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 2.10 nm in diameter? Take the mass of an electron to be 9.11× 10–31 kg.

=_____J

To calculate the kinetic energy (KE) of the electrons required to resolve the protein molecule, we can use the de Broglie wavelength equation:

λ = h / p

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

p = mv

where m is the mass of the electron and v is the velocity of the electron.

To find the velocity, we can use the equation for kinetic energy:

KE = 0.5mv^2

We can rearrange this equation to solve for v:

v = sqrt(2KE / m)

Substituting this value for velocity back into the momentum equation, we have:

p = m * sqrt(2KE / m)

By equating this expression for momentum to the de Broglie wavelength formula, we can solve for KE.

λ = h / (m * sqrt(2KE / m))

Rearranging the equation to solve for KE:

KE = (h^2 / 2m) / λ^2

Given that the diameter of the protein molecule is 2.10 nm, we can calculate the wavelength (λ) using the formula:

λ = 2 * diameter

λ = 2 * 2.10 nm

First, we need to convert nm to meters. 1 nm is equal to 1 × 10^-9 meters.

λ = 2 * 2.10 × 10^-9 m

Substituting this value into the KE equation:

KE = (h^2 / 2m) / (2 * 2.10 × 10^-9)^2 m^2

Next, we substitute the known values for Planck's constant (h) and the mass of an electron (m):

KE = ((6.626 × 10^-34 J·s)^2 / 2 * 9.11 × 10^-31 kg) / (2 * 2.10 × 10^-9)^2 m^2

Simplifying the expression:

KE = (4.373476 × 10^-67 J·s^2 / 2 * 9.11 × 10^-31 kg) / (4.41 × 10^-18) m^2

KE = (2.186738 × 10^-67 J·s^2) / (2 * 9.11 × 10^-31 kg) / (4.41 × 10^-18) m^2

Simplifying further:

KE = (2.186738 × 10^-67 J·s^2) / (2.82 × 10^-17) kg·m^2

KE = 7.75205 × 10^-51 J

Therefore, the kinetic energy of the electrons needed to resolve the protein molecule is approximately 7.75205 × 10^-51 J.

To resolve an object in an electron microscope, the wavelength of the electrons must be close to the diameter of the object. The de Broglie wavelength of an electron is given by the equation:

λ = h / p

where λ is the wavelength, h is Planck's constant (6.63 x 10^-34 J⋅s), and p is the momentum of the electron.

The momentum of the electron can be calculated using the equation:

p = √(2mE)

where m is the mass of the electron (9.11 x 10^-31 kg) and E is the kinetic energy of the electron.

We can rearrange the equation for the momentum to solve for the kinetic energy:

E = p^2 / (2m)

Substituting the equation for the wavelength into the equation for the momentum:

E = (h^2) / (2mλ^2)

Now we can substitute the given values into the equation to calculate the kinetic energy:

E = ((6.63 x 10^-34 J⋅s)^2) / (2 * (9.11 x 10^-31 kg) * (2.10 x 10^-9 m)^2)

Calculating this expression will give us the answer. Let's do the math:

E ≈ 1.43 x 10^-18 J

Therefore, the kinetic energy of the electrons must be approximately 1.43 x 10^-18 Joules to resolve a protein molecule with a diameter of 2.10 nm in an electron microscope.

λ=h/mv

2.10*10^-9=(6.63*10^-34)/(9.11*10^-31)v
(1.91*10^-39)v=6.63*10^-34
v=347120.42

k=1/2mv^2
k=1/2(9.11*10^-31)(347120.42)^2
k=5.49*10^-20

wavelength = h/mv

Substitute and solve for v, then substitute v into KE = 1/2 mv^2 and solve for KE.