TOPIC 4: ATOMIC PHYSICS | PHYSICS FORM 6

TOPIC 4: ATOMIC PHYSICS | PHYSICS FORM 6

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 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.  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.  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

rn= radius of the nth​​ orbit

vn= velocity of electron in the nth​​ 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=0.53×10-10m   

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 En​​ of the electron in the nth​​ orbit is the sum of kinetic and potential energy in the nth​​ orbit.

–          The K.E of electron in the nth​​ orbit is​​

The potential energy of electron in the nth​​ orbit is

Total energy of electron in the nth​​ 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→∞  En​​ =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 n1,​​ 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​​

RH​​ is Rydberg constant. The value of RH​​ 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  nth​​ 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

WORKED EXAMPLES

1.      1. Find the radius of the first orbit of hydrogen atom. What will be the velocity of electron in the first orbit? Hence find the  size of hydrogen atom​​

Solution

The radius of nth​​ orbit of it atom is given by

Radius of first orbit of it atom n=1

Velocity of electron in the nth orbit of hydrogen atom is given by

  =​​ 

Velocity of electron in the first orbit of hydrogen atom is given by

Since there is one electron in hydrogen atom the size hydrogen atom is equal to double the radius of the first orbit

Size of the atom

                              = 2

                              = 2 x 0.53Å

Size of an atom =1.06Å

2        2. (a) The hydrogen atom is stable in the ground,  state why?

     (b) The ionization energy of hydrogen is 13. 6eV what does it mean?

     (c) Calculate the wavelength of second line of Lyman series

    ​​ Solution

If the hydrogen atom is in the ground state (n=1) there is no state of lower energy to which a down ward transition  can occur thus a hydrogen atom in the ground state is stable

a)   It means that energy required to remove the single electron from the lowest energy state of hydrogen atom to becomes free electron is 13.6eV

b)     Second line of Lyman  series is obtained when electron jumps from  third  orbit​​ =3 to the first orbit n=1

According to Bohr’s theory the wavelength of emitted radiation is given by

 = 

  =​​  ​​

  =  x​​ 

                              ​​ 

3        3.( a) What is the meaning of negative energy of orbiting electron?

     (b) What would happen if the electron in atom were stationary?

   ​​ 

 Solution
a)   The negative total energy means that it is bound to the nucleus. If it acquires enough energy from some external source (a collision for example) to make its total energy zero the electron is no longer bound it is free.
b)     If the electrons were stationary they would fall into the nucleus  due to electrostatic force of attraction so atom would be unstable i.e. it would not exist
c)     For  Paschen series we have longest wavelength line​​ =4

This is a wavelength in the infrared part.  Other lines in this series have shorter wavelength bad approach series limit of wavelength to given by This wavelength is also in the hydrogen part. This the range or centre series (820.4nm to 1875nm) is the infrared

                 ​​
4. a)  If an electron jumps from first orbit to third orbit will it absorb  energy?
b)  Name the series of hydrogen spectrum lying in the infrared region
c)  Calculate  the shortest wavelength  of the Balmer series
d)  What is the energy possessed by an electron for n=?

Solution

a)    Yes it is because the energy level of third orbit is  more than that of the first orbit

b)    * Paschen series

            * Bracket series

            * P fund series

                 Solution
In Balmer series the radiation of shortest wavelength (i.e. of highest of highest energy) is emitted when electron jumps from infinity orbit​​ = to the second orbit​​ =2 of hydrogen atom.

5        5.  a) The ionization potential of hydrogen is 13.6V what does it mean?

b) Find the longest wavelength in Lyman series

c)  How much is the ionization potential of hydrogen atom?

d) The energy of the hydrogen atom in the ground state is 13.6eV. Determine the energies of those energy levels whose quantum numbers are 2 and 3.

Solution

a)     The ionization energy of hydrogen is 13.6eV. Therefore, if an electron which has been accelerated from rest through a p.d of 13.6V collides with a hydrogen atom it has exactly the right amount of energy to produce ionization.

This is a common method of producing ionization and therefore the term ionization potential is often used.

b)           Solution

In Lyman series the radiation of longest wavelength (i.e. lowest energy) is emitted when electron jumps from second orbit​​ =2 to first orbit n=1 of hydrogen atom

c)     The energy of hydrogen atom in the ground state is – 13.6eV. therefore its ionization energy is 13.6eV and ionization potential =13.6V

d)           Solution

The energy of an electron in the nth​​ orbit of hydrogen atom is given by

6
6.  a) Name the series of hydrogen spectrum lying in the

i) Visible region

ii) Utraviolet region of electromagnetic spectrum

b) Write the empirical relation for Paschen series lines of hydrogen spectrum

c) What are the values of first and second excitation potential of hydrogen atom?

d) Calculate the radii and the energy of three lowest energy allowed orbits for the electron in Lithium ion.  What is the energy of a photon that when absorbed causes an electron in Lithium ion to be excited from n=1 to n=3 state?

Solution

a)     i)  Balmer series

ii) Lyman series

 b)     The wavelength of the spectral lines in paschen series are  given by

c)  Excitation energy for first excited state   = –3.4 – (-13.6)

=10.2eV

For second excited state​​

= – (1. 51 – (-13.6)

= 12.1eV

       Solution

d)     (I ) for a single electron atom or ion the radius of the nth​​ orbit is given​​

           ​​ 

For a single electron atom or ion the energy of electron in the nth orbit is given by

Thus the energy of n=1, 2 and 3 orbits. The photo energy must be equal to the energy needed to excite the electron

E3​​ – E1​​ =hf

(-13.6) – (-122.4) =photon’s energy

Photon’s energy = 108.8eV

7. The ionization energy of hydrogen like atom is 4rydbergs

                  (a) What is the wavelength of radiation emitted when electron jumps from first excited state to the ground state?

          (b)What is the radius of the first orbit for this atom?
​​ 

         (d)     According to Bohr’s  theory what  is the angular momentum of a electron in the third orbit

           Solution

The energy electron in the nth​​ orbit of hydrogen like atom is

          

The energy required to excite the electron from n=1 level to n=2

If is the wavelength of the emitted radiations then, Radius of first orbit for this atom

Solution

( b) Radius of nth​​ orbit

(c)   ​​ Solution

Angular momentum L of an electron in nth​​ orbit is

L = n

Here   n=3

 Then ​​

L= 3​​ 

L=

7         8. The energy levels of an atom are shown in figure below.

        ​​ 

(a)Which one of these transitions will result in the emission of photon of wavelength 275nm?

(b) An electron orbiting in hydrogen atom has energy level of 3.4eV what will be its angular momentum
(c) The total energy of an electron in the first excited state of hydrogen atom is about​​
– 3. 4eV what is the wavelength?
solution
(a) Energy of emitted photon E

Therefore photon of wavelength 275nm will be emitted for transition B

Solution

 

(d)     K.E of electron = -(total energy  of electron)

    K.E of electron =3.4eV

ii)   P.E of electron = 2xtotal energy

P.E of electron = -6.8eV

10.  (a) How many lines can be drawn the energy level diagram of hydrogen atom?

       (b) Use Bohr’s model to determine the ionization energy of the He ion also calculate the minimum wavelength a photo must have to cause ionization

    ​​ 

      (d)  i) In neon atom the energies of the 3s and 3p states are respectively 16.70eV and 18.70eV. What wavelength corresponds to 3p -3s transitions in neon atom?

          ii) The wavelength of the first member of the Balmer series in hydrogen spectrum is 6563Å. Calculate the wavelength of first member of Lyman series in the same spectrum

PLANCK’S QUANTUM THEORY OF BLACK BODY RADIATION

The findings in the black body radiation led Max plank in 1901 to postulate that radiant energy is quantized i.e. it is radiated in form of energy packets.

BASIC POSTULATES OF THE PLANK’S THEORY

1.      Any radiation is associated with energy.

2.      Radiant energy is emitted or absorbed in small packets known as quanta.

3.      The energy associated with a quantum is proportional to the frequency​​ f  of the radiation

4.      The energy is absorbed or emitted only in whole number of quanta

    Black Body Radiation

A blackbody is a substance that absorbs all light fall on it and does not reflect any light.

It is not easy to get a black body however a sealed metal box with a very small hole on it is very close to a black body.​​

From law of physics it follows that a good absorber of radiation also is a good radiator.  A black body is supposed to be the best radiator.

When a black body is heated it emits light. The colour of light emitted changes from red  to yellow then to white as the temperature is increased.

The change in colour with temperature shows that the frequency changes with temperature.

This in contradiction with the classical wave theory since in the classical wave theory energy is uniformly distributed over the wave form when heating the black body the colour of radiation should stay the same

Only the intensity is supposed to increase with temperature.

DISTRIBUTION OF ENERGY IN THE SPECTRUM OF A BLACK BODY

Lamer and Pringshein investigated the distribution of energy amongst the different wavelength of a thermal spectrum of a back body radiation​​

The results obtained by Lamer and Pringshein are shown in figure below

Results:​​ 

1.   At a given temperature the energy is not uniformly distributed in the radiation spectrum of a hot body

2.   At a given temperature the intensity of radiations increases with increased in wavelength  and at a particular  wavelength λ its value is maximum with further increase in wavelength   the intensity of heat  radiations decreased​​

3.    With increase in temperature wave length  increases, wavelength emission of energy takes place.

The points on the dotted line represent wavelength at various temperatures​​

4.     For all  wavelength an increase in temperature causes an increase in the energy emission The area  under each curve represents the total energy  emitted for the complete spectrum at a  particular temperature

5.    This area increases in temperature of the body. It is found that the area is directly proportional to fourth power of the temperature of the body​​

                         ​​ 

This represents Stefan’s Boltzmann’s law.

Plank’s constant

Plank’s constant is a fundamental constant equal to the ratio of the quantum energy to the frequency of the radiation.

ELECTRON EMISSION

This is the liberation of electron from the surface of a substance.

For electron emission metals are used because they have many free electrons.

If a piece of metal is investigated at room temperature the random motion of free electrons is as shown in figure below

However these electrons are free only to the extent that they may transfer from one atom to another within the metal but they cannot leave the metal surface to provide electron emission.​​

It is because the free electron that start at the surface of metal find behind them positive nuclei pulling them back and none  pulling forward.

This at the surface of a metal a free electron encounters force that prevents it to leave the metal

 In other words the metallic surface  offers a barrier to free electrons and is known as surface barriers

However if sufficient external energy is given to the free electron its kinetic energy is increased and thus electron will cross over the surface barrier to leave the metal.

​​ WORK FUNCTION OF THE METAL

This is the additional energy required by an electron to overcome the surface barrier of the metal.

Or​​

This is the minimum energy required by an electron to just escape from the metal surface.

The work function depends on​​

i)  Nature of metal​​

ii) Conditions of the metal surface​​

 It is measured by a smaller unit of energy called electron volt. (eV) because this is the conventional unit of energy i.e. joule is very large for computations in atomic and nuclear physics.

Electron volt.

One electron volt is the amount of energy acquired by an electron when it is accelerated through a potential difference of IV

Since potential difference V

Work done = QV​​

For an electron​​

 

The electron volt is the kinetic energy gained by an electron being accelerated by a potential difference of one volt.

The work function of pure metal varies roughly from​​ V to​​ V as shown in table below

It is clear from the table above that the work function of platinum is the highest while it is lowest for Cesium.

It is desirable that metal used for electron emission should have low work function so that a small amount of energy is required to cause emission of electrons

PHOTOELECTRIC EFFECT

Photoelctric effect is the phenomenon of emission of electron from a metallic surface when radiation of suitable frequency falls on it or is the phenomenon where electromagnetic radiation of certain frequency when incident on certain material liberates electron from the surface of the material.

Photo emission

Photo emission is the emission of electron by electromagnetic radiation.

Photo electrons

These are emitted or rejected electrons from the surface of the cathode.

Photo electric effect is a general phenomenon exhibited by all substances but is most easily observed with metals.

 When radiation of suitable frequency the threshold frequency is incident on a metallic surface electrons are emitted from the metal surface.

The threshold frequency is different for metals.

Certain alkali, metals e.g. sodium potassium, calcium show photo electric effect when visible light falls on them.​​

However, metal like zinc, calcium, magnesium etc show photo electric effect to ultra violet light.

Threshold Frequency

The threshold frequency is the minimum frequency of the incident radiation which is just sufficient to eject photo electron from surface of a metal Or is the minimum frequency of radiation below which no photo electron emission occurs.

It is denoted by​​ 

Illuminating a metal surface with light of frequency less than​​  will not cause ejection of photo electrons, no matter now great is   the intensity of radiation.

But illumination with a frequency greater than ​​   causes emission   of photo electrons even if the radiation intensity is very small

 Threshold wavelength​​ 

The threshold wavelength is the maximum wavelength of the incident radiation at which photo electric emission occurs.

It is the wavelength corresponding to threshold frequency

If the wavelength of the incident radiation is greater than threshold wavelength then there will be no photo electric emission.

Photoelectric current is the photo-electrons emitted per second

EXPERIMENTAL STUDY OF PHOTO ELECTRIC EFFECT

Figure below shows the experimental set up for studying the photoelectric effect.

The arrangement consists of an evacuated glass or quartz tube inclusive a photosensitive cathode C and metallic A.

 A transparent window W is sealed onto the glass tube which can be covered with different filters to obtain the desired frequency.

The anode and cathode  are connected to a battery through a potential divided by which potential difference between anode and cathode can be changed.

 The reversing switch RS tends to make anode positive or negative with  respect  to cathode.​​

 The P.D between anode and cathode is measured by the voltmeter V while photoelectric current is indicated   by the micro ammeter.

1.     ​​
1.Effect of intensity of light on photo electric current

 The anode A is maintained at positive potential with  respect  to cathode C and a radiation of suitable frequency (above threshold frequency) is incident on cathode C.

 As a result photo electric current is set up.

 Keeping the frequency of incident radiation and accelerating potential fixed the intensity of the incident radiation is changed in steps.

 For each value of intensity of radiations the corresponding value of photo electron current is noted.

 If we plot a graph between  intensity of radiation and photoelectric current it is found to be a straight line passing through the origin O as shown in figure below​​

This shows that photoelectric current is directly proportional to the intensity of incident radiation

The intensity of radiation can be changed by changing the distance between cathode C and the source of radiation.

Effect of potential of anode with  respect  to cathode on photoelectric current

We keep the anode at some positive accelerating potential with  respect  to cathode C and illuminate the cathode with radiation of fixed frequency f above threshold frequency and fixed intensity I

 If we increase the positive potential on anode gradually, it is found that photo electric current also increases a stage comes when the photo electric current becomes maximum.

If we increase the positive potential on anode further the photo electric current does not increase.

This maximum value of photo electric current is called saturation current and corresponds to the photo electrons emitted by the cathode reach the anode A

 

Now saturation current is of higher value as shown in figure below

This is expected because the greater the intensity of incident radiation the greater is the photo electric current​​

Stopping potential

Stopping potential is the minimum retarding potential at which photoelectric current becomes zero Or  is the potential difference when no electrons are able to reach the anode.

It is also known as stopping voltage or cut-off potential.

Stopping potential is a measure of the maximum kinetic energy of the photo electrons.

    Since potential difference V

           ​​

For stopping Voltage Vo

For an electron​​

At V0, even the photo electrons having maximum kinetic energy K.E​​ max​​ (i.e. fastest photo electrons) cannot reach the anode A.

Therefore, the stopping potential V0​​ is a measure of the maximum kinetic energy K.E​​ max​​ of the photo electrons.

eV0​​ is the work done by the retarding force to stop the photo electron with maximum kinetic energy and is therefore equal to K.Emax.

 At V0, it is found that the photoelectric current cannot be obtained even if we increase the intensity of radiation. It is same for different intensities I1, I2​​ and I3​​ of incident radiation.

3.   Effect of frequency of incident radiation on stopping potential.

We now study the relation between the frequency​​ f​​ of the incident radiation and the stopping potential V0.

For this purpose, we take the radiations of different frequencies but of the same intensity.

For one frequency say f1, of the incident radiation, we plot the graph between photoelectric current and potential of anode A with  respect  to cathode C at a constant intensity of incident radiation​​

 Keeping the intensity of incident radiation the same, we repeat the experiment for frequency​​ f2​​ of the incident radiation.​​

The following is the resulting graph
                          

Observations from the graphs

1.      The value of stopping potential is different from radiation of different  frequencies

2.      The value of stopping potential is move in low higher frequency. This implies that the value of maximum kinetic energy depend on the frequency of incident radiation. 

The greater the frequency of incident radiation, the greater is the kinetic energy of emitted photo electrons.​​

3.       The value of saturation current depends on the intensity of incident radiation but is independent of the frequency of incident radiation.

If we draw a graph between the frequency of incident radiation (f) and the stopping potential (V0) at constant intensity of radiation, it will be a straight line AB as shown in figure below

     ​​ 

From the graph​​

At​​ fo, stopping potential V0​​ = 0. It means that at​​ fo, the photo electric current is just zero (i.e. photo electrons and emitted with zero velocity) and there is no retarding potential.

 V0​​ = 0

This limiting frequency fo​​ is called threshold frequency for the cathode material.

It is a minimum frequency of the incident radiation which is just sufficient to eject photo electrons (i.e. with zero velocity) from the surface of a metal.​​

Stopping potential is directly proportional to the frequency of incident radiation.

V0​​ α​​ f

The greater the frequency of incident radiation, the higher is the stopping potential and vice versa. ​​

Experiments show that photo electric emission is an instantaneous process.

As soon as light of suitable frequency (equal to or greater than fo) is incident on the surface of the metal, photo electrons are emitted from the metal surface.

The time delay is less than 10-9​​ second

LAWS OF PHOTOELECTRIC EMISSION

 The above experimental study of photoelectric effect leads to the following laws of photoelectric emission.

        i.          For a given metal, there exists a certain minimum frequency of incident radiation below which no emission of photo-electrons takes place. This cut off frequency is called threshold frequency fo.

      ii.         For a given metal and frequency of incident radiation (>fo) the photo electric current is directly proportional to the intensity of incident radiation.

    iii.        Above the fo, the maximum kinetic energy of the emitted photo-electron is independent of the intensity of the incident radiation but depends only upon the frequency of the incident radiation. ​​

    iv.        The photoelectric emission is an instantaneous process.

The above laws of photoelectric emission cannot be explained on the basis of light or radiation. This gave death blow to the wave theory of light or radiation.

FAILURE OF WAVE THEORY/CLASSICAL PHYSICS TO EXPLAIN PHOTOELECTRIC EFFECT

The wave theory of radiation  failed to explain photoelectric effect. This will become clear from the following discussion.

                            I. According to wave theory of radiation the greater the intensity of the wave the greater  the energy of the wave.

     So wave theory does explain why the number of emitted photoelectrons increase as the intensity of radiation is increased.

     But it fails to explain the experimentally observed fact that the velocity or kinetic energy of the emitted photoelectron is independent of the intensity of incident radiation.​​

     According to the wave theory, an increasing in the intensity of radiation should increase the kinetic energy of the emitted photoelectrons but it is contrary to the experimentally observed fact.

  II.  According to wave theory, intensity of radiation is independent of it is frequency it depends upon the amplitude of electric field vector.

       Therefore, an increase in the frequency of radiation should not affect the velocity or kinetic energy of the emitted electrons.

        But it is observed experimentally that if the frequency of the incident radiation is increased, the kinetic energy of the emitted electrons also increases.

   ​​
III. According to the wave theory, electrons should always be emitted from a metal by radiation of any frequency if the incident been is strong enough.

      However experiments show that no matter how great is the intensity of the incident radiation; no electrons are emitted from the metallic surface if the frequency of radiation is less than a particular value i.e.  threshold frequency.

IV.  According to the wave theory the energy of radiation is spread continuously over the wave fronts of the radiation.

      Therefore, a single electron in the metal will intercept only a small fraction of the wave’s energy.

      Consequently considerable time should be needed for an electron to absorb enough energy from the wave to escape the metal surface.
But experiment show that electron are emitted as soon as radiation of suitable frequency falls on the metallic surface.

In other words photoelectric emission is instantaneous there is no delay.

 The above discussion is a convincing proof of the inability of the wave theory to explain the photoelectric effect.

EINSTEIN QUANTUM THEORY OF LIGHT​​ 

Einstein explained photoelectric effect on the basis of Planck’s quantum theory.

According to Einstein light radiation consist of tiny packets of energy called quanta.​​

Photon

Photon is the single quantum of light radiation which travels with the speed of light.

The energy of a photon is given by E.

E=hf

where

f –frequency of light radiation​​

h – Plank’s constant.

Further, Einstein assumed that one photon of suitable frequency (=fo​​ or >fo) can eject only one photoelectron from the metal surface.

He suggested that the energy of a single photon cannot be shared among the free electron in the metal.

Only one electron can absorb the energy of a single photon.

EINSTEIN’S PHOTOELECTRIC EQUATION

According to Einstein, one photoelectron is emitted from a metal surface if one photon of suitable frequency is incident on the metal.

Suppose a photon of suitable frequency f (< than the fo​​ for the metal) is incident on the metal.

                                                                                                                                                                                       ​​

 The energy hf of the photon is spent in two ways​​

       I.            A part of photon energy is used in liberating the least tight bound electron from the metal surface which is equal to the work function W0​​ of the metal.

    II.             The rest of the energy of photon appears as maximum kinetic energy of the emitted electron.

Einstein summarized this idea in what is called the Einstein photoelectric equation.

Photon = work function + maximum kinetic energy

The above equation is known as​​ Einstein’s photoelectric equation

If the frequency of the incident radiation is fo then the emitted photoelectron will have zero velocity.

Maximum kinetic Energy of emitted photoelectrons is​​

If  f < fo, then from above equation K.Emax​​ is negative which is impossible therefore, photoelectrons emission cannot occur if the frequency of incident radiation is less than fo.

If f > fo, then equation above K.Emax​​ α​​ f. This means that max kinetic energy of photoelectrons depends only on the frequency (f) of the incident radiation.

                                                                                                                               NUCLEAR PHYSICS

RADIOACTIVITY

This is the emission of radiations from heavily elements such as uranium whose nuclei are unstable.

Radiations emitted are called alpha  , Beta (β) particles and gamma​​ rays.

RATE OF DISINTEGRATION

 The number of atoms of radioactive elements disintegrating per second​​  is directly proportional to the number of atom present at that instant.

                  

       Where λ = decay constant

  The negative sign (-) indicates that N decreases as time (t) increases.

If N0​​ is the number of atoms at time t= 0 and N is the number of atoms at time t. then:-

         ​​ HALF LIFE

The half life time​​ of a radioactive element is the times taken for the atoms disintegrate to half their initial number.

         NECTA 1994/1/19

Draw a graph of  

          SOLUTION:

This is the graph of radioactive decay in time.

                 

X-RAYS

 X-rays are produced by bombarding a target of heavy metal with high energy electron.

NECTA 1984/2/8

 The emission of X-rays may be regarded as the inverse of photo electric effect.Explain

 SOLUTION

 X-Rays which are waves are produced by bombarding a hard metal with electrons(particles)where as in photoelectric effect electron(particles) are liberated from ametal surface by incident radiation(waves)

 MODERN X-RAY TUBE

 

 Electron  are obtained from the filament by thermionic emission and are accelerated to the anode having small target of high melting point such as tangstem.

 X-RAY QUANTITY

 Refers to the intensity of X-rays which increases with the number of electrons limiting the target.This depends on the cathode temperature controlled by the heating current

X-RAY QUALITY

  Refer to the penetrating power of X-ray and determined by velocity whith which electrons strikes the target.

SOFT X-RAY:
Are those which can penetrate soft objects such as flesh

HARD X-RAY
Are those which can vibrate much more solid material

PROPERTIES OF X-RAY

 1.They travel in a straight line.

 2.They readily penetrate matter.

 3.They affect photographic plates

 4.They are not deflected by electric or magnetic field,because they have no charge

  They are wave of wavelength 10A

X-RAY SPECTRA

 

 CONTINUOUS SPECTRUM

  Explained by electromagnetic theory.

 It is due to electrical interaction between the bombarding electrons and the nuclei of the target atom

 An electron approaching a positively charged nucleus is accelerated and according to electromagnetic theory accelerated motion is accompanied by emission of radiations in this case x-ray

DISCRETE SPECTRA

 These are explained by quantum theory.They are produced when the incident electron interact with electron close to nucleus of the target atom.The bombarding electrons knock off electron in their orbits and the created gaps are filled by electrons in the high energy levels,when they moves in these gaps they produce x-ray photon.

The cut off wavelength or short wave limit(λ min)

 This is the minimum wavelength of x-radiations which corresponds to the maximum energy of the X-rays produced by electrons which have given out all their K.Ɛ   on a single encounter with the target nucleus

 ​​ ENERGY OF X – RAY:

  Energy of an electron striking the atom of the target is eV where e = electronic charge.

 V = p.d across the X-ray tube.

 If a direct collision is made with a target atom and the energy is absorbed then on quantum theory X-rays produced hence a maximum energy hV

Therefore

             eV = 1/2mV2 = hV

             eV = hc/ λ minimum

             λmin = hc/eV

    ​​ 

 NECTA 1989/1/18

Calculate the wavelength of most energetic X-rays produced by a tube operating at 1.0 x 105V.

USES OF X – RAYS

1.in investigation of suspected bone fractures

2.Detection of lung tuber culosis (feased tissue is denser than a heatthy one).

3.Treatment of cancer (cancer cells are killed by X-rays)

4.Casting and welding joints can be inspected for internal imperfection.

5.It is used in the study of crystal structure refer to Bragg’s law.

NUCLEAR FISSION FUSSION

Nuclear fission is a splitting disintegrating of heavy nucleus such as uranium into two other lighter nuclei (with several neutrons)

           ​​ Example:

Nuclear fusion

This is the combination of light nuclei to form a heavier nucleus.

       Example:

   ​​ NECTA 1990/2/8

  (1)Both fission and fusion are source of energy

     Explain.

         ​​ SOLUTION:

  ​​ Fission-During disintegration of heavy nucleus such as uranium, energy is released which is

    transformed into K.E of fragments.

  ​​ Fussion-When light nuclei are fused together a large amount of energy is released in term of heat.(eg  in the sun)

 2T1