The resolution limit of a microscope is roughly equal to the wavelength of light used in producing the image. Electron microscopes use an electron beam (in place of photons) to produce much higher resolution images, about 0.16 nm in modern instruments.
Assuming that the resolution of an electron microscope is equal to the de Broglie wavelength of the electrons used, to what speed must the electrons be accelerated to obtain a resolution of 0.16 nm?
w = wavelength
w = h/mv
w = 0.16E-9m
m in kg. Solve for v in m/s.
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To determine the speed at which electrons must be accelerated to achieve a resolution of 0.16 nm in an electron microscope, we can use the de Broglie wavelength formula:
λ = h / p
Where:
λ is the de Broglie wavelength
h is the Planck's constant (6.626 x 10^-34 J·s)
p is the momentum of the electron
Since the resolution is given as 0.16 nm, we need to convert it to meters:
λ = 0.16 nm = 0.16 x 10^-9 m
Substituting the values into the equation, we have:
0.16 x 10^-9 m = (6.626 x 10^-34 J·s) / p
Solving for the momentum (p):
p = (6.626 x 10^-34 J·s) / (0.16 x 10^-9 m)
p ≈ 4.14125 x 10^-25 kg·m/s
The momentum (p) of an electron is given by:
p = mv
Where:
m is the mass of the electron (9.10938356 × 10^-31 kg)
v is the velocity of the electron
Substituting the values, we can solve for the velocity (v):
4.14125 x 10^-25 kg·m/s = (9.10938356 × 10^-31 kg) v
v ≈ (4.14125 x 10^-25 kg·m/s) / (9.10938356 × 10^-31 kg)
v ≈ 4.54 x 10^6 m/s
Therefore, the electrons must be accelerated to a speed of approximately 4.54 x 10^6 m/s to achieve a resolution of 0.16 nm in an electron microscope.
To find the speed at which the electrons must be accelerated, we first need to determine the de Broglie wavelength of the electrons. The de Broglie wavelength is given by the equation:
λ = h / p
Where:
λ is the de Broglie wavelength,
h is the Planck's constant (h = 6.626 x 10^-34 J·s),
p is the momentum of the particle.
Since we are given the resolution of the electron microscope (0.16 nm), we can assume this value represents the de Broglie wavelength. Therefore, we have:
λ = 0.16 nm = 0.16 x 10^-9 m
We can rearrange the de Broglie wavelength equation to solve for momentum:
p = h / λ
Substituting the values, we have:
p = (6.626 x 10^-34 J·s) / (0.16 x 10^-9 m)
p = (6.626 x 10^-34) / (0.16 x 10^-9) kg·m/s
Now, we know that momentum (p) is equal to the product of mass (m) and velocity (v):
p = m × v
Rearranging the equation to solve for velocity, we have:
v = p / m
Since we know the mass of an electron (m = 9.10938356 x 10^-31 kg), we can substitute the values and solve for velocity:
v = [(6.626 x 10^-34) / (0.16 x 10^-9)] / (9.10938356 x 10^-31)
v = (6.626 / 0.16) / (9.10938356 x 10^-31) m/s
Evaluating this expression, we find that the velocity required to achieve a resolution of 0.16 nm is approximately 4.37 x 10^6 m/s.