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 8.10 nm in diameter? Take the mass of an electron to be 9.11× 10–31 kg.
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To resolve an object in an electron microscope, we can use the de Broglie wavelength formula, which relates the wavelength of a particle to its kinetic energy and mass.
The de Broglie wavelength formula is given by:
λ = h / p
Where:
λ is the wavelength of the particle
h is Planck's constant (6.626×10^(-34) Js)
p is the momentum of the particle
p = mv, where m is the mass of the particle and v is its velocity.
In this case, we need the wavelength to be close to the diameter of the protein molecule, which is 8.10 nm or 8.10×10^(-9) m.
First, let's calculate the momentum of the electron using its kinetic energy.
The kinetic energy of an electron is given by:
KE = 0.5 * m * v^2
We need to rearrange this equation to find the velocity of the electron.
v = sqrt((2 * KE) / m)
To resolve the protein molecule, we want the wavelength of the electron to be close to the diameter of the molecule. Therefore, the wavelength should be equal to or smaller than the diameter, so we can use an approximation and set the wavelength to be equal to the diameter:
λ ≈ 8.1×10^(-9) m
Now, substitute the values into the de Broglie wavelength equation:
8.1×10^(-9) m = h / (m * v)
Solve for v:
v ≈ h / (8.1×10^(-9) m * m)
Now, substitute the values for Planck's constant and the mass of the electron:
v ≈ (6.626×10^(-34) Js) / (8.1×10^(-9) m * 9.11×10^(-31) kg)
Calculate the value of v:
v ≈ 9.169×10^6 m/s
Next, substitute this value of velocity into the kinetic energy equation:
KE = 0.5 * m * v^2
KE = 0.5 * (9.11×10^-31 kg) * (9.169×10^6 m/s)^2
Calculate the value of KE:
KE ≈ 3.87×10^(-18) J
Therefore, the kinetic energy of the electrons must be approximately 3.87×10^(-18) J in order to resolve a protein molecule with a diameter of 8.10 nm using an electron microscope.