The resolution of an electron microscope is approximately equal to the de Broglie wavelength of the electrons. To find the speed at which electrons must be accelerated to achieve a resolution of $0.20 \ \text{nm}$, we will use the de Broglie wavelength formula:

$\lambda = \frac{h}{mv}$

Where:

• $\lambda$ is the wavelength (in meters),
• $h$ is Planck's constant ($6.626 \times 10^{-34} \ \text{J·s}$),
• $m$ is the mass of an electron ($9.109 \times 10^{-31} \ \text{kg}$),
• $v$ is the speed of the electron.

Step 1: Convert the given resolution to meters:

$\lambda = 0.20 \ \text{nm} = 0.20 \times 10^{-9} \ \text{m}$

Step 2: Rearrange the de Broglie equation to solve for $v$:

$v = \frac{h}{m\lambda}$

Step 3: Plug in the known values:

$v = \frac{6.626 \times 10^{-34}}{(9.109 \times 10^{-31})(0.20 \times 10^{-9})}$

Now, let's calculate $v$.The speed at which the electrons must be accelerated to obtain a resolution of $0.20 \ \text{nm}$ is approximately $3.6 \times 10^6 \ \text{m/s}$.

Do you need any further explanation, or have any other questions?

Here are 5 related questions to expand your understanding:

1. What is the de Broglie wavelength, and how is it related to the momentum of a particle?
2. How does increasing the speed of electrons affect the resolution of an electron microscope?
3. What is the significance of Planck’s constant in quantum mechanics?
4. Can you describe how electron microscopes differ from optical microscopes?
5. What role does relativistic speed play in high-resolution electron microscopes?

Tip: As electrons approach relativistic speeds, their mass increases, affecting the de Broglie wavelength and the accuracy of non-relativistic formulas.