# Bohr Model of the Atom - Michigan State University

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Bohr Model of the Atom

The Hydrogen atom consists of a relatively massive positively charged ( e ) nucleus (proton) around which an electron of charge – e moves.

The force attracting the electron to the proton is given by Coulombs law f

G

= e r

ˆ

G

4

πε

0 r 2 r

ε

0

is the magnitude of the position vector from the proton to the electron and

= −

12 C N m

2 is the permittivity of free space.

The velocity of the electron is equal to the time derivative of the position vector

G

V= dr dt

and if the orbit is circular r

G

= r cos

θ i ˆ + r sin

θ j ˆ where i

ˆ & ˆ j are unit vectors. The velocity becomes

G

V= d r

G dt

= r d

θ dt

(

− sin

θ i

ˆ + cos

θ j

ˆ

) and the magnitude is

V= r d

θ dt

= r

ω

We need the acceleration which is

G d V d

= dt dt

2

2 r

G

= r d

2

θ dt

2

(

− sin

θ i ˆ + cos

θ ˆ

) j r

⎛ d

θ ⎞ 2

⎝ dt ⎠

( cos

θ i ˆ + sin

θ j ˆ

)

Identifying r

G

ˆ =

( cos

θ i

ˆ + sin

θ j

ˆ

)

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means that the counterbalancing force is rm

⎛ d

θ

⎝ dt ⎠

2

= rm

ω

2

Equating the two forces gives

4 e

πε

2

0 r 2

= rm

ω

2

= m V 2 r

= m V 2 r

Or

4

πε

0 r

=

( m V

)

2 = p 2 =

= 2 n 2 r 2

Where Bohr used the de Broglie relationship p

= h

λ and assumed that

2

π r

= n

λ n

=

1, 2,3, . This results in the radius of the electrons orbit being r =

=

4

πε

0 = a n 2 n

0

=

1,2,3,

"

Where a

0

=

= 2 4

πε

0 is equal to

10 m . Note that it is the smallest radius possible for the electron (

One often refers to n

=

1 n

=

1 ) and is called the Bohr radius.

as the first Bohr orbit, n

=

2 as the second Bohr orbit and so on.

The energy of the electron in the H atom is the sum of its kinetic ( T ) and potential energies ( V )

E T V

= 1

2 m V 2

4 e

πε

2

0 r

= −

8

π e

ε

2

0 r

And using the value of the radius found above the energy becomes

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E n

= −

2

( me

4

πε

0

)

2

4

=

2 n

2

=

E

1 n 2 n

=

1, 2,3,

"

Where

E

1

= −

2

4

( me

4

πε

0

)

2 =

2

= − −

18

J

= −

13.606

e V

E

1

is the lowest possible energy for the H atom. As with the radius of the allowed orbits one refers to E

1

as the energy of the electron in the first Bohr orbit, E

2

as the energy of the electron in the second Bohr orbit and so on.

It’s useful to remember the possible energies of the H atom as

E n

= −

13.606

n 2 eV n

=

1,2,3, "

How fast is the electron moving in the allowed orbits? We can estimate this as follows

Since

2

π r

= n

λ = nh p

We have p

=

2 nh

π r

=

2

π nh a n

0

2

= m v

And so

2

π h

=

= a mn a mn

0 0

= v

So the speed (magnitude of the velocity) is v =

= a mn

0

= n x

6

It’s insightful to express this speed relative to the speed of light c. v

= c

137.0

J. F. Harrison Michigan State University 3