Transmission electron microscopes that use high-energy electrons accelerated over a range from 40.0 to 100 kV are employed in many applications including the study of biological samples (like a virus) and nanoscience research and development (alloy particles and carbon nanotubes, for example). What would be the spatial limitation (in pm) for this range of electrons

Explanation:

Using the concept of de Broglie wavelength under relativistic conditions to determine the spatial limits for electron range.

de Broglie wave-length

where;

h = plank's constant

p = momentum of particle which is expressed as:

replacing the expression for (p) into , we have:

here;

c = velocity of light = 3 × 10⁸ m/s

h = 4.13 × 10⁻¹⁵ eV.s

i.e.

hc = (4.13 × 10⁻¹⁵ eV.s)(3 × 10⁸ m/s)

hc = 1240 eV.m

The electron's rest energy (mc²) = 0.511 × 10⁶ eV

For the elctrons;

the minimum accelerated voltage = 40kV

the maximum accelerated voltage = 100 kV

the minimum K.E of the electron K = eΔV

K = e × 40 kV

K = 40 KeV

K = 40 × 10³ eV

From equation (1);

The maximum K.E of the electron K = eΔV

K = e × 10 kV

K = 10 KeV

K = 100 × 10³ eV

From equation (1);

As a result, the spatial limits for the electron's range span from 6.02 pm to 3.70 pm.