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 4.40 nm in diameter? take the mass of an electron to be 9.11x10^-31kg.

Answer 1

Expression of De Broglie wavelength can be written as follows:

λ = h / mv

Where, λ = De Broglie wavelength = 4.40nm = 4.40 x 10^-9 m

h = Plank’s constant = 6.626 x 10^-34 J.s

m = mass of electron = 9.11 x 10^-31 kg

v = velocity of electron

Thus, v can be calculated as shown below:

v = h / mλ

v = (6.626 x 10^-34 J.s) / (9.11 x 10^-31 kg)( 4.40 x 10^-9 m)

v = 0.1653 x 10⁶ m/s

Expression for kinetic energy can be shown as follows:

Kinetic energy = 1/2 m v²

Kinetic energy = 1/2 (9.11 x 10^-31 kg) (0.1653 x 10⁶ m/s)²

Kinetic energy = 0.12446 x 10^-19 kg.m²/s²

Kinetic energy = 0.12446 x 10^-19 J

Thus, in order to resolve given protein molecule electron should have 0.12446 x 10^-19 J kinetic energy.