Calculate the deBroglie wavelength of the electron in a trip

Calculate the deBroglie wavelength of the electron in a triply ionized beryllium atom when it is orbiting in the following state: n = 5.

Solution

deBroglie wavelength,  = h/p = h/mv

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

p is the momentum of the orbiting electrons

From the Bohr\'s quantization rule,

mvr = nh/2

m, v and r are the mass, velocity and radius of an electron orbiting the nucleus.

h/mv = 2r/n ...(1)

The radius of the nth orbit of an electron is give by the formula

r = n²a/Z

a is the Bohr radius, a = 0.0529 x 10^-9 m.

n = 5, Z = 4

r = 25 x (0.0529 x 10^-9) / 4

0.330625 x 10^-9 m

Substituting r in (1)

h/mv = [2 x (0.330625 x 10^-9)] / 5

= 0.4155 x 10^-9 m