Question

To resolve an object in an electron microscope, the wavelength of the electrons must be close to the diameter of the object. What kinetic energy must the electrons have in order to resolve a protein molecule that is 5.60 nm in diameter? Take the mass of an electron to be 9.11× 10–31 kg.

Answer 1

The wave nature of the electron is described by the DeBroglie equation:

                             λ = h/mv
where h is Planck's constant = 6.6261 x 10^-34 Js
m is the mass of a particle (electron) = 9.11 x 10^–31 kg
λ is wavelength which is close to diameter of object ≈ 5.60 nm or = 5.6 x 10^-9 m
v is the velocity of electron

     Thus from equation,   λ = h/mv
          v = h/mλ

Substituting all above values, we get

                    v = (6.6261 x 10^-34 Js)/[(9.11× 10^–31 kg)(5.6 x 10^-9 m)]
                   v = 1.298 x 10⁵ m/s
Kinetic energy, E = 1/2(mv²)

                         = 1/2[(9.11× 10^–31 kg)(1.298 x 10⁵ m/s)²]

                         = 1/2 (1.53486 x 10^-20)

                     E = 7.67 x 10^-21 J