Model-of-Hydrogen-Atom

PHYSICS CHAPTER 1113  The Bohr’s radius is defined as the radius of the most stable the radius of the most stable lowest orbit or ground state n =1 in the hydrogen atom in the hydrogen

PHYSICS CHAPTER 11

At the end of this chapter, students should be able to:

 Explain Bohr’s postulates of hydrogen atom.

PHYSICS CHAPTER 11

3

11.1.1 Early models of atom

Thomson’s model of atom

that consists of homogenous positively charged spheres with tiny negatively charged electrons embedded throughout the sphere

as shown in Figure 11.1

atom

11.1 Bohr’s atomic model

positively charged sphere

electron

Figure 11.1

PHYSICS CHAPTER 11

Rutherford’s model of atom

 In 1911, Ernest Rutherford performed a critical experiment that showed the Thomson’s model is not correct and proposed his new atomic model known as Rutherford’s planetary model of the atom as shown in Figure 11.2a

 According to Rutherford’s model, the atom was pictured as electrons orbiting around a central nucleus which concentrated

PHYSICS CHAPTER 11

5

around the nucleus

electron orbits will be decreased steadily

hence the atom would collapse as shown in Figure 11.2b

 According to Bohr’s Model, he assumes that each electron each electron

moves in a circular orbit which is centred on the nucleus,

the necessary centripetal force being provided by the centripetal force being provided by the

electrostatic force of attraction between the positively charged nucleus and the negatively charged electron as shown in Figure 11.3

11.1.2 Bohr’s model of hydrogen atom

Figure 11.3

PHYSICS CHAPTER 11

7

 On this basis he was able to show that the energy of an energy of an

orbiting electron depends on the radius of its orbit

 This model has several features which are described by the

postulates (assumptions) stated below :

1 The electrons electrons move only in certain circular orbits, called

STATIONARY STATES or ENERGY LEVELS ENERGY LEVELS When it is

in one of these orbits, it does not radiate energy does not radiate energy

2 The only permissible orbits are permissible orbits are those in the discrete set

for which the angular momentum of the electron angular momentum of the electron L

equals an integer times h/ 2 π Mathematically,

of radius :

r

electron the

of mass :

m

, ,

,

n : principal quantum number  1 2 3

PHYSICS CHAPTER 11

3 Emission or absorption Emission or absorption of radiation occurs only when an

electron makes a transition from one orbit to another The frequency f of the emitted (absorbed) radiation is

given by

i

f E E

Planck' :

h

state energy

change :

E



state energy

PHYSICS CHAPTER 11

9

At the end of this chapter, students should be able to:

 Derive Bohr’s radius and energy level in hydrogen atom.

a n

En

PHYSICS CHAPTER 11

11.2.1 Bohr’s radius in hydrogen atom

 Consider one electron of charge –e and mass m moves in a circular orbit of radius r around a positively charged nucleus with a velocity v as shown in Figure 11.3

 The electrostatic force between electron and nucleus electrostatic force between electron and nucleus

contributes the centripetal force as write in the relation below:

11.2 Energy level of hydrogen atom

c

F  centripetal forceelectrostatic force

r

mv r

Q

2

2 1 0

0

2 2

4 

PHYSICS CHAPTER 11

11

 From the Bohr’s second postulate:

By taking square of both side of the equation, we get

 By dividing the eqs (11.4) and (11.3), thus



2

nh mvr 

(11.4)

2

2

2 2

2 2

4 

h

n r

h n mv

r v m

0 2 2

2 2

2

2 2 2

2 2

PHYSICS CHAPTER 11

h

n r



12

2 2

(11.5)

3 , 2 , 1

;

4 2 2

2 2

rn



0

2a n

rn 

2 2

2 0

4 mke

h a





(11.6)

and

PHYSICS CHAPTER 11

13

 The Bohr’s radius is defined as the radius of the most stable the radius of the most stable

(lowest) orbit or ground state (n =1) in the hydrogen atom in the hydrogen atom

and its value is

 Unit conversion:

 The radii of the orbits associated with allowed orbits or states

n = 2,3,… are 4a0,9a0,…, thus the orbit’s radii are orbit’s radii are

quantized

 31 9 19 2 2

2 34 0

10 60

1 10

00 9 10

11 9 4

10 63

31

a OR 0.531 Å (angstrom)

1 Å = 1.00 1010 m

PHYSICS CHAPTER 11

 is defined as a fixed energy corresponding to the orbits in a fixed energy corresponding to the orbits in

which its electrons move around the nucleus

 The energy levels of atoms are quantized quantized

 The total energy level total energy level E of the hydrogen atom is given by

 Potential energy U of the electron is given by

11.2.2 Energy level in hydrogen atom

K U

2

a n

ke

U   (11.8)

nucleus electron

PHYSICS CHAPTER 11

15

 Kinetic energy K of the electron is given by

 Therefore the eq (11.7) can be written as

22

0

2 2

0

24 2

1

04

22

1

a n

ke K

2 0

ke a

PHYSICS CHAPTER 11

 In general, the total energy level E for the atom is

 Using numerical value of k, e and a0, thus the eq (11.10) can

2

Z a

2 19

10 31

5 2

10 60

1 10

00

eV 10

60 1

10 17

Eqs (11.10) and (11.12) are valid for energy level of the hydrogen atom

where Z : atomic number

where En : energy level of nth state (orbit)

PHYSICS CHAPTER 11

17

 The negative sign negative sign in the eq (11.12) indicates that work has to work has to

be done to remove the electron from the bound of the atom

to infinity , where it is considered to have zero energy zero energy

 The energy levels of the hydrogen atom are when

n=1, the ground state ground state (the state of the lowest energy level lowest energy level) ;

n=2, the first excited state first excited state;

n=3, the second excited state second excited state;

n=4, the third excited state third excited state;

n=, the energy level is

6

132

E    

6

132

E    

6

an atom.

is defined as

the energy levels that higher than the ground state.

is defined as the energy the energy

required by an electron that raises it to an excited state from its ground state.

Figure 11.4

PHYSICS CHAPTER 11

19

The electron in the hydrogen atom makes a transition from the energy state of 0.54 eV to the energy state of 3.40 eV Calculate the wavelength of the emitted photon

(Given the speed of light in the vacuum, c =3.00108 m s1 and Planck’s constant, h =6.631034 J s)

Solution :

The change of the energy state in joule is given by

Therefore the wavelength of the emitted photon is

Example 1 :

eV 40 3 eV;

54

58

58





m 10

34







The ionization energy in joule is given by

Therefore the frequency of the photon required to ionize the atom is

Example 2 :

0 eV;

6

18

2  18





 E hf

E 



 34  f

18 6 63 10 10

18

29

3  15



f

4 mke

h n

31 2

2 34 2

2

10 60

1 10

00 9 10

11 9 4

10 63

.

6 2



r

m 10

12

r

PHYSICS CHAPTER 11

Solution :

b By applying the Bohr’s 2nd postulate, thus

c The kinetic energy of the orbiting electron is given by



34 10

10 12

2 10

09

h mvr 

22

1 10

11

9 2

41

5 19





K

PHYSICS CHAPTER 11

23

A hydrogen atom emits radiation of wavelengths 221.5 nm and 202.4 nm when the electrons make transitions from the 1st excited state and 2nd excited state respectively to the ground state

Calculate

a the energy of a photon for each of the wavelengths above,

b the wavelength emitted by the photon when the electron makes a transition from the 2nd excited state to the 1st excited state

(Given the speed of light in the vacuum, c =3.00108 m s1 and Planck’s constant, h =6.631034 J s)

Solution :

a The energy of the photon due to transition from 1st excited state

to the ground state is

Example 4 :

m 10

4 202 m;

10 5

1

10 5

221

10 00

3 10

63

98

 E

PHYSICS CHAPTER 11

24

Solution :

a The energy of the photon due to transition from 2nd excited state

to the ground state is

b

Therefore the wavelength of the emitted photon due to the transition from 2nd excited state to the 1st excited state is

m 10

4 202 m;

10 5

2

10 4

202

10 00

3 10

63

83

50

50

34







PHYSICS CHAPTER 11

25

At the end of this chapter, students should be able to:

 Explain the emission of line spectrum by using energy

level diagram.

 State the line series of hydrogen spectrum.

 Use formula,Learning Outcome:

PHYSICS CHAPTER 11

 The emission lines correspond to the photons of discrete energies that are emitted when excited atomic states in the gas make transitions back to lower energy levels

 Figure 11.5 shows line spectra produced by emission in the visible range for hydrogen (H), mercury (Hg) and neon (Ne)

11.3 Line spectrum

Figure 11.5

PHYSICS CHAPTER 11

27

 Emission processes in hydrogen give rise to series, which are sequences of lines corresponding to atomic transitions

 The series in the hydrogen emission line spectrum are

 Lyman series involves electron transitions electron transitions that end at the end at the

ground state of hydrogen atom It is in the ultraviolet ltraviolet

(UV) range

 Balmer series involves electron transitions electron transitions that end at end at

the 1 st excited state of hydrogen atom It is in the visible visible

light range

 Paschen series involves electron transitions electron transitions that end at end at

the 2 nd excited state of hydrogen atom It is in the

infrared (IR) range

 Brackett series involves electron transitions electron transitions that end at end at

the 3 rd excited state of hydrogen atom It is in the IR IR

range

 Pfund series involves electron transitions electron transitions that end at the end at the

4 th excited state of hydrogen atom It is in the IR range IR range

11.3.1 Hydrogen emission line spectrum

E

0 0 54 0



85 0



51 1



39 3



6 13



n

4 3 2

1

 5

PHYSICS CHAPTER 11

 If an electron makes a transition from an outer orbit of level ni to

an inner orbit of level nf, thus the energy is radiated

 The energy radiated energy radiated in form of EM radiation (photon) form of EM radiation (photon) where the wavelength is given by

 From the Bohr’s 3rd postulate, the eq (11.13) can be written as

11.3.2 Wavelength of hydrogen emission line spectrum

2

f

n a

2 1 2

i

n a

ke

En

2 f 0

2

1 2

1

1

n a

ke n

a

ke hc

2

1

n n

a

ke hc

22

1 1

1

n n

1 1

1 2

1 3

1 4

1 5

PHYSICS CHAPTER 11

33

 The Bohr’s model of hydrogen atom

 predicts successfully the energy levels of the hydrogen atom

but fails to explain the energy levels of more complex fails to explain the energy levels of more complex

atoms

 can explain the spectrum for hydrogen atom but some

details of the spectrum cannot be explained especially cannot be explained especially

when the atom is placed in a magnetic field

 cannot explain the Zeeman effect (Figure 11.7)

 Zeeman effect is defined as the splitting of spectral the splitting of spectral

lines when the radiating atoms are placed in a magnetic field.

11.3.3 Limitation of Bohr’s model of hydrogen atom

a the longest wavelength, and

b the shortest wavelength of the photon emitted in this series

(Given the speed of light in the vacuum c =3.00108 m s1 ,Planck’s constant h =6.631034 J s and Rydberg’s constant RH = 1.097 

10 m )

Example 5 :

) eV (

n

E

0

0 38 0



85 0



51 1



40 3

3

2

54 0



Figure 11.8

00 3 10

63 6

10 60

1 51

1 40

3

58

1 1

1

n n

1 10

097

1

1



m 10

56

3 10

63 6

10 60

1 0

40 3

66

1 1

1

n n

1 10

097

1

1



m 10

65

1 1

1 10

097

1

m 10

03







PHYSICS CHAPTER 11

Exercise 11.1 :

Given c =3.00108 m s1, h =6.631034 J s, me=9.111031 kg,

e=1.601019 C and RH =1.097107 m1

1 A hydrogen atom in its ground state is excited to the n =5

level It then makes a transition directly to the n =2 level before returning to the ground state What are the

wavelengths of the emitted photons?

ANS : 4.34107 m; 1.22107 m

2 Show that the speeds of an electron in the Bohr orbits are

given ( to two significant figures) by

.



