PHOTO ELECTRIC EFFECT - DEFINITION, LAW AND EINSTEIN'S THEORY

 Photo electric effect

Consider a negativity charged zinc plate connected to a gold leaf electroscope (Fig. 1). The divergence in the leaves indicates the potential of the plate. Let a beam of ultraviolet light be incident in this plate. It is observed that the divergence decreases gradually. This is due to the ejection of electrons from the plate. 

Fig. 1. Ejection of electrons from surface of metal by light. 

Definition:

     "Photo electric effect is the phenomenon of emission of electrons from the surfaces of certain substances, mainly metals, when light of shorter wavelength is incident upon them."

Experiment study of photo electric effect

The apparatus required for the experimental study of photo electric effect is shown in Fig. 2. It consists of an evacuated glass tube 'T' fitted with two electrodes. Electrodes 'E' known as emitter is coated with a photo sensitive material. Light from a source 'S' after passing through a quartz window 'W' is made to fall upon 'E'. Collecting electrodes can be provided with a potential whose value can be changed with the help of a potentiometer 'P' fed with a 10 volt battery through a reversing key 'R'. Key 'R' enables us to provide both positive and negative potential to the collecting electrodes. Potential difference between two electrodes is noted with a voltmeter while the current is given by the ammeter 'A'. 

Fig. 2. Experimental setup for the study of photo electric effect. 

     (a) Effect of collector's potential on photo electric current. Keeping the intensity and frequency of light from 'S' fixed, note the value of current in the circuit by changing potential of collector from positive to zero and to negative value. Plot a graph between the potential (along X-axis) and photo electric current (along Y-axis). Two such curves, one for low intensity (i) and second foe high intensity (ii) are shown in Fig. 3.

Fig. 3. Variation of photo electric current due to potential of anode. 

     Each curve shows that the current changes continuously with a change in potential. Beyond some positive potential of collector the current attains a saturation value while for a certain negative value 'V₀' it is required to zero. 'V₀' is called the stopping potential. Following conclusions can be drawn from above observations :

     (i) Presence of current for zero value potential indicates that the electrons are ejected from the surface of emitter with some energy. 

    (ii) A gradual change in the number of electrons reaching the collector due to change in its potential indicates that the electrons are ejected with a variety of velocities. 

   (iii) Current is reduced to zero for some negative potential of collector indicating that there is some upper limit to the energy of electrons emitted. 

   (iv) For any potential of collector the current in case of curve (ii) is greater than that in case (i). This means the current depends upon the intensity of incident light. 

   (v) Both the curves meet at same point on X-axis indicting that stopping potential in independent of the intensity of light. 

     (b) Effect of intensity of light. Keeping potential of collector fixed note the value of current in the circuit for different intensity of light. A graph between intensity of light and photo electric current is found to be a straight line (Fig. 4) indicating that the photo electric current is directly proportional to the intensity of incident radiation. 

Fig. 4. Intensity of light. 

     (c) Effect of frequency of light. Repeat the experiment as described in case (a) for different sources of frequencies f₁, f₂, f₃, giving same intensity of light. Three curves (i), (ii) and (iii) as shown in Fig. 5 are obtained. The curves shows that :

Fig. 5. Variation of current with anode potential for different frequencies. 

     (i) Stopping potential depends upon the frequency of light. Greater the frequency of light, greater is the stopping potential. 

    (ii) Saturation current is independent of frequency. Note the value of stopping potentials V₀₁, V₀₂, V₀₃, ......., for various frequencies f₁, f₂, f, ....... respectively. A straight line as shown in Fig. 6 is obtained the straight line has an intercept 'f₀' on f-axis indicating that for frequency 'f₀' there will be no current when collector is at zero potential. This is the minimum frequency capable of producing photo electric effect and is called threshold frequency. Greater the frequency of incident light, greater the value of negative potential required to stop the current. Hence, maximum energy of the emitted electrons depends upon the frequency of light. 

Fig. 6. Variation of stopping potential with frequency. 

Law of photo electricity

The conclusions drawn from the experimental study of photo electric effect can be summed up as laws of Photoelectricity. 

     (i) Photo electric effect is an instantaneous process. 

    (ii) Photo electric current is directly proportional to the intensity of incident light and is independent of its frequency. 

  (iii) The stopping potential and hence the maximum velocity of the electrons depends upon the frequency of incident light and is independent of its intensity. 

   (iv) The emission of electrons stop below a certain minimum frequency known as threshold frequency. 

Einstein's theory of photo electric effect

Einstein explained photo electric effect on the basis of Planck's quantum theory. According to Max Planck's radiation was composed of energy bundles only on the neighborhood of the emitter. Einstein suggested that these energy bundles known as 'photons' preserve their identity throughout their life. Energy contained in bundle or packet is

                      E = hf

     It is matter of common observation that the electrons are ejected only when light is incident on a metal. This is due to the existence of a potential barrier all around the surface of metal. The electron must possess certain minimum energy 'W₀' in order to cross the barrier, 'W₀' is known as work function. It is defined as the minimum energy required to pull an electron out from the surface of metal. 

     An incident photon supplies whole of its energy 'hf' to the electron, which consumes an energy 'W₀' against the work function and comes out with the remaining energy as its kinetic energy [Ek = ½ (mc²)]. 

                 ½ mv²max = hf - W0                       ... (1) 

     If 'f₀' is the shareholder frequency, the energy 'hf₀' of the photons will be just sufficient to help the electron in overcoming the potential barrier. 

                    hf₀ = W₀                                 ...(2)

    Substituting for 'W₀' in equation (1), we get

                  ½ mv²max = hf - hf0

or ½ mv²max = h (f - f0) ....(3)

     Equation (3) is known as Einstein's equation of photo electric effect. 

     It is clear from equation (3) that greater the frequency of incident radiation, greater is the kinetic energy of the electron and hence greater negative potential is required to stop it. 

     An increase in intensity of light results in an increase in the number of photons which is turn results in ejection of more number of electrons. Hence, photo electric current is proportional to the intensity of light. 

     Electrons coming from the surface, spend energy only on overcoming the potential barrier. Therefore, kinetic energy of these electrons is maximum. The incident radiation penetrates the metal to thickness of about 10-6 cm. Thus, it will be able to eject the electrons lose some of the energy as they rise to the surface and hence are emitted with a lesser kinetic energy. Different electrons lose some amounts of energies. This is the reason that electrons are ejected with a variety of velocities. 

     If the frequency of incident radiation is less than 'f₀' given by equation (2), it is unable to help the electron in overcoming potential barrier. This is the reason for the existence of threshold frequency. 

     If         V₀ = stopping potential

             ½ mv²max = eV0                                 ... (4)       

     From equation (1) and (2), we get

                 eV0 = hf - W0,     V0 = (h/e) f - (W0/e) 

Let h/e = A    and    W0/e = B

V0 = Af - B                                   ...(5) 

     It is the equation of a straight line between f (along X-axis) and V₀ (along Y-axis) shown in Fig. 6 

     Calculation of f₀ (threshold frequency) 

     If V₀ = 0, from equation (5), 

                 0 = Af - B

or    f = B/A = W₀/e × e/h = W₀/h = f₀   [From equation (2)]

      Thus, the intercept of 'V₀ - f' curve of f-axis gives the value of threshold frequency. 

     Calculation of 'W₀' (work function) 

     If         f = 0,   V₀ = V' 

     From equation (5) V' = - B = - W₀/e

                 W₀ = - eV' = e (- V') 

or       Work function = e × (intercept on V₀-axis) 

     Calculation of 'h'

     Slope 'm' of straight line given by equation (5) us

              m = A = h/e      ∴     h = em

or     Planck's constant = e × slope of 'V₀ - f' curve. 

Photo electric cell

     It consists of an evacuated bulb B, whose inside is coated with an alkali metal 'P', leaving a clear portion 'W' in the form of a window (Fig. 7). The bulb is made up of glass if it is to be used for white light and is made up of quartz if it is to be used in case of ultraviolet light. It has am electrode 'C' which is given a positive potential with the help of a battery. Light from a source 'S' is focussed into a metal P with the help of a convex lens 'L'. An ammeter connected in the circuit indicates the photo electric current. 

Fig. 7. Photo electric cell. 

Application of photo electric cell

Photo electric cell has found a number of uses in various fields of science. 

     (i) It plays an important role in television studio. It converts light and shade of the picture into electrical waves which after a proper processing are transmitted to distant stations. 

    (ii) It is used for reproduction of sound in films. Microphone converts sound into electric waves which after amplification are fed to an electric lamp. Intensity of light from this lamp records lines of varying transparency on the film. During reproduction, a  beam of light falls on a photo cell after crossing through this film. Photo cell converts light back into electrical oscillation which produces sound when fed to the receiver. 

   (iii) It is used for triggering fire alarm. In factories using chemical or explosive materials, photo cells are fitted at selected places. Light from any accidental fire falls on the cell. This produces a current which after amplification is fed to an alarm. 

   (iv) It is used in operating burglar's alarm. Ultraviolet light from a source is incident constantly on a cell. Any unwanted person entering a room cuts that beam unknowingly, thus stopping the current for a fraction of a second. This triggers an alarm. 

   (v) It is used for automatic switching of street light. Light from sun during day falls constantly on the cell making a current to flow through the circuit continuously. This current after amplification is fed to an electromagnet which keeps the key of the street light circuit open. In the evening, after sunset intensity of light and hence the current decreases. The electromagnet loses its strength. As a result of this the key is closed and the street light is automatically switched on. 

   (vi) A photo cell coupled with an electronic counter can be used to count automatically. The number of persons entering and leaving a hall. Each person will cut a beam of ultraviolet light once, thus producing a kick in the counter. 

 (vii) It is used to compare the illuminating power of the two sources. Current produced is directly proportional to intensity which I turns is proportional to the illuminating power of the source. Ratio of deflection in ammeter with two sources of light gives the ratio of their illuminating powers. 

Important Notes

Photo electric effect is the phenomenon of ejection of negativity charged particles (electrons) due to incidence of light on metals.  Photo electrons are ejected with an initial velocity which varies between zero and certain maximum value.  Photo electric current falls to zero for some negative potential of opposite electrodes. This negative potential is called stopping potential.  Stopping potential is independent of intensity of incident light.  Stopping potential depends upon frequency of incident light.  Saturation current is independent of frequency of light.  Saturation current varies directly as the intensity of incident light.  Instantaneous photo electric current varies directly as the intensity of incident light.  Photo electric current stops below a particular frequency of incident light. The frequency is known as threshold frequency.  Work function is the characteristic of a photosensitive material.  Threshold frequency is equal to (1/h) times the work function.  Work function is equal to electric charge times the intercept of (V0 - f) graph on (V0 - f) graph.  Electronic charge is equal to the multiplication of Planck's constant and slope of (V0 - f) graph. Photo electric effect establishes the quantum nature of radiation. This can be considered to be a proof in favour of particle nature of light.

Keywords

1. Photo electric cell. Device which converts light into electricity. 

2. Photo electric current. Current flowing in the circuit due to the emitted photo electrons. 

3. Photo electric effect. Phenomenon of conversion of light into electricity. 

4. Photo sensitive material. Material which emits electrons due to incidence of light. 

5. Potential barrier. Electrostatic obstruction which restrict the outer orbital electrons to go away. 

6. Quartz. Material which allows ultraviolet light to pass through it. 

7. Stopping potential. Negative potential on the collector which reduce the photo electric current to zero. 

8. Threshold frequency. Minimum frequency capable of producing photo electric effect. 

9. Ultraviolet light. Light of wavelength just smaller than that of violet light. 

10. Wavelength. Distance between two consecutive crestes or between two consecutive throughs. 

11. Work function. Minimum energy required by an electron to overcome the potential barrier. 

Key Formulae

Equation of photo electric effect

½ mv²max = hf - hf0

½ mv²max = h (f - f0)

Bohr's atom model and Hydrogen spectrum

 Bohr's atom model

Following some serious objections to Rutherford's atom model, Bohr suggested some modifications in the atom model. The modified atom model is called Bohr's atom model whose basic postulates are described below :

     (i) The central part of the atom called nucleus, contains whole of positive charge and almost whole of the mass of atom. Electrons revolve round the nucleus in fixed circular orbits. 

    (ii) Electrons are capable of revolving only in certain fixed orbits, called stationary orbits or permitted orbits. In such orbits they do not radiate any energy. 

   (iii) When revolving in permitted orbit an electron possesses angular momentum 'L' ( = mvr) which is an integral multiple of h/2π

i.e.,               L = n . h/2π

where 'n' is an integral and 'h' is planck's constant. 

    (iv) Electrons are capable of changing the orbits. On absorbing energy they move to a higher orbit while emission of energy takes place when electrons move to a lower orbit. If 'f' is the frequency of radiant energy, 

                   hf = W₂ − W₁

     Here 

          W₁ = enerɡy of electron in lower orbit

and

         W₂ = enerɡy of electron in hiɡher orbit

    (v) All the laws of mechanics can be applied to electron revolving in a stable orbit while they are not applicable to an electron in transition. 

Bohr's theory of atomic structure

Consider an electron of mass 'm' moving round a nucleus having a charge + Ze (Fig. 1) where 'Z' is the atomic number. If 'r' is the radius of orbit, centripetal force 'Fe' required by the electron. 

Fig. 1. Electron revolving around a nucleus. 

                    Fe = k (mv²/r)                          ... (1) 

     The force is provided by the electrostatic attraction between the nucleus and the electron. Electrostatic force 'Fe' is given by

                     Fe = (1/4πɛ₀) × (q₁ × q₂/r²) 

     Here      q₁ = Ze   and   q₂ = e, 

                    Fe = (1/4πɛ₀) × (Ze × e/r²) 

or                Fe = (1/4πɛ₀) × (Ze²/r²)         ... (2) 

     From equations (1) and (2), we get

             mv²/r = (1/4πɛ₀) × (Ze²/r²) 

     ∴      mv² = (1/4πɛ₀) × (Ze²/r)             ... (3) 

     According to the basic postulate of bohr's atomic model, 

             L = mvr = n . h/2π                      ... (4) 

(i) Orbital velocity of electron

    Dividing equations (3) and (4), we get

        mv²/mvr = (1/4πɛ₀) × [(Ze²/ r) × (2π/nh)] 

     ∴       Vn = (1/4πɛ₀) × (2π Ze²/nh)      ... (5) 

     'Vn' denotes the velocity of electrons in nth orbit. 

     Equation (5) indicates that :

     (i) For a particular orbit (n = constant), orbital velocity of electron varies directly as the atomic number of the substance, i.e., Vn ∝ Z. 

    (ii) For a particular element (Z = constant), orbital velocity of the electron varies inversely as the order of the orbit

i.e.,               Vn ∝ 1/n

     Let v₁ be the velocity of electron in the first orbit (n = 1) 

               v₁ = 1/4πɛ₀                                   ... (6) 

     Dividing equations (5) by (6), we get

                       Vn/v₁ = 1/n

     ∴                     Vn = v₁/n                        ... (7) 

(ii) Radius of orbit of electron

     From equation (4) 

                           v = nh/2πmr

     Substituting for 'v' in equation (3), we get

           m (n²h²/4π²m²r²) = (1/4πɛ₀) × (Ze²/r) 

     ∴            r = 4πɛ₀ × (n²h²/4π²mZe²)    ... (8) 

                   r = (ɛ₀/π) × (n²h²/mZe²)        ... (9) 

     Since 'n' can have only integral values, only those orbits are possible which have radii corresponding to n = 1, 2, 3, ...... Thus, the orbits are quantised. 

     Equation (8) indicates that :

     (i) Radius of a particular orbit of electron (n = constant) varies inversely as the atomic number i.e., r ∝ 1/Z. It means heavier the element, shorter is the radius of orbit. 

    (ii) For a particular element (Z = constant) radii of different orbits vary directly as the square of the order of orbit i.e., r ∝ n². It means the outer radii will be more spaced apart than the inner ones. 

   (iii) Energy of electron. As an electron revolves around the nucleus in a stable orbit, it possesses energy which is composed of two energies. 

     (a) Kinetic energy. It is the energy possessed by the electron by virtue of its motion in the orbit. 

    If 'v' is the velocity of electron,   

                   K.E. = 1/2 (mv²) 

     Substituting for 'mv²' from equation (3), we get

                   K.E. = 1/2 (1/4πɛ₀) × (Ze²/r) 

     (b) Potential energy. It is the energy possessed by the electron by virtue of its position near the nucleus. Potential energy of two charges q₁ and q₂ separated a distance 'r' apart is 

                 P.E. = (1/4πɛ₀) × (q₁q₂/r)

     Here       q₁ = Ze    and    q₂ = -e

     ∴           P.E. = (1/4πɛ₀) × (Ze × (-e)/r)

or              P.E. = - (1/4πɛ₀) × (Ze²/r) 

     (c) Total energy. Total energy 'W'of an electron revolving round the nucleus is

W = K.E. + P.E.

     = ½ (1/4πɛ₀) × (Ze²/r) - (1/4πɛ₀) × (Ze²/r)

     = (1/4πɛ₀) × (Ze²/r) × (½ - 1) 

     = - ½ × (1/4πɛ₀) × (Ze²/r)

     

   Substituting for 'r' from equation (8), we get 

 

 W = - ½ (1/4πɛ₀) × [Ze²/(ɛ₀/n) (n²h²/mZe²)]

  W = - (1/8ɛ₀²) × (Z²me⁴/n²h²) ... (10)

     Energy of an electron revolving in an orbit is negative. It means that electron is bound to the nucleus.  

     Equation (10) indicates that :

     (i) energy of an electron revolving in a particular orbit (n = constant) varies directly as the square of the atomic number of the atom. Because of the negative sign an electron in nth orbit of a heavier element is less energetic than that of lighter element in nth orbit. 

    (ii) Energy of an electron of a particular element (Z = constant) varies inversely as the square of the order of the orbit. Again, because of negative sign the electrons in he outer orbits of an element are more energetic than those in inner orbits. 

   (iii) Frequency, wavelength and wave number of radiation. When an electron jumps from a higher orbit to a lower one, the difference of the energy is emitted in the form of a radiation. If the electron jumps from an orbit (n = n₂) of energy 'W₂' to one (n = n₁) of energy 'W₁', according to the basic postulate of Bohr's theory, 

             hf = W₂ - W₁

           W₁ = - (1/4πɛ₀)² × (2π²Z²me⁴/n₁²h²) 

and    W₂ = - (1/4πɛ₀)² × (2π²Z²me⁴/n₂²h²) 

     ∴     hf = [- (1/4πɛ₀)² × (2π²Z²me⁴/n₂²h²)] - [- (1/4πɛ₀)² × (2π²Z²me⁴/n₁²h²)]

     ∴       f = (1/4πɛ₀)² × (2π²Z²me⁴/h²) × [1/n₁² - 1/n₂²]

or          f = (1/4πɛ₀)² × (2π²Z²me⁴/h³) × [1/n₁² - 1/n₂²]                                                          ... (11) 

     If 'λ' is the wavelength of radiation, 

              f = c/λ

where 'c' is the velocity of radiation (light). 

     ∴     c/λ = (1/4πɛ₀)² × (2π²Z²me⁴/h³) × [1/n₁² - 1/n₂²]

      ∴    1/λ = (1/4πɛ₀)² × (2π²Z²me⁴/ch³) × [1/n₁² - 1/n₂²]                                                       ... (12) 

     Since 1/λ = ̄f (wave number of radiation) 

     ∴   ̄f  = (1/4πɛ₀)² × (2π²Z²me⁴/ch³) × [1/n₁² - 1/n₂²]                                                         ... (13)

or     ̄f = RZ² (1/n₁² - 1/n₂²)

where   R = (1/4πɛ₀)² × (2π²me⁴)/ch³ is called Rydberg's constant. 

     ∴      R = (1/4πɛ₀)² × (2π²me⁴/ch³) 

or         R = (1/8ɛ₀²) × (me⁴/ch³) 

     Value of R is 10973731 m-1 or 1.0973731 ×107 m-1

     In S.I. (k = 1/4πɛ₀)

 ̄f = (1/8ε₀²) × (Z²me⁴/ch³) (1/n₁² - 1/n₂²)  ... (14) 

     Equation (14) indicates that :

     (i) Wave number and hence the wavelength of emitted radiation depends upon the order of two orbits between which the transition takes place. 

    (ii) For a particular transition (n₁ and n₂ constant) the radiation emitted by a heavier element possesses greater wave number and hence the smaller wavelength. 

Important Notes

(i) Orbital velocity of electron :       It is smaller in outer orbits vn = v1/n. In a particular order it is greater for heavier elements.  (ii) Radius of orbit :   It varies directly as square of order of orbit r ∝ n². For a particular order it is smaller for heavier elements.  (iii) Energy of electron : Electrons in the outer orbits are more energetic.  For a particular order, electrons in higher elements are more energetic.  (iv) Wave number/ Wavelength of emitted radiation :   It depends upon the orders of two orbits between which the transition takes place.  For a particular transition emitted by a heavier element possesses greater wave number and hence the smaller wavelength.

Bohr's theory of hydrogen atom

Hydrogen atom has atomic number 'Z' as one. It contains one electron revolving round the nucleus. Putting Z = 1, we get quantities connected with hydrogen atom. 

     (i) Radius of orbit. From equation (9), we get

                   r = (ɛ₀/π) × (n²h²/me²) 

    (ii) Energy of electron. From equation (10), we get

                  W = - (1/8ɛ₀²) × (me⁴/n²h²) 

   (iii) Frequency, wavelength and wave number of radiation. 

     Putting Z = 1 in equation (11), (12) and (13), 

            f = (1/8ɛ₀²) × (me⁴/h³) × (1/n₁² - 1/n₂²)

       1/λ = (1/8ɛ₀²) × (me⁴/ch³) × (1/n₁² - 1/n₂²)

           ̄f = (1/8ɛ₀²) × (me⁴/ch³) × (1/n₁² - 1/n₂²)

Hydrogen Spectrum

As transition of electron takes place from a higher orbit to a lower orbit, difference of energy is radiated in the form of radiation. The wavelength of the radiation depends upon the initial and final orbit within which the transition takes place. Accordingly a number of series are emitted. Each series is composed of a number of lines (Fig. 2).

Fig. 2. Production of hydrogen spectrum. 

     (i) Lymen series. This is a series in which all the lines correspond to transition of electrons from a higher excited state to orbit having n = 1, i.e., n₁ = 1 and n₂ = 2, 3, 4 ... wave number of lines constituting 'Lymen series' are given by

                   ̄f = R [1/(1²) - 1/n²]

where       n = 2, 3, 4, ..... 

     and 'R' is the Rydberg's constant. 

     This series lies /observed in ultraviolet region 

    (ii) Balmer series. This is a series in which all the lines correspond to transition of electrons from higher excited state to the orbit having n = 2 

i.e.,           n₁ = 2,   n₂ = 3, 4, 5, .... 

     Therefore, wave numbers of lines constituting 'Balmer series' are given by

               ̄f = R (1/2² - 1/n²) where n = 3, 4, 5, ... 

     First member of this series corresponds to the transition of electron from 3rd to 2nd orbit. 

     ∴       n₁ = 2,     n₂ = 3

     ∴      1/λ = R (1/2² - 1/3²) = R (1/4 - 1/9) 

or         1/λ = 5R/36          ∴     λ = 36/5R

     Substituting       R = 1.09737 × 10

                  λ = 6563 ̊A

     The limiting case of this series is given by n₂ = ∞ 

     ∴        1/λ = R (1/2²) = R/4

     ∴           λ = 4/R = 3646  ̊A

     The value of wavelength indicates that the series lies in the visible region. 

   (iii) Paschen series. This is a series in which all the lines correspond to transition of electrons from a higher excited state to the orbit having n = 3,

i.e.,        n₁ = 3,  and   n₂ = 4, 5, 6, 7, ..... 

     ∴     Wave-numbers of lines constituting 'Paschen series' are given by, 

         ̄f = R (1/3² - 1/n²) where n = 4, 5, 6, 7, .... 

     This series lies/observed in infrared region 

   (iv) Brackett series. This is a series in which all the lines correspond to transition of electrons from a higher excited state to the orbit having n = 4,

i.e.,        n₁ = 4,   and    n₂ = 5, 6, 7, ...... 

     Therefore, wave number of lines constituting 'Brackett series' are given by, 

             ̄f = R (1/4² - 1/n²) where n = 5, 6, 7, .... 

     This series lies/observed in infrared region. 

    (v) P-fund series. This is a series in which all the lines correspond to the transition of electrons from a higher excited state to the orbit having n = 5,

i.e.,          n₁ = 5   and    n₂ =6, 7, 8, ..... 

           ̄f = R (1/5² - 1/n²) where n = 6, 7, 8, .... 

     This series lies/observed in infrared region. 

Energy Level of Hydrogen Atom 

The energy 'W' of an electron revolving round the nucleus is

          W = - [(1/8ɛ₀²) × (me⁴/n²h²), 

          m = 9.1 × 10-31 kg

           e = 1.59 × 10-19 C,    

           h = 6.67 × 10-34 joule sec. 

     For the innermost orbit n = 1. Energy 'W₁' of electron in the innermost orbit is given by

              W₁ = - (9 × 109)²

2 × (3.142)² × 9.1 × 10-31 × (1.59 × 10-19)⁴/(6.67 × 10-34)²

 

                      = - 21.78 × 10-19 J. 

     

     Since    1eV = 1.6 × 10-19 J. 

  ∴           W1 = - (21.78× 10-19/1.6 ×10-19) 

                         = - 13.6 eV

     For the 1st excited state, 

                    n = 2

                 W₂ = W₁/4 = - (13.6/4) 

                       = - 3.4 eV

     For 2nd excited state, 

                    n = 3

                 W₃ = W₁/9 = − 13.6/9 eV

                       = − 1.51 eV.

     Similarly, for other excited states

                 W₄ = − 0.85 eV

     and     W₅ = − 0.54 eV

Fig. 3. Energy level diagram for hydrogen atoms. 

     Various energy levels are shown in Fig. 3. The set of spectral lines is also shown in figure. 

Excitation and Ionisation Potentials

     (i) Excitation potential. Under normal conditions, the orbital electrons keep on revolving around the nucleus in fixed orbits. When exposed to incident energy from outside, the electrons absorb energy and have a tendency to shift to higher orbits. In this state the atom is said to be existing in excited state. 

     Excitation potential of a particular state is the minimum accelerating potential to which if the electron is subjected, it acquires just the required amount of energy to reach a desired higher orbit. 

     Illustration. Let us suppose we want to take the electron of hydrogen atom from first to second orbit. 

 Energy of electron in first orbit = - 13.6 eV 

 Energy of electron in second orbit = - 3.4 eV

 Required amount of energy = (-3.4 eV) - (-13.6 eV) = 10.2 eV

     This energy can be acquired by the electron if it is accelerated between two electrodes having a potential difference of 10.2 eV. 

     Therefore, the excitation potential for hydrogen atom from first to second orbit is 10.2 V. 

     Similarly, it can be calculated that the excitation potentials of hydrogen atom from first to third, fourth and fifth orbits are respectively 8.69 V, 9.66 V. 

    (ii) Ionisation potential. An electron is said to be just free if its energy has zero value. This will happen for n = ∞ only. The atom, in this stage is said to be ionised. 

     Ionisation potential of particular orbital electron is the minimum accelerating potential to which if the electron is subjected, it acquires just the required amount of energy to be displaced to the outermost orbit (n = ∞). 

     Since hydrogen possesses only one electron and it requires 13.6 eV of energy to be displayed to the outermost orbit, therefore, ionisation potential of hydrogen is 13.6 V. Atoms which possess large number of electrons possess a variety of ionisation potentials varying with the variation in the order of orbit initially occupied by the electron. 

Limitations to Bohr's Theory

A close examination of hydrogen spectrum reveals that a line, in any of the series consists of a number of lines packed very close to each other. These lines can only be seen with the help of high resolving power microscope. 

     Presence of these lines could not be explained on the basis of Bohr's theory which assumes that orbits of electrons around the nucleus are circular. Some field modified the theory by assuming the orbits to be elliptical and was able to explain the fine structure of atom. The theory was further modified by taking into account the spin of electron and relativistic variation of mass. 

Important Notes

Lyman series lies in ultraviolet region, Balmer series in visible region while Paschen, Brackett and P-fund series lies in infrared region.  A line having longest wavelength in a particular series corresponds to the transition of electron from the nearest higher orbit.

Key Formulae

1. Bohr's theory of atomic structure

     (i) Orbital velocity of electron

                vn = (1/4πɛ₀) × (2πZe²/nh) 

    (ii) Radius of orbit of electron

               r = (ɛ₀/π) × (n²h²/mZe²) 

   (iii) Energy of electron

               W = - [(1/8ɛ₀²) × (Z²me⁴/n²h²)]

   (iv) Frequency, wavelength and wave number of radiation

              f = (1/4πɛ₀)² × (2π²Z²me⁴/h³) × [1/n₁² - 1/n₂²]

              ̄f = 1/λ = (1/4πɛ₀)² × (2π²Z²me⁴/ch³) × [1/n₁² - 1/n₂²]

2. Hydrogen Spectrum

     (i) Lymen series :

                   ̄f = R [1/(1)² - 1/n²]

                      where n = 2, 3, 4, ..... 

    (ii) Balmer series :

                  ̄f = R [1/(2)² - 1/n²]

                     where n = 3, 4, 5, ..... 

   (iii) Paschen series :

                  ̄f = R [1/(3)² - 1/n²]

                      where n = 4, 5, 6, .....

  (iv) Brackett series:

                 ̄f = R [1/(4)² - 1/n²]

                      where n = 5, 6, 7, ..... 

   (v) P-fund series:

                ̄f = R [1/(5)² - 1/n²]

                    where n = 6, 7, 8, ..... 

3. Energy level of hydrogen atom

         W₁ = - 13.6 eV,        W₂ = -3.4 eV, 

         W₃ = -1.51 eV,         W₄ = -0.85 eV, 

         W₅ = -0.54 eV

Atomic models - Thomson's and Rutherford's atom model

 Atom Model

Man had always been curious to know the details of the constituent of matter. Structure of atom, which is the smallest particle of matter which takes part in a chemical reaction has always been one of the main targets of scientists. 

     An explanation regarding the structure of atom is called an atom model. 

     Faraday, during this experiments on electrolysis showed that each atom, irrespective of the nature of the element gave up or received a fixed quantity of charge equal in magnitude to 1.59 × 10-19 C. This led to the electrical nature of matter. Every atom is electrically neutral and is found to be stable. Every atom model must be based upon these experimentally observed facts. 

Thomson's Atom Model

J.J. Thomson gave the first idea regarding structure of atom. The model is known after him as Thomson's atom model. According to this atom model whole of the positive charge is distributed uniformly in the form of a sphere. Negativity charged electrons are arranged within this sphere here and there (Fig. 1). The model is popularly known as plum-pudding model. Every electron is attached towards the center of uniformly charged sphere while they exert a force of repulsion upon each other. The electrons get themselves arranged in such a way that the forces of repulsion are exactly balanced. When disturbed, electrons vibrate to and fro within the atom and cause emission of visible, infrared and ultraviolet light. 

Fig. 1. Thomson's atom model. 

     Thomson's atom model satisfied the requirements of the atom and the demands of electromagnetic theory. According to this model, hydrogen can give rise to a single spectral line. Experimentally, hydrogen is found to give several series, each series consisting of several lines. This indicated that, "Thomson's atom model needed magnifications". 

Rutherford's Experiment

Rutherford performed experiments on the scattering of alpha particles by extremely thin metal foils. 

     A radioactive source (radon) of ∝-particles was placed in a lead box having a narrow opening (Fig. 2). This source emits ∝-particles in all possible directions. However, only a narrow beam of alpha particles emerged from the lead box, the rest being absorbed by the lead box. This beam of ∝-particles is made incident on a gold foil whose thickness is only one micron, i.e., 10-6 m. When passing through the metal foil, the ∝-particles get scattered through different angles. These particles fall on a fluorescent screen producing a tiny flash of light on the screen. This can be easily viewed by a low power microscope in a dark room. 

Fig. 2. Rutherford's ∝-scattering.

     In 1913, Geiger and Marsden performed a more sensitive experiment on the scattering of ∝-particles on the guidelines suggested by Rutherford. This experiment is described below :

     Apparatus consists of an air tight chamber 'C' which can be evacuated by a tune 'T' (Fig. 3). The chamber is capable of rotation inside a jacket 'R' about a vertical axis. The source 'S' of ∝-particles, radon, is placed inside a lead cavity 'L'. ∝-particles, after coming out of the narrow opening in the lead cavity strike a thin foil 'F'. This foil is made of some metal of high atomic weight like gold, silver or platinum. The foil is placed at the centre of chamber 'C'. Scattered ∝-particles are viewed through low power microscope 'M', which is provided with a fluorescent screen. As the chamber rotates about a central axis, the microscope M rotates along with it. But the cavity 'L' and foil 'F' remain fixed with the tube. 

Fig. 3. Experimental set up for study of ∝-scattering.

     The ∝-particles on striking the atom of the toilet scattered in different directions. By rotating the chamber, the number of particles scattered along different directions can be recorded by observing the scintillations on the fluorescent screen. 

     The above experiment gave the following results :

     (i) Most of the ∝-particles either passed straight through the metal foil or suffered only small directions. This could be explained by Thomson's atom model. 

    (ii) A few particles were deflected through angles which were less than or equal to 90°.

   (iii) Very few particles were deflected through angles greater than 90°. It was observed that only 1 in 8000 particles was found to be deflected greater than 90°. Sometimes a particle was found to be deflected through 180°. In other words, it was sent back in the same direction from where it came. The large angle of scattering came as a greater surprise. It could not be explained by Thomson's atom model. It was one of the main reasons for rejecting Thomson's atom model. 

    (iv) If 'Φ' is the angle made by a scattered particle with its original direction of motion and 'N' is the number of particles available in that direction, it was found that, 

    (v) If 't' is the thickness of the foil and 'N' is the number of ∝-particles scattered in a particular direction (Φ = constant) it was observed that

                    N/t = constant. 

Conclusions

     (i) The fact that most of the ∝-particles passed undeviated led to the conclusion that an atom has a lot of empty space in it. 

    (ii) ∝-particles are heavy particles having high initial speeds. These could be deflected through large angles only by a strong electrical force. This led Rutherford to the conclusion that whole of positive charge and nearly the entire mass of the atom were concentrated in a tiny central core. Rutherford named this core as nucleus. 

   (iii) The difference in deflection of various particles can be explained as follows :

     ∝-particles which pass at greater distances away from the nucleus shown as A and A' in (Fig. 4)  suffer a small deflection due to smaller repulsion exerted by the nucleus upon them. The particles like B and B' which pass close to the nucleus experiences a comparatively greater force and hence get deflected through greater angles. A particle 'C' which travels directly towards the nucleus is first slowed down by the repulsive force. Such a particle finally stops and then is repelled along the direction of its approach. Thus, it gets repelled back after suffering a deviation of 180°.

Fig. 4. Different deviations for different ∝-particles.

Rutherford's atom model

On the basis of conclusions drawn from Rutherford's experiment, a new atom model was proposed. This atom model known as Rutherford's atom model has following characteristics : 

     (i) An atom consists of equal amounts of positive and negative charge so that atom as a whole is electrically neutral. 

    (ii) The whole of positive charge of the atom and practically whole of its mass is concentrated in a small region which forms the core of the atom called nucleus. 

   (iii) The negative charge, which is contained in the atom of electrons distributed all around the nucleus but separated from it. 

   (iv) In order to explain the stability of electron at a certain distance from the nucleus, it was proposed by 'Rutherford' that the electron resolve round the nucleus in circular orbits. The electrostatic force of attraction between the nucleus and the electron provides the centripetal force. 

    (v) The nuclear diameter is of the order of 10-14 metres. This can be shown as follows :

     Let an ∝-particles having velocity 'v' approaches a nucleus (head-on) having a charge '+Ze' upon it (Fig. 5). 

Fig. 5. An ∝-particle approaching the nucleus. 

     The velocity of ∝-particle decreases till it comes to rest at a distance 'ro' from the nucleus. It is then repelled back along the direction of approach 'ro' gives the radius of nucleus. 

         Initial K.E. of ∝-particle = 1/2 (mv²) 

         Initial P.E. of ∝-particle = 0

              [∝-particle is supposed to start from infinity]

        Final K.E. of ∝-particle = 0

        Final P.E. of ∝-particle = 1/4πε₀ × q₁q₂/r₀

     According to law of conservation of energy, 

          1/2 (mv²) = 1/4πε₀ × q₁q₂/r₀

     ∴                r₀ = 2/4πε₀ × q₁q₂/mv²

     For ∝-particle,  q₁ = 2e, q₂ = Ze

     ∴               r₀ = 2/4πε₀ × 2Ze²/mv²

                          = 4/4πε₀ × Ze²/mv²

                          = 4 × 9 × 109 Ze²/mv²

     In one of the experiments, ∝-particles of velocity 2 × 107 ms-1 were bombarded upon gold foil (Z = 79). 

     Here 'Z' = 79, e = 1.59 × 10-19 C 

         m = 4 × 1.67 × 10-27 kg, v = 2 × 107 ms-1

r0 = 4 × 9 × 109 × [79 × (1.59 × 10-19)²/4 × 1.67 × 10-27 × ( 2 × 107)²]

  

 = 2.69 × 10-14 m

     This gives the radius of nucleus. 

Failure of Rutherford's atom model

     (i) According to electromagnetic theory, a charged particle in accelerated motion must radiate energy in the form of electromagnetic radiation. As the electron revolves in a circular orbit, it is constantly subjected to centripetal acceleration v²/r. So it must radiate energy continuously. As a result of this, there should be a gradual decrease in the energy of electron. The electron should follow a spiral path and ultimately fall into the nucleus (Fig. 6). Thus, the whole atomic structure should collapse. This is contrary to the actual fact that atom is very stable. 

Fig. 6. Electron spiralling inwards. 

    (ii) According to Rutherford's model, electrons can revolve in any orbit. If so, it must emit continuous radiations of all frequencies. But atoms emits spectral lines of only definite frequencies. 

Key Words

1. ∝-particles. Helium nucleus

2. Atom model. An explanation to the structure of atom. 

3. Atom. Smallest particle of matter which can take part in a chemical reaction. 

4. Fluorescent screen. A screen coated with such a material which causes glow as charged particle strike against it.