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(ii), , (ii), , ral, 9, , Source of a, particles, , , , 15. Structure of Atoms and Nuclei, , , , DUCTION, , According to Greek philosophers Leucippus and, Democritus matter is made of indivisible parts called, , atoms. Dalton gave his atomic theory in nineteenth, century, , {i), , , , . According to this theory,, Matter is made up of indestructible particles,, Atoms of a given elements are identical., , Atoms can combine with other atoms to form, new substances., , That atoms were indestructible was shown to be, wrong by J.J. Thomsons experiment who discovered, electron in 1887. Thomson’s model had some, deficiencies and was later improved by Ernest, Rutherford and Niels Bohr., , THOMSON’S ATOMIC MODEL:, (i) According to this model an atoms is a sphere, having a uniform postive charge in which, electrons are embedded., , This model is referred to as Plum-pudding, model. The total positive charge is equal to the, total negative charge of electrons in the atom,, rendering it electrically neutral., , (iii) As the whole solid sphere is positively charged,, ~ the positive charge cannot come out and only, the negatively charged electrons can be emitted., The model also explained the formation of ions, and ionic compounds., , However further experiments on structure of, atoms which are described below, showed the, distribution of charge to be very different than, what was proposed in Thomson’s model, , , , (ii),, ~S, , (iv, , (v), , GEIGER - MARSDEN EXPERIMENT:, The experimental arrangement is as shown in fig., 15.1. Alpha particles from a source were collimated,, i.e, focused into a narrow beam, and were made to, , fall on a gold foil., , , , ooo, ooo, , Collimator, , , , , Detector, 5.1 Geiger - Marsden Experiment, , duced scintillations on, e scintillations which, ‘ed through a microscope which, , Fig. 1, , particles pro, , ttered, Oe screen. Thi, , the surrounding, , could be observ, , could be ee, inci eam., , to the incident es passed straight, , erved that alpha particl ‘, ae foil while a few were deflected through, , cover different angles with respet, , various scattering angles., , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (iv) A typical scattering angle is shown by ® in the, fig. 15.1. Only about 0.14% of the incident alpha, particles were scattered through angles larger than, 0.1° and most were deflected through very smal], angles. :, About one alpha particle in 8000 was deflected |, through angle larger than 90°and few were deflected, through angles as large as 180°, , 4. RUTHERFORD’S ATOMIC MODEL:, yeeros states that the entire positive charge, , and most of the mass of an actom is concentrated, in the central nucleus and the electrons revolve in, circular orbit around the nucleus. This is similar to, the revolution of the planets around the sun in the, solar system., , oe The revolution of the electrons was necessary as, without it, electrons would fall into the nucleus and, , the atom would collapse. The space between orbit, , and nucleus is mostly empty., , Thus, most alpha particles pass through this empty —, , space undeflected and few are which in direct line, , with nucleus or close to it get repelled and get, , deflected through large angles., , fy This model also explains, why no positively charged, particles are emitted by atoms while negatively, charged electrons are. This is because of the large, mass of the nucleus which does not get affected, when force is applied on the atom., , * Difficulties with Rutherford's model:, es According to Rutherford theory electron revolves in, , : a circular orbit around the nucleus, with constant, velocity. Its direction changes continuously and, so the motion is an accelerated motion, Thus the, electron should emit electromagnetic radiation —, continuously, its energy would decreases and, consequently, the radius of its orbit decreases, continuously. Hence the electron would move along, a spiral path and ultimately fall into the nucleus., Thus atom would not be stable while in fact the, atom has most stable structure., , (8) the revolving electrons emit the radiation, continuously with varying frequencies. Hence the, spectrum is continuous spectrum, but they do, not constantly emit electromagnetic radiation and, definitely not varying frequency. The atomic spectra, are found to be a line spectra., , Rutherford’s model failed on all these counts., , 5. ATOMIC SPECTRA, When a metallic object is heated, it emits radiation —, of different wavelengths. When this radiation is, passed through a prism, we get a continuous, spectrum. However when hydrogen gas is heated, inside a glass tube to high temperature the case is _, , (vy), , (iii), , , , different. The emitted radiation has only few, , , , , , wavelengths and when passed through prism we g' at, , , , Scanned with CamScanner

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y, , (2) Balmer series:-n = 2 and m= %, , (4) Brackett serie, , (5) Pfund series:- n = 5 and, , structure of Atoms and Nucte}, geureofAtomsand Nuctei, , a line spectrum as shown ;, , shows that hydrogen coe paanible, 410, 434, 486 and 656nm 3, radiation with wavelengths, lengths. The lines seen in tl, emission ‘lines, Hydrogen, wave- lengths in the UV, I, wave- lengths., , range. It, ation of wavelengths, and does not emit any, in between these wavehe spectrum are called, also emits radiation at, R radiation and at longer, , Eee EE € Visible gir, fe ¢ gir, ot ©, = 3 2, Sf 2, , S, Fig. 15.2 Hydrogen spectrum, , The spectral lines can be classified into several, , series, starting with shorter wavelength as follows:, , (i) Lyman series (ii) Balmer series, , (ii) Paschen series (iv) Brackett series, , (v) Pfund series etc, , In each series, the separation between successive, , lines decreases as we go towards shorter wave, , length and they reach a limiting value, , Schematic diagram for first three series are shown, , in fig given below. Limiting value of the wavelength, , is shown by dotted line., , , , , , , , , , , , , , , , , , , , , , , , , , , , 90 nm 100 nm 110nm 120nm ),, ae : Lyman, limit — >} series, , 400 nm 500 nm 600 nm X, i Balmer, , 0.5 um 1,0 um 1,5 um 2.0um, Pas i Paschen, aaa ee, , , , , , , , , , , , , , , , , , , , , , , , , , Fig. 15.3 Lyman, Balmer and Paschen series, , The observed wavelengths of the emission lines are, , found to obey the relation., 1 1, , nay me, , , , eek, , , , i line., where A is the wavelength ofa line, f, n and m be the lower and higher orbit respectively, , R= constant, Conditions:, , (l) Lyman series :- n= 1 andm DiS ear, This series lies in UV region of oS las, , i ctrum This series lies in visible region of the spe, , , , '3) Paschen series:- n =3 and m Ss, . aes ion of the spe 2, This series lies in IR reg 56,7), , , , n=4and m=, , y ectrum., This series lies in near IR region of oe, Siecle ame, , , , 155, , This series lies in far IR region of the spectrum., Rutherford model could not explain the atomic, spectra., , BOHR’S ATOMIC MODEL:Niels Bohr modified Rutherford's model by applying, idea of quantum physics which were being, developed at that time. He made three postulates, which defined his atomic model., , —Pestulate-1:, The electrons revolve around the nucleus in circular, orbit., This is the same assumption as in Rutherford’s, model and the centripetal force necessary for the, circular motion is provided by the electrostatic force, of attraction between the electron and the nucleus, , Centripetal force = Electrostatic force, , mv? i @, , (0), , , , +Pestulate-2:, The radius of the orbit of an electron can only take, certain fixed values such that the angular momentum, of the electron in these orbit is an integral multiple of, , 4 h being the Planck's constant., , Such orbits are called stable orbit or state of the, electrons and electrons in these orbits do not emit, radiation as is demanded by classical physics. Thus, different orbits have different and definite values of, angular momentum and therefore different values, of energies., , Angular momentum = =, Benn, lo = On, = nh, or mvr = 57 ol}, , ~ Pestulate-3:, , An electron can make a transition from one ofits orbit, , to another orbit having lower energy. In doing so, it, , emits a photon of energy equal to the difference in its, , energies in the two orbits., E,,-E, = hv, , Radii of the Orbits:, ...(3), , Consider an electron revolving around the nucleus, in a circular orbit of radius r,,, , Let m, -be the mass of electron., , v, - be the velocity of electron in n® orbit., , By 2" postulate,, nh, , , , MUI, = Om err (h), , where, nis principal quantum number., nh, , , , v,, Pier nret a, 2, , mde «1.(2), , Scanned with CamScanner

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By 1" postulate, for Z number of atoms., , , , , , , , mu; _1 Ze, % ane Te, Ze, ie. v ai (3), , , , Comparing eq" (2) and (3), Ze, , 4ne,7,m,, Ze, , Tee, , , , , , n ho, , Th emZe, , , , (4), , = h'to, ror where, ss = constant, , i.e. Radius of the orbit is directly proportional to the, square of principal quantum number., * Energy of the Electrons:, Total energy of an orbiting electron is the sum of its, K.E and its P.E., , -. Total energy (E,) = K.E + P.E., , where E, - is the total energy of an electron in n*, orbit, K.E of an orbiting electron is, , 1, K.E=3 mv,, By 1st postulate,, mv? 1 Ze, , , , , , , , 7, 48) Ty, , 1 Zé, aie ess, ory Ane. a7, 1 me eee, Yn ~ Diane, 7, |, 1 Zé, , HOS anekart, , Similarly P.E = Potential x charge of eletron, , , , The negative value of the energy indicates that the, electron is bound inside the atom., , Substituting the values of the constants acta, , , , UTTAM’s Physics Papers Solution ~ Xi, , and ¢, in eq” (1) we get, 2, EB, =-13.6 ar eV n/t), , where n = 1, 2, 3, ..., , For hydrogen atom Z = 1, , 3.6, , Sara eV +++.(3), , , , For first orbit n = 1, E, =-13.6 eV, , For 2" orbit n = 2, E, = -3.4 eV, , For 3” orbit n = 3, B,=-1.51 eV, , For 4 orbit n = 4, E, =—0.85eV and so on, , The first energy orbit which has minimum energy, is called the ground state of the atom. The higher, energy orbits i.e. E,, E,, E, ... are called the excited, states of the atom. An atom is most stable in ground, state, The energy levels come closer and closer as n, increases and their energy reaches a limiting value, of zero as n goes to infinity., , The energy level diagram of hydrogen atom is as, given below., , Ionized atom, , E,=0, , , , n=2 %, n=5 E, =-0.54 eV, n=4 B, =-0.85 eV, E,, n=3 =-151ev |F, e Taser, (infrared), n=2 B,=-3.4ev, Balmer series, (visible region), ak See BE, =-13.6 eV, (ultraviolet), , Fig. 15.4 Energy level diagram, , The energy required to take an electron from, the ground state to an excited state is. called the, excitation energy of the electron., , For hydrogen atom minimum excitation energy is, (-3.4) - (-13.6) = 10.2 ev., , In order to remove the electron in the ground state, from a hydrogen atom, we have to supply 13.6 eV, energy to it. This energy is called the ionization, energy of an hydrogen atom. It is the binding energy:, The binding energy of an atom is the minimum, amount of energy required to be given to an electron, in the ground state of that atom to set the electro”, free., By 3 postulate of Bohr model, , ny ese eb, -mZe') |-mZe', hy = Ze = —_, , mee (1 1, VE Fee he eae a, , , , ipa oe Sea, , Scanned with CamScanner

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“a, , structure of Atoms and Nuclei, a, , (), , (i), , (iii), , ...(4), , , , Micra Rum, where R,, = 8e,ch® ~ Constant called the Rydberg, , constant, Its value is 1.097 x 107m, , Equation (4) is called the Rydberg’s formula,, , For hydrogen Z=1 . Eqn (4) becomes, Hi, me], , , , a(5), , Limitations of Bohr’s model:, It could not explain the line spectra of atoms other, than hydrogen. Even for hydrogen, more accurate, study of the observed spectra showed multiple, components in some lines which could not be, explained on the basis of this model., , The intensities of the emission lines seemed to, differ from line to line and Bohr’s model had no, explanation for that., , On theoretical side also the model was not entirely, satisfactory as it arbitrarily assumed orbits following, a particular condition to be stable. There was no, theoretical basis for that assumption, , De Broglies Explanation:, Debroglie suggested that instead of considering the, orbiting electrons inside atom as particles we should, view them as standing waves similar to the case of., standing waves on string or in pipes, the length of, the orbit of an electron has to be an integral multiple, of its wavelength. Thus the length of the first ore, will be equal to one debroglie wavelength, the secon, orbit will be twice the debroglie wavelength of the, electron in that orbit and so on. This is shown for, , the 4" orbit in figure., , (/, , Fig. 15.5 Standing electron, , i tron, wave for 4" orbit of an ele, , Ih 8eneral, we can write,, , , , 2nr,= nd, where n= 1,2,3, --, 6, =n md), ® n, , , , (a), , 157, , The de Broglie, wave length is related to the linear, momentum p,, of the particle by, , , , Substituting this in eq" (1), , lip enin, , P,, a, mente, , Bee Qrr,, , Thus, the angular momentum of the electron in n™, orbit is, nh, Lie inmaoe, which is the 2"! postulate of Bohr’s model. Therefore, considering electron as wave gives some theoretical, basis for the 2" postulate made by Bohr., , ATOMIC NUCLEUS:, , Constituents of a nucleus:, The atomic nucleus is made up of subatomic, particles called protons and neutrons. The proton, has positive charge and neutron is neutral., Together, protons and neutrons are referred as, nucleons. Mass of a proton is about 1836 times that, of an electron. Mass of a neutron is nearly same as, that of proton but is slightly higher. The magnitude, of its charge is equal to the magnitude of charge of, an electron, The number of protons in an atom is, called its atomic number and it is denoted by Z., The number of electrons in an atom is also equal, to Z. Thus, the total positive and negative charges, in an atom are equal in magnitude and the atom, as a whole is electrically neutral. The number of, neutrons in a nucleus is written as N. The total, number of nucleons in nuelens is called the mass, number. It is denoted by A., , » A=Z4N, , Thus a nucleus of an element X is symbolically, expressed as, “x where, A = mass number, , Z = atomic number, , For e.g. symbol for Gold, Uranium and carbon are, , written as, 197, 238. 12, zo, 4.U and ,C respectively, , Isotopes:- The atoms having the same number of, protons but different number of neurons are called, isotopes. = Z 4, e.g. The deuterium ae tritium “H are the isotopes, of hydrogen. Similarly jHe and iHe are the istopes, of helium. Z, , Isobars:- The atom having the same mass number A,, are called isobars ., , For e.g., _H and jHe are isobars., , , , Scanned with CamScanner

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160, , fe), , In this type mass number is unchanged and atomic, number increases by one and N decreases by one., , A A, 2X > ,,Y +e +antineutrino — ...(4), , For example, , 60 COR ae P re, ace > ae + e+ antineutrino, , There is another type of beta decay called as beta, plus decay in which proton gets converted to a, neutron by emitting a positron and neutrino., A positron is a particle with the same properties as, an electron except that its charge is positive. It is, called as the antiparticle of electron., , pt+n+e’+ neutrino st 65}), The mass number remains unchanged during the, decay but Z decreases by one and N increases by, one., , AX a a Y + e* +neutrino, , Z Za le), , ©.g.5- vNa > TONE + e* + neutrino, , In beta decay, the total mass of the product of the, decay is less than mass of the parent atom. The, excess mass is converted into kinetic energy of the, product. The Q- value is, Q= [m, — m, —m_]c* ef), Here, mass of the neutrino is negligible compared to, the masses of the nuclei, Gamma(y) Decay:It is a type of decay in which high energy photons, are emitted. When 7 particles are emitted then mass, number and atomic number remains unchanged., Ax => ay, Z Zz, A nucleon can make a transition from a higher, , energy level to a lower energy level, emitting photon, , in the process. The difference between atomic and, , nuclear energy levels is in their energies and energy, , , , , , , , , , , , separation., ‘Usually, the nucleons in a nucleus are in the, i ergy state. They can not easily, amount of energy is required for, nucleon however may end up in, ; a result of the parent nucleus, alpha or beta decay. Thus gamma, a cur after one of these decays., , lergoes beta plus decay to form, is “Fe which is in an excited, , , , , , , , , UTTAM’s Physics Papers Solution — XII, , 10. LAW OF RADIOACTIVE DECAY:, * Radioactivity:, , _Phe substance which emit spontaneous emission, of radiation from their nuclet is called radioactive, substance., , e.g. Uranium, Radium, Thorium., , he phenomenon of spontaneous emission of, radiation from radioactive substance is called as, radioactivity., , Henry Becquerel discovered that heavy element, having A > 82 are unstable and emit highly, penetrating radiation., , 7 Statement of the law:, , He number of nuclei undergoing the decay per unit, time is proportional to the number of unchanged, nuclei present at that instant., , Let N, - be the number of nuclei present at any, instant t., dN - be the number of nuclei that disintegrated, in short interval of time dt, According to law of radioactive decay, , aN, Gen,, dN, = ON, oxx( Ch), , where 2. - is a constant called as decay constant., The negative sign indicates that N, is decreasing, , with time., Rearranging the eq? (I), dN, ye > hat 2), , ‘, Integrating equation (2), , Ne, Gets, , Here, N, is the number of parent atoms at time, , t=0, [log NJN = -At, N,, feelers, log N, At, N, —= et, N,, N, = Nie* -(3), , ~/This is dpe : decay law of radioactivity. The rate of, decay TaEK , is also called as the activity (A) can be, written using eq" (1) and (3),, , A= =. AN,, A, = AN, e* --(A), At t = 0 eq* (4) becomes, Ay = AN,, -. Eqn(4) implies, A, = Aye -..(5), , Scanned with CamScanner