Modern Physics: Mastering Physics: The Bohr Atom

Learning Goal: To understand the Bohr model of the hydrogen atom.

In 1913 Niels Bohr formulated a method of calculating the different energy levels of the hydrogen atom. He did this by combining both classical and quantum ideas. In this problem, we go through the steps needed to understand the Bohr model of the atom.

Part A
Consider an electron with charge and mass orbiting in a circle around a hydrogen nucleus (a single proton) with charge . In the classical model, the electron orbits around the nucleus, being held in orbit by the electromagnetic interaction between itself and the protons in the nucleus, much like planets orbit around the sun, being held in orbit by their gravitational interaction. When the electron is in a circular orbit, it must meet the condition for circular motion: The magnitude of the net force toward the center, , is equal to . Given these two pieces of information, deduce the velocity of the electron as it orbits around the nucleus.

Part A

Consider an electron with charge

and mass

orbiting in a circle around a hydrogen nucleus (a single proton) with charge

. In the classical model, the electron orbits around the nucleus, being held in orbit by the electromagnetic interaction between itself and the protons in the nucleus, much like planets orbit around the sun, being held in orbit by their gravitational interaction. When the electron is in a circular orbit, it must meet the condition for circular motion: The magnitude of the net force toward the center, F_c

, is equal to mv^2/r

. Given these two pieces of information, deduce the velocity

of the electron as it orbits around the nucleus.

Express your answer in terms of

, and

, the permittivity of free space.

ANSWER:

$\frac{e}{\sqrt{mr4{\pi}{\epsilon}_{0}}}$

Part B

The key insight that Bohr introduced to his model of the atom was that the angular momentum of the electron orbiting the nucleus was quantized. He introduced the postulate that the angular momentum could only come in quantities of $nh/(2 \pi)$ , where

is Planck's constant and

is a nonnegative integer ( $0, 1, 2, 3, \dots$ ). Given this postulate, what are the allowable values for the velocity

of the electron in the Bohr atom? Recall that, in circular motion, angular momentum is given by the formula L=mvr

Express your answer in terms of

, Planck's constant

, and

ANSWER:

$\frac{nh}{2{\pi}mr}$

Part C

In Parts A and B you found two different expressions to describe the allowed electron velocities

. Equate these two values (eliminating

) and solve for the allowable radii

in the Bohr model.

Express

in terms of

, and

ANSWER:

$\frac{n^{2}h^{2}{\epsilon}_{0}}{{\pi}me^{2}}$

As the electron orbits the nucleus with a speed

at a radius

, it has both kinetic energy and electric potential energy. The total energy

of the electron can be expressed as

$E=\frac {1}{2} m v^2 - \frac{e^2}{4\pi \epsilon_0 r}$ .

Part D

In Parts B and C you saw that, according to Bohr's postulate, the electron radius

and the electron velocity

only have certain allowable values. Plug the values obtained for these two quantities into the energy statement given above to arrive at a new statement for the allowed energy levels in the Bohr atom.

Express your answer in terms of

, and

ANSWER:

$-\frac{e^{4}m}{8n^{2}h^{2}{\epsilon}_{0}{^{2}}}$

The standard formula for energy in the Bohr model is

$E_n = - \frac{e^4m}{8n^2h^2\epsilon_0^2}$ .

Despite the serious flaws of the Bohr model, such as casually mixing quantum and classical ideas with little if any justification, this formula turns out to be equivalent to the energy formula for hydrogen obtained from quantum mechanics. To adequately deal with atoms other than hydrogen, however, requires the full quantum theory.

Modern Physics

Sunday, February 13, 2011

Mastering Physics: The Bohr Atom

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