Suppose the velocity of an electron in an atom is known to an accuracy of (reasonably accurate compared with orbital velocities). What is the electron's minimum uncertainty in position, and how does this compare with the approximate 0.1-nm size of the atom?

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SumantLv10

31 Dec 2020

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The uncertainty in an electron's position is 0.008 nm.

a) What is the minimum uncertainty Δp in its momentum?

b) What is the kinetic energy of an electron whose momentum is equal to this uncertainty (Δp=p)?

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Some of the earliest atomic models held that the orbital velocityof an electron in an atom could be correlated with the radius ofthe atom. Assume that the radius of the hydrogen atom is 7.51Â·10-11m and that the electrostatic force is responsible for the circularmotion of the electron.

a) What is the electron's orbital velocity?

b) What is the kinetic energy of this orbital electron in eV?

An electron of momentum p is at a distance r from a stationary proton. The electron has kinetic energy . The atoms has potential energy and total energy . If the electron is bound to the proton to form a hydrogen atom, its average position is at the proton but the uncertainty in its position is approximately equal to the radius r of its orbit. The electron's average vector momentum is zero, but its average squared momentum is approximately equal to the squared uncertainty in its momentum as given by the uncertainty principle. Treating the atom as a one-dimensional system: (a) estimate the uncertainty in the electron's momentum in terms of r. Estimate the electron's (b) kinetic energy and (c) total energy in terms of r. The actual value of r is the one that minimizes the total energy, resulting in a stable atom. Find (d) that value of r and (e) the resulting total energy. (f) State how your answers compare with the predictions of the Bohr theory.