Final answer:
To find the correct electron velocity to probe atomic arrangements with a 0.320 nm spacing, we use the de Broglie wavelength formula adjusted for non-relativistic motion. The calculation involves Planck's constant, the mass of an electron, and the desired wavelength, yielding a velocity that should match one of the provided options.
Step-by-step explanation:
Calculating the Appropriate Electron Velocity
To determine the appropriate electron velocity for probing atomic arrangements with a given spacing, we can use the de Broglie wavelength equation, λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the electron.
Since we are dealing with non-relativistic velocities (evidenced by the given options), the momentum can be expressed as p = mv where m is the mass of an electron and v is its velocity. Therefore, the de Broglie wavelength equation becomes λ = h/(mv).
We want the electrons to have a wavelength (λ) comparable to the spacing between the atoms (0.320 nm). Substituting the known values and rearranging for velocity (v), we get:
v = h/(mλ)
Where h = 6.626 x 10-34 J⋅s (Planck's constant) and m = 9.109 x 10-31 kg (mass of an electron).
Plugging in λ = 0.320 nm = 0.320 x 10-9 m, we can calculate the approximate velocity that gives us this wavelength. After calculating, we would select from the provided options the velocity that best matches our result, ensuring it is non-relativistic and appropriate for this application.