FORM 6 PHYSICS: ATOMIC PHYSICS PART 1

 

1.   1. Structure  of the ATOM
  -Describe the Rutherford and bohr’s models of the atom.
  -Analyze atomic energy levels.
  -Discuses the hydrogen energy levels, and derives expressions for the energy levels.
  -Perform experiment to determine wavelength in the Balmer series of the hydrogen spectrum.  

2. Quantum physics
  – Describe failures of classical physics.
  – Explain Planck’s quantum theory of blackbody radiation.
  – Spectral nglish-swahili/distribution” target=”_blank”>distribution of black body radiation.
  – Explain Einstein’s quantum theory of light.
  – Perform experiment to determine the Planck’s constant.
  – Account for the photoelectric effect phenomenon.
  – Deduce stopping potential threshold frequency and work function of a metal.
  – Explain the photo electric effect.
  – Deduce de Broglie wave length for electron.
  – Discus the wave- particle duality of electron.
  – Derive de Broglie’s wavelength for the electron.
  – Describe production and uses of x- rays.
  – Uses in medicine, industry and in sample analysis.

3.   3. LASER
   – Describe production of laser light.
   – Explain properties of laser light.
   – Distinguish types of lasers.
   – Discuss methods of pumping in laser production.
   – Identify application of laser light.  

4.  4. Nuclear Physics
   – Describe   the structure of the nucleus.
    * Review Rutherford experiment.
  – Determine half life and the decay constant (λ) of a radioactive substance.
  – Explain the relation of nuclear mass and binding energy.
     * Discuss Einstein’s mass energy equation.
     * Apply Einstein’s mass energy relation to determine the biding energy of nuclei.
 – Identify criteria for stable and unstable nucleus.
    * Analyze the neutron and proton ratio and plot  N against Z for radioactive elements.
    * Establish criteria for stable and unstable nuclei
 – Identify uses and hazards of radioisotopes
    * Application
    * Hazards
 – Distinguish between fission and fusion processes
   * Meaning of fission and fusion
   * Calculate the energy released in a nuclear fission
   * Calculate the energy absorbed in nuclear fission
   * Describe the application of nuclear fission and fusion
  – Describe operation of a nuclease reactor
   * Construction and operation of nucleus reactor for safe application

THOMSON’S MODEL OF ATOM

According to Thomson an atom is a positive charged sphere in which the entire mass and positive charge of the atom is uniform distributed with negative electrons embedded in it as shown.

The number of electrons is such that their negative charge is equal to the positive charge of the atom. This atom is electrically neutral.

This model was called Thomson’s plum pudding model because the negatively charge electrons (the plums) were embedded in a sphere of uniform positive charge (the pudding).

Drawbacks of this Model
1.It could not provide stability to the atom it is because the positive and negative charges are stationary and will be drawn towards each  other, thus destroying the individual negative and positive charges.
2.  It could not explain the presence of discrete spectral lines emitted by hydrogen and other atoms.

 RUTHER FORD’S MODEL OF ATOM
The salient features of this model are
   (i)Every atom consist of a tiny central core, called the nucleus which contains all the atom’s positive charge and most of its mass (99.9%).

   (iii) The electrons occupy the space outside the nucleus. Since an atom is electrically neutral the positive charge on the nucleus is equal to the negative charge on electrons surrounding the nucleus.

   (iv)  Electrons are not stationary but revolve around the nucleus in various circular orbits as do the planets around the sun.

In this way Rutherford provided stability to the atom. It is because the centripetal force required by the electrons for revolution is provided by the electrostatic force of attraction between electrons and the nucleus.

   
             

e = charge on electron

z =total number of protons in the nucleus

m=mass of the electron

r =distance of electron from the nucleus

v= linear velocity of the electron

 
 

Force of attraction between electron and the nucleus is

    

     

where  Ze is a nuclear charge

The centripetal force required to keep the electron moving in circular path is

   

Since the atom is stable      

 
 

 
 

Kinetic energy of electron

 

From equation (1)

                                         
                                              
                                             

 
Potential energy of electron

           

 Total energy of electron

  

 

 
 

The total energy of electron in the orbit is negative hence the electron is bound to the positive nucleus

 For hydrogen Atom

For   hydrogen atom z= 1. Therefore K. E and P.E OF electron in hydrogen atom are

                                                                      
       

 
The total energy of electrons hydrogen atom is

 Limitations of Rutherford’s model of atom

1.      According to Maxwell’s theory of electromagnetism a charge that  is accelerating radiates energy as electromagnetic waves
The electron moving around the nucleus is under constant accelerating radiates energy as electromagnetic waves.

–          Due to   this continuous loss of energy the electrons in Rutherford’s model were bound to spiral towards the nucleus and fall into it when all of their rotational energy were radiated 

–          Hence Rutherford’s atomic model cannot be stable while in actual practice, an atom is stable 

This shows that Rutherford’s model is not correct

1.      During inward spiraling the electron’s angular frequency continuously increases 

–          As result electrons will radiate electromagnetic waves of all frequency i.e. the spectrum of these waves will be continuous in nature because these are continuous loss of energy.

–         But this is contrary to observation experiments shows that an atom emits line spectra and each line corresponds to a particular frequency   or wavelength. 

Rutherford’s model failed to account for the stability of the atom. It was also unable to explain the emission of line spectra.

BOHR’S MODEL OF ATOM
According to Bohr’s atomic model, the revolving electrons in the atom do not emit radiations under all conditions. They do so under certain conditions as expalined by him in his model.

BASIC POSTULATES OF BOHR’S MODEL OF ATOM

1.    The electrons revolve around the nucleus of the atom in circular orbits. The centripetal force required by electrons for revolution is provided by the electrostatic force of attraction between the electrons and the nucleus.

2.      An  electron can revolve only in those circular orbits in which its angular momentum is an integral multiple of

           

 
 

h= Plank’s constant.

 Radius of orbit r

From,   
                

   

 Since n is a whole number only certain value of r is allowed.

 Thus according to Bohr, an electron can revolve only in certain orbits of definite radii not in all these are called stable orbits (stationary orbit)

 According to this postulate the angular momentum of the electron does not have continuous range i.e. the angular momentum of the revolving electron is quantized.

 While revolving in stable or stationary orbits the electrons do not radiate energy inspite of their acceleration towards the centre of the orbit.

–          For  this reason these permitted orbits are called stable or stationary orbits.

e= charge on electron

m= mass of electron

r_n= radius of the n^th orbit

v_n= velocity of electron in the n^th orbit

Z= number of positive charge (protons)
   
 Positive charge on nucleus Ze

RADIUS OF BOHR’S STATIONARY ORBITS

As the centripetal force is provided by the electrostatic force of attraction  between the nucleus and electron.

According to Bohr

Consider equation

Take equation (ii) square it

 
 

Take equation (iii) equation (i)

 = .

 

It is clear that    n², radii of the stationary orbits are in ratio 1²: 2²:3² ………..clearly the stationary orbits are not equally spaced.

For hydrogen atom

For hydrogen atom z= 1, so that equation become

Now    = 0.53 x10^-10m

e = electronic charge          = (0. 53 x 10^-10)
 metres

Thus the radii of the first, second and third stationary orbits of hydrogen atom are 0.53 Å, 2.12 Å and 4.77Å respectively.

2.    VELOCITY OF ELECTRON IN BOHR’S STATIONARY  ORBIT

From equation below, we have

Putting the value of   into that equation

 

It is clear that in other words, electrons move at a lower speed in higher orbits and vice versa.

For hydrogen atom

Z =1

Then

 
 

3.      FREQUENCY OF ELECTRON IN STATIONARY ORBIT

The number of revolution completed per second by the electron in a stationary orbit around the nucleus

Velocity of electron in the  orbit

For hydrogen atom

Z = 1

Then,

Frequency of electron in the first orbit of hydrogen atom is n=1, r₁=0.53×10^-10_m   

 
 

Electron in first orbit of hydrogen atom will have a frequency of 6.57x 10¹⁵revolutions per second.

4. TOTAL ENERGY OF ELECTRON IN STATIONARY ORBIT

The total energy E_n of the electron in the n^th orbit is the sum of kinetic and potential energy in the n^th orbit.

–          The K.E of electron in the n^th orbit is

 
 

The potential energy of electron in the n^th orbit is

Total energy of electron in the n^th orbit is

But

 

Thus as n increases i.e. electron moves to higher orbit, the total energy of the electron increases i.e. total energy becomes less negative.

For hydrogen atom z=1

 
 

Thus the total energy of electron in a stationary orbit is negative which means that the electron is bound to the nucleus and it is not free to leave the atom.

We can find the total energy of electron in the various orbits of hydrogen atoms as under.

 
 

The total energy of electron increases i.e. becomes less negative as the electron goes to higher orbits

When n→∞  E_n =0 and the electron becomes free

Ground state/ normal state

This is the state of atom when the entire electrons in it occupies their lowest energy levels as required by their n and l values.

The energy of an atom is least i.e. largest negative value when n=1 i.e. when electron revolves in the first orbit.

The energy of hydrogen atom in the ground state is 13.6eV.

Excited state

This is the state of an atom when electrons in an atom occupy energy levels higher than those permitted by the values of n and l values.

At room temperature most of the hydrogen atoms are in the ground state

If hydrogen atom absorbs energy i.e. due to rise in temperature it may be promoted to one of the higher orbits (i.e. n=2, 3, 4…..)

The atom is  said to be in the excited state.

WAVE LENGTH
OF EMITTED
RADIATION.

  When an electron jumps from a higher orbit (n₂) to the lower orbit (n₁) the energy difference between the two orbits is released because the energy of electron in the higher orbit is more than in the lower orbit. Consider two orbits having principle quantum numbers n₂ and n₁ where n₂>n₁

Then energy of electron in the two orbits is given by

As the electron jumps from orbit n₂ to n_1, energy is  released in the form of electromagnetic radiation.

where

f= frequency of the emitted radiation

                               

The wavelength of the emitted radiation is given by

c=λf

  =

This equation gives the wavelength of emitted radiation.

Now,

   == wave number

Wave number

These are the number of waves in a unit length.

For hydrogen atom

For hydrogen atom z = 1

This gives the mathematical  formula for the wavelength of radiation emitted by hydrogen atom when electron jumps from outer orbit to inner orbit.

 
 

where

R_H is Rydberg constant. The value of R_H can be calculated as the value of e, m, h and c are known

 HOW TO CALCULATE THE RYDBERG CONSTANT USING CALCULATOR

From

 
 

Clearly, wavelength/frequency of radiation emitted from the excited atom is not continuous. They have definite value depending upon the values of , and

SPECTRAL SERIES OF HYDROGEN ATOM

Bohr gave a mathematical explanation for the spectrum of hydrogen atom.

 The whole hydrogen spectrum can be divided into district groups of lines each group of lines is called spectral series.

 The wavelength of the lines in each group can be calculated from Bohr’s formula

  = 

The following are spectral series of hydrogen atom

i)    Lyman series

ii)   Balmer series

iii)  Paschen series

iv)  Bracket series

v)   Pfund series

i)  Lyman series

The Lyman series is obtained when electron jump to first orbit n₁=1 from outer orbits (=2, 3, 4…)

 Therefore the formula for calculating the wavelength of the lines in this series is,

 
 

where

This series lies in the ultraviolet region which is the invisible region.

ii) Balmer series

Therefore the formula for calculating the wavelength of the lines in this series is

where

This series lies in the visible spectrum and was found first of all in the hydrogen series

iii) Paschen series

Therefore the formula for calculating the wavelength of the lines in this series is

where

This series lies in the infrared region.

iv) Brackett series

Therefore the formula for calculated the wavelength of the lines in this series is

This series lies in the infrared region.

v) Pfund series 
The Pfunds series is obtained when electrons jump to fifth orbit n₁ = 5 from outer orbits (n₂
=6, 7, 8…..)

Therefore the formula for calculating the wavelength of the lines in this series is

where

(n₂
=6, 7, 8…..)

This series also lies in the infrared region

ENERGY LEVEL DIAGRAM

Energy level diagram is a diagram in which the total energies of electron in different stationary orbit of an atom represented by parallel horizontal lines drawn according to some suitable energy scale

 In order to draw energy level diagram of an atom we must know the total energy of electron in different stationary orbits.

The total energy of an electron in the  n^th orbit of hydrogen atom is given by

By putting value of n=1, 2, 3…..  we can find the total energy of electron in various stationary orbits of hydrogen atom as

Similarly we can find the total energy of electron in the higher orbits

The table below gives the total energy of electron of hydrogen atom in different stationary orbits.

                                                                                                                                                                                                                                                                                                                                                                                                                         The energy level diagram of hydrogen atom is shown below

Total energy of electron in a stationary orbit is represented by a horizontal line drawn to some suitable energy scale.

 
 

(i)  The hydrogen atom has only one electron and this normally occupies the lowest level and has energy of -13.6eV

         When the electron is in this level the atom is said to be in the ground state.At room temperature nearly all the atoms of hydrogen are in ground.

(ii)  If hydrogen atom absorbs energy (due to rise in temperature )the electron may be promoted into one of the higher energy levels

 The atom is now said to be in an excited state.Thus when the electron occupies other than the lowest energy level the atom is said to be in the excited state.

(iii) Once in an excited state the atom is unstable after a short time interval the electron falls back into the lowest state so that the atom is again in the ground state.

          The energy that was originally impacted is emitted as electromagnetic waves.

(iv)   The total energy of electron for (n=) it becomes free of atom.

      The minimum energy required to free the electron from the ground state of an atom is called ionization energy

            For hydrogen atom ionization energy is +13. 6eV

(v)    The difference between the adjacent energy goes on decreasing as the   value of n increases.

             So much so that when n>10 the energy difference is almost zero this is show by closeness of energy level lines at higher levels.

(vi)    Note that region is labeled continuous at energy above zero n= level, the electron is free from the atom and is at rest

       Higher energy represents the translation kinetic energy of the free electron

       This energy is not quantized and so all energies above n = are allowed

IMPORTANT TERMS

It is desirable to discuss some important terms much used in the study of structure of atom.

(i) EXCITATION ENERGY

Excitation energy is the minimum energy required to excite an atom in the ground state to one of the higher stationary state.

Hydrogen atoms are usually in their lowest energy state where n=1

 In this state (ground state) they are said to be unexcited.

However if you bombard the atoms with particles such as electron or proto collision can excite them

In other words a collision may give an atom enough energy to change it from ground state to some higher stationary state.Consider the case of hydrogen atom we know that = -13.6eV (ground state = -3.4eV (first excited state) =1.51eV (second excited state) and =0

In order to lift an electron from ground state n =1 to the first excited state n=2 energy required is E

E =–

E= -3.4 – (- 13.6)

E = 10.2eV

Therefore the bombarding particle must provide an energy of 10.2eV to excite the atom from n =1 state to n=2 state

Similarly to excite the atom from n=1 state to  n=3 state energy required is

   E = – 1.51 – (-13.6)

    E = 12.1eV

 
 

We say that first and second excitation energies of hydrogen are 10.2eV and 12.1eV respectively

(ii) EXCITATION POTENTIAL

Excitation potential is the minimum accelerating potential which provide an electron energy sufficient to jump from the ground state n=1 to one of the outer orbits

Energy required to lift a electron from ground state n=1 to n=2 state is

Hence excitation potential for the first excited state of hydrogen is 10. 2V

Similarly energy required to lift an electron from ground state n=1 to n=2

 The value of excitation potential depend upon the state to which the atom is excited to which the atom is excited from the ground state

 (iii) IONIZATION ENERGY

Ionization energy is the minimum energy needed to ionized an atom

Consider the case of hydrogen atom it has only one electron and this normally occupies the ground state.

The energy of the electron for n= state is zero and if the electron is lifted to this level (n=) it becomes free of hydrogen atom i.e. hydrogen atom is ionized

(iv) IONIZATION POTENTIAL

Ionization potential is the minimum accelerating potential which would provide electron energy sufficient to just remove the electron from the atom.

The ionization potential of one electron atom or ion is given by

(v) QUANTIZATION OF ENERGY

  Quantization of energy is the existence of energy radiated by atoms in a specific amount which is are integral multiples of a constant (hf).

SUCCESS OF BOHR’S THEORY

The success of bohr’s theory is not to be attributed so much to the mechanical picture of atom he proposed but rather to the development of mathematical explanation that agrees exactly with experimental observations. Bohr’s theory achieved the following successes.

i)   MADE ATOM STABLE

Bohr’s theory made the atom stable according to this theory an electron  moving in the formatted (quantum)  orbits cannot lose energy even though under constant acceleration. This provided stability to the atom.

ii)   INTRODUCED QUANTUM MECHANICS

 Bohr’s theory introduced quantum mechanics in the realm of atom for the first time

 Bohr’s explained that sub- atomic particles e.g. electrons are governed by the laws of quantum mechanics and not by classical laws of electron hydrogen as assumed by Rutherford

 This completely changed our thinking and was the major step towards the discovery of the rudiment laws of the atomic world

iii)   GAVE MATHEMATICAL EXPLANATION OF HYDROGEN SERIES

The hydrogen series found by various scientists were based on empirical relation but had no mathematical explanation

However these relations were easy derived by applying Bohr Theory

Further the size of hydrogen atom as calculated from this theory agreed very closely with the experimental value.

LIMITATIONS OF BOHR’S THEORY

 Bohr’s simple theory of circular orbits inspire of its many successes was found inadequate to explain many phenomena observed experimentally.

This theory suffered from the following drawbacks.

(i)     It could not explain  the difference in the intensities  of emitted radiations.

(ii)     It is silent about the wave  properties of electron

(iii)     It could not explain experimentally observed phenomena such as Zeeman Effect, Stack effect etc.

(iv)     Bohr’s model does not explain why the orbit are circular while elliptical path is also possible

(v)     It could only partially explain hydrogen atom. For example this theory does not explain the fine structure of spectral lines in the hydrogen  atom