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 5.40 nm in diameter? Take the mass of an electron to be 9.11× 10–31 kg.
I am having difficulty determining what would be the wavelength that I use to determine the velocity using de Broglie's equation.
I think you want to use
E = hc/wavelength and solve for E. Then KE = 1/2 m*v^2 and solve for v.
But what is the wavelenght? Do I use 5.40 nm?
To calculate the kinetic energy of the electrons needed to resolve a protein molecule, we can use the de Broglie wavelength equation. This equation relates the wavelength of a particle to its momentum. The equation is as follows:
λ = h / p
Where λ is the wavelength, h is the Planck's constant (6.626 × 10^(-34) Js), and p is the momentum of the particle.
The momentum of a particle can be calculated as:
p = mv
Where p is the momentum, m is the mass of the particle, and v is the velocity of the particle.
Since we are given the diameter of the protein molecule, we need to find the velocity of the electron required. We can rearrange the equation for wavelength to solve for velocity:
v = h / (mλ)
Now we can substitute the given values and solve for the velocity:
λ = 5.40 nm = 5.40 × 10^(-9) m
m = 9.11 × 10^(-31) kg
v = (6.626 × 10^(-34) Js) / ((9.11 × 10^(-31) kg) × (5.40 × 10^(-9) m))
After calculating the value for v, we can calculate the kinetic energy using the formula:
KE = (1/2)mv^2
Substitute the values of m and v from above, and calculate the kinetic energy to find the answer.