Title MODERN PHYSICS III.pdf ✅

Modern Physics  Digital www.allendigital.in [ 245 ] Various Models for Structure of Atom Dalton's Atomic Theory Every material is composed of minute particles known as atom. Atom is indivisible i.e. it cannot be subdivided. It can neither be created nor be destroyed. All atoms of same element are identical physically as well as chemically, whereas atoms of different elements are different in properties. The atoms of different elements are made up of hydrogen atoms. (The radius of the heaviest atom is about 10 times that of hydrogen atom and its mass is about 250 times that of hydrogen). The atom is stable and electrically neutral. Thomson's Atom Model The atom as a whole is electrically neutral because the positive charge present on the atom (sphere) is equal to the negative charge of electrons present in the sphere. Atom is a positively charged sphere of radius 10–10 m in which electrons are embedded in between. The positive charge and the whole mass of the atom is uniformly distributed throughout the sphere. Shortcomings of Thomson's model: (i) The spectrum of atoms cannot be explained with the help of this model. (ii) Scattering of –particles cannot be explained with the help of this model. Rutherford atomic model Rutherford experiments on scattering of –particles by thin gold foil: Ernst Rutherford, H. Geiger and E. Marsden Performed this experiment. In this experiment a beam of -Particles (Helium nucleus) of energy 5.5 MeV, emitted by a 214 83 Bi radioactive source was directed at a thin metal foil made of gold. They studied the -particles scattered at various angles. The experimental arrangement is shown in figure. –particles are emitted by some radioactive material (polonium), kept inside a thick lead box. A very fine beam of –particles passes through a small hole in the lead screen. This well collimated beam is then allowed to fall on a thin gold foil. While passing through the gold foil, –particles are scattered through different angles. A zinc sulphide screen was placed out the other side of the gold foil. This screen was movable, so as to receive the –particles, scattered from the gold foil at angles varying from 0° to 180°. When an –particle strikes the screen, it produces a flash of light and it is observed by the microscope. It was found that : Modern Physics-3 [Atomic Structure and X-Ray] lead box source of -particle lead screen vacuum ZnS screen most -pass through  some are deviated through large angle  about 1 in 8000 is repelled back beam of -particle gold foil 10–7m N() cosec4 ቀ θ 2 ቁ 90° 180°  N() Electron Positively charged matter
NEET : Physics [ 246 ] www.allendigital.in  Digital • Most of the –particles went straight through the gold foil and produced flashes on the screen as if there were nothing inside gold foil. Thus the atom is hollow. • Few particles collided with the atoms of the foil which have scattered or deflected through considerable large angles. Few particles even turned back towards source itself. • The entire positive charge and almost whole mass of the atom is concentrated in small centre called a nucleus. • The electrons could not deflected the path of a –particles i.e. electrons are very light. • Electrons revolve round the nucleus in circular orbits. So, Rutherford 1911, proposed a new type of model of the atom. According to this model, the positive charge of the atom, instead of being uniformly distributed throughout a sphere of atomic dimension is concentrated in a very small volume (Less than 10–13cm is diameter) at it centre. This central core, now called nucleus, is surrounded by clouds of electron makes the entire atom electrically neutral. According to Rutherford scattering formula, the number of –particle scattered at an angle  by a target are given by, ( ) 2 2 0 2 2 2 2 4 0 0 2 N nt(2Ze ) 1 N 4(4 ) r (mv ) sin   =   Where, N0 = number of –particles that strike the unit area of the scatter n = Number of target atom per m3 t = Thickness of target Ze = Charge on the target nucleus 2e = Charge on –particle r = Distance of the screen from target v0 = Velocity of –particles At nearer distance of approach the size of a nucleus or the distance of nearer approach is given by, 2 2 0 0 0 K 2 0 1 ) 1 (2Ze) (2Ze r 4 4 E 1 mv 2 =  =         (Where, EK = K.E. of –Particle) Failure of Rutherford's Atomic model (i) It couldn't explain the stability of atom. (ii) It couldn't explain discrete nature of hydrogen spectra. target nucleus  b area = b2 -particle r0 Ze nucleus
Modern Physics  Digital www.allendigital.in [ 247 ] Bohr Model of Hydrogen Atom Bohr's theory of hydrogen atom is based on the following assumption: An electron in an atom moves in a circular orbit about the nucleus under the influence of coulomb's force of attraction between the electron and nucleus. As the atom as a whole is stable the coulombian force of attraction provides necessary centripetal force. 2 2 2 0 e mv 4 r r =  ...(i) Only those orbits are possible for which the angular momentum of the electron is equal to an integral multiple of h 2 i.e. h mvr n 2 =  ...(ii) Where, h is Planck's constant. The electron moving in such allowed orbits does not radiate electromagnetic radiations. Thus, the total energy of the electron revolving in any of the stationary orbits remains constant. Electromagnetic radiations are emitted if an electron jumps from stationary orbit of higher energy E2 to another stationary orbit of lower energy E1. The frequency n of the emitted radiation is related by the equation. E2 – E1 = h ...(iii) Drawbacks of Bohr Model ➢ This model could not explain the fine structure of spectral lines, Zeeman effect and Stark effect. ➢ This model is valid only for single electron systems. (can not explain electron-electron interaction) ➢ This model is based on circular orbits of electrons whereas in reality there is no orbit. ➢ Electron is presumed to revolve round the nucleus only whereas in reality motion of electron can not be described. ➢ This model could not explain the intensity of spectral lines. ➢ It could not explain the doublets obtained in the spectra of some of the atoms. ➢ Bohr's model is semi quantum model, it means, it includes two quantum numbers (E and L) but unfortunately it consider circular motion of electron. Radii of Orbit and Velocity of Revolving Electron Radii of orbits From equation, nh v 2 mr =  , Here n is number of orbit. Substituting value of v in equation 2 2 2 2 2 2 2 0 0 2 2 2 2 2 2 0 e m nh mn h .4 r n h r 4 r 4 m r e me r 2 mr     =  = =         In general, 2 2 0 n 2 n h r me  =  ...(iv) Equation (iv) shows that the radii of the permitted orbits vary as the square of n. For the smallest orbit n = 1 substituting the values of h, 0, m and e we have radius of first orbit r1 = 0.529 × 10–10 m = 0.529 Å This calculations shows that the atom is about 10–10 meter in diameter. r Ze nucleus v e –
NEET : Physics [ 248 ] www.allendigital.in  Digital Velocity of Revolving Electron To obtain the velocity of the revolving electron, we substitute the value of r from eq. (iv) in eq. (ii), we have 2 2 2 2 0 2 2 2 0 0 n h h nh me 1 e mv n v . . me n h 2 2 m 2nh       =  = =        ...(v) This expression shows that the velocity of the electron is inversely proportional to n i.e. the electron in the inner most orbit has the highest velocity. Frequency of Electron in an orbit: Frequency of electron is given by, 2 2 4 2 2 2 3 3 0 0 0 1 v e 1 me me T 2 r 2nh 2 n h 4 h n   = =   =   =      ...(vi) This expression shows that the frequency of an electron is inversely proportional to the cube of n. Electron Energy The electron energy consist of two types: (i) Kinetic energy and (ii) Potential energy (i) Kinetic energy is due to the motion of electron and its value is 1 2 mv2 where v is the velocity of the electron,  K.E = 1 2 mv2 = 2 2 0 1 e m 2 2nh        from equation K.E. = 4 2 2 2 0 me 8n h  (ii) Potential energy is due to the fact that electron lies in the electric field of positive nucleus. We know that potential at a distance r from the nucleus is :- 0 e V 4 r =  The potential energy of electron of charge e is. P.E. = V × (–e) = 2 2 2 4 2 2 2 2 2 0 0 0 0 e e me me 4 r 4 n h 4n h − −  − = =     So, total energy in nth orbit, En = K.E. + P.E.  En = 4 4 2 2 2 2 2 2 0 0 me me 8n h 4n h −    En = 4 2 2 2 0 me 8n h −  Frequency of emitted radiation The frequency of emitted radiations can be found from the following relation when electron jumps from higher orbit n2 to lower orbit n1. n n 2 1 h E E  = −  4 2 3 2 2 0 1 2 me 1 1 8 h n n    = −      ...(vii) 4 2 3 2 2 0 1 2 1 me 1 1 8 h c n n   = −       where R = 4 2 3 0 me 8 h c  ; R = Rydberg's constant = 10.97 × 106 m–1  1.1 × 107 m–1 2 2 1 2 1 1 1 R n n    = = −     