc²                 (2)

m= mass of photon

c= Velocity of radiation

From eq. (1) & (2)

hv    =mc²

v   =c/ λ

hc/ λ =mc²

λ =h/mc

Solved Numerical Problems

Question 1.     An electron of mass 9.1×10^–28 gm. is moving with velocity of 3×10¹⁰ cm/sec. Find its wave length?

Ans.    m=9.1×10^–28 gm.                     h= 6.6×10^–27 erg-Sec

v= 3×10¹⁰ cm/Sec

λ =h/mc

=6.6×10^–27/9.1×10^–28 ×3×10¹⁰ = 0.2417 ×10^–9

λ =2.42×10^–10cm  Ans.

Question 2.   Mass of Electron is 9.1×10^–31kg. If its kinetic energy is 3×10^–25 Joule, then calculate its wave length?

Ans.    De Broglie Equation ,

λ =h/mc

m=9.1×10^–31 kg

E= 3×10^–25 Joule

K.E. or E =mv²/2

putting the values of m & E,

v = √ 3×10^–25 x 2 / 9.1×10^–31

v = 0.812 x 10³

h= 6.6×10^–34 Joule-Sec

λ =h/mc

λ =6.6×10^–34/ (9.1×10^–31 x 0.812 x 10^{3 )}

= 0.893×10^–6

wave length = 8.93×10^–7 m