Question

In 1913 Niels Bohr formulated a method of calculating the differentenergy levels of the hydrogen atom. He did this by combining bothclassical and quantum ideas. In this problem, we go through thesteps needed to understand the Bohr model of the atom.Consider an electron with charge −e−e and mass mmm orbiting in a circle around a hydrogen nucleus (a single proton) with charge +e+e. 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, FcFcF_c, is equal to mv2/rmv2/r. Given these two pieces of information, deduce the velocity vvv of the electron as it orbits around the nucleus.

Answer 1

The velocity v of the electron orbiting the nucleus in the Bohr model of the hydrogen atom can be calculated using the balance of electrostatic force and centripetal force, leading to the expression v = √(kee2/mr).

Step-by-step explanation:

In the classical Bohr model of the hydrogen atom, an electron with charge −e and mass m is orbiting in a circular path around a nucleus with a positive charge +e. According to the condition for circular motion, the centripetal force Fc which keeps the electron in orbit is provided by the electrostatic force of attraction between the electron and the nucleus and is equal to the necessary centripetal force for circular motion, mv2/r. We can use Coulomb's law to determine the magnitude of the electrostatic force, which states that the force F between two point charges is given by F = kee2/r2, where ke is Coulomb's constant.

Setting the electrostatic force equal to the centripetal force, we have:

kee2/r2 = mv2/r

This simplifies to:

v2 = kee2/mr

Taking the square root of both sides, the velocity v of the electron can be expressed as:

v = √(kee2/mr)

Thus, we have derived the expression for the electron's velocity v in a circular orbit within the Bohr model of the hydrogen atom.

Answer 2

v = √ k e²/m r

Step-by-step explanation:

From the classical point of view the force between the nucleus and the electron is electrostatic

F = q E

q = e

we apply Newton's second law

F = m a

where the centripetal accelerations

a = v² / R

we substitute

e E = m v² / r

v² = ( e/m E r)

the electrioc field is

E = k q/r²

v² = e/m k e/r

v = √ k e²/m r