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 7.90 nm in diameter? Take the mass of an electron to be 9.11*10^-31kg

_____J

KE = 1/2 mv^2

2.8 x 10^-29

To calculate the kinetic energy of electrons in order to resolve a protein molecule, you need to use the de Broglie wavelength equation:

λ = h / p

where λ is the wavelength, h is the Planck's constant (6.626 × 10^-34 J·s), and p is the momentum. The momentum can be calculated using the equation:

p = mv

where m is the mass and v is the velocity.

First, let's calculate the velocity of the electrons using their kinetic energy (KE) equation:

KE = (1/2)mv^2

Rearranging the equation to solve for v:

v = √(2KE / m)

Now, substituting the known values:

m = 9.11 × 10^-31 kg (mass of an electron)

Now, let's calculate the de Broglie wavelength using the velocity:

λ = h / (mv)

Substituting the known values:

λ = (6.626 × 10^-34 J·s) / (m * v)

We need to find the kinetic energy (KE) required to resolve a protein molecule with a diameter of 7.90 nm. We can start with assuming an initial velocity for the electrons and calculate the de Broglie wavelength. If the calculated wavelength is close to 7.90 nm, then the assumption of velocity is correct. Otherwise, we need to adjust the kinetic energy and repeat the calculation until we get a wavelength close to 7.90 nm.

Let's assume an initial kinetic energy, for example, 1 eV (electron-volt), which is equal to 1.602 × 10^-19 J.

1 eV = 1.602 × 10^-19 J (conversion factor)

Now, we can substitute the initial kinetic energy and calculate the velocity and de Broglie wavelength:

v = √(2KE / m)

λ = (6.626 × 10^-34 J·s) / (m * v)

Repeat these calculations using various initial kinetic energy values until the calculated wavelength is close to 7.90 nm.

Please note that this calculation assumes non-relativistic electron velocities, as relativistic effects would need to be taken into account for higher electron energies.