Electrons can be used to probe the arrangement of atoms on a solid surface if the wavelength of the electrons is comparable with the spacing between the atoms. Which of the following electron velocities would be appropriate for use in this application if the atoms are separated by 0.320 nm? A)2.27 * 10⁶ m/s B)1.24 * 10³ m/s C)3.00 * 10⁸ m/s D)4.41 * 10⁶ m/s E)8.06 * 10³ m/s

Question

Quizplus · Accepted Answer

The de Broglie wavelength of an electron is given by λ = h/p, where h is the Planck's constant and p is the momentum of the electron. For the electrons to probe the arrangement of atoms on a solid surface, their wavelength should be comparable to the spacing between the atoms, i.e., λ ≈ 0.320 nm. Equating this with the de Broglie wavelength equation, we get p = h/λ. Now, the kinetic energy of the electron is given by K = p^2/2m, where m is the mass of the electron. Substituting the values, we get K = (h^2/2mλ^2). Taking the square root of K/m, we get the velocity. Substituting h, m, and λ, we get:

v = (2K/m)^0.5 = [(2*(6.626 x 10^-34 J s)^2)/(9.109 x 10^-31 kg*(0.320 x 10^-9 m)^2)]^0.5 = 2.27 x 10^6 m/s

Therefore, choice A) is the best choice as it corresponds to the appropriate velocity of electrons for probing the arrangement of atoms on a solid surface where the atoms are separated by 0.320 nm.

Electrons Can Be Used to Probe the Arrangement of Atoms