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 8.80 nm in diameter? take the mass of an electron to be 9.11× 10–31 kg. you should first solve for the velocity of an electron that would have a de broglie wavength equal to the diameter of the protein. then calculate the energy of an electron with this velocity, using the equation for kinetic energy given in the hint below.

Answer 1

By De Broglie concept of dual nature of light we know that

$\lambda = \frac{h}{mv}$

here we know that for this validity the wavelength must be equal to the diameter of the molecule

so here we have

$\lambda = 8.80 nm$

now from above equation

$8.80* 10^{-9} = \frac{6.6 * 10^{-34}}{9.1*10^{-31}*v}$

$v = \frac{6.6 * 10^{-34}}{9.1 * 10^{-31} * 8.80 * 10^{-9}}$

$v = 8.24 * 10^4 m/s$

Now the kinetic energy of the electron is given as

$KE = \frac{1}{2}mv^2$

$KE = \frac{1}{2}*9.1*10^{-31}*(8.24* 10^4)^2$

$KE = 3.1 * 10^{-21} J$

so above is the KE of electron