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NEET-2022 Ultimate Crash Course PHYSICS Atoms

POINTS TO REMEMBER 1. The deviation (or scattering) of an − particle depends o its impact parameter. The smaller the impact parameter, the closer the − particle passes to the centre of the atom and larger is the deviation. 2. For each value of impact parameter (b), there is a distance of closest approach at which the − particle is nearest to the nucleus. The minimum distance occurs for a head-on collision when b = 0. 3. When Bohr's theory was proposed, only the Balmer and Paschen series for hydrogen were known. The results of Bohr's theory predicted the existence of more series for hydrogen whose wavelengths could be calculated discovered. calculated even before they were discovered 4. Bohr's efforts to explain the hydrogen spectrum led him to assume that angular momentum is quantised just as Planck's efforts to explain black body radiation led him to assume that energy is quantised. 5. The Bohr's theory of the atom was important historically in providing some ideas about atomic structure. But it has been replaced by a description using quantum mechanics, i.e., Bohr model has been replaced by quantum mechanical model. According to this model, the electrons in an atom do not move around the nucleus in definite orbits, but are free to move anywhere in the orbit. In quantum mechanics, we talk in terms of probabilities of things to happen or probabilities of finding them at certain locations at given times. One essential difference between quantum-mechanical model and the Bohr model is clear. Even though the probability of: finding the electron is high near the Bohr orbit radius in the quantum mechanical model, the probability is not zero between these orbits. As a matter of fact, we can say that Bohr model is a special case of quantum-mechanical model.' 6. We see the signature of an atom in the spectral lines it produces. Analysis of this signature was the key to the understanding of atomic structure for physicists at the turn of the nineteenth century. (Signature : means a distinctive quality by which something can be recognised) 7. Bohr radius (ao = 5.292 x 10-11 m) corresponds to the radius of the hydrogen atom in its ground state (n=1). Radius (r) of an allowed orbit for any other atom is given by ( ) 2 0 r a / Z n = . But it must be emphasized that this value of r must not be taken too literally. According to quantum mechanics, it should be considered only an indication of the order of magnitude of the region in which the electron is most likely to be found. 8. To account for the fine and hyperfine structures of observed spectral lines and the effects of electric and magnetic fields, it is necessary to look for the existence of more energy levels in these situations. Since energy levels arise as a consequence of quantum conditions imposed on electrons, it was found that in all four quantum numbers are required; three more in addition to the principal quantum number (n) postulated by Bohr : n defines the mean separation of the electron from the nucleus. The second quantum number, called the orbital quantum number (l) characterises the orbital angular momentum of the electron. The third quantum number, called magnetic quantum number (m), accounts for the behaviour of spectral lines in externally applied magnetics The fourth quantum number, called spin quantum number (s), specifies one of the two possible 'spin states of the electron. Spin is simply an intrinsic property of an electron. Pauli exclusion principle which states that no two electrons can have the same set of quantum numbers governs how electrons occupy various quantum states. This simple principle was given by Austrian physicist, Wolfgang Pauli in 1925. 9. The predictions of quantum theory must correspond to the predictions of classical physics in the region of sizes where classical theory is known to be valid. This is called Bohr's correspondence principle. When the size becomes large, quantum number (n) becomes large and we may state this principle as n lim → [quantum physics] = [classical physics] According to this principle, the quantum condition for emission (E E hv i f − = ) and the Maxwell's classical radiation theory (i.e., electronic charges with orbital frequency f radiate light waves of frequency f) must simultaneously hold for the case of extremely large electronic orbits. Calculations show that for n > 10,000,f is different from v by less than 0. 015V 10. Rydberg constant, R is equal to me' / 8 €2° ch3 under the assumption that the 'nucleus is at rest, being infinitely heavier than the electron. Actually, this assumption is incorrect since the nucleus has a slight motion and its mass is a finite multiple of the mass of an electron. 11. The value of Rydberg constant varies slightly from one atom to another. The error is removed by replacing the mass m of the electron by its reduced mass, mM/ (m+M)=m/ (1+ m/M), where M is the mass of the nucleus. Thus,
R H (Rydberg constant for hydrogen) = + R / 1 m / M ( H ) He R + (Rydberg constant for singly ionised helium, He+ )= R/(1+ m/ 4MH ) (as M 4M He H = ) It can easily be seen that H H He He H M R R / 4 m R R + + − = − The most accurate spectroscopic measurements of RH and RH, were made by W.V. Houston in 1927 RH= 1.09678 x 107 1 m − and He R + = 1.09722 x 7 1 10 m− which give MH/ m =1869. This result is close to the value 1836 determined earlier and represents another triumph of Bohr's theory. 12. A positronium "atom" is a system that consists of a positron and an electron that orbit each other. The reduced mass of the system is m' = mM/ (m+M) = m2 / 2m = m/ 2. This implies that the Rydberg constant for such an atom is R/2. As such the wavelengths of positronium spectral lines are all twice those of the corresponding lines in hydrogen spectrum 13. We can obtain Lyman series in both emission as well as absorption spectrum. But other spectra are obtained in the emission spectrum. 14. At minimum potential energy, an atom achieves most stable state. 15. Spectrum of hydrogen has fine structure. For example, each spectral line of hydrogen atom consists of a large number of fine lines. 16. For hydrogen atom, binding energy is the sum of kinetic energy and potential energy of the orbital electron. 17. In hydrogen atom, when an electron Jumps from the excited state to the ground state, its kinetic energy increases but potential and total energy decreases. 18. On increasing the principal quantum number, the energy difference between the two successive energy level decreases, but wavelength of spectral line increases. 19. Bohr's theory is applicable for hydrogen and hydrogen like atoms or ions. For example the atoms or ions having number of electron = 1 but atomic number z may be different. SOME IMPORTANT FORMULAE 1. ( ) e 0 Ze 2e K k r  = or 2 0 e 2Ze r k K = where 0 r is the distance of closest approach of an a-particle of kinetic energy %moving towards a nucleus of charge Ze 9 2 2 e 0 1 k 9 10 Nm / C 4     = =     2. 0 2b cot 2 r      =   where  is the angle of scattering and b is the impact parameter. 3. (a) 2 2 0 n 2 n h r me  =  (n 1, 2,.... = ) where n r represents the radii of permitted orbits of the electron in case of H-atom When n = 1, 2 0 0 1 2 h r 0.53A me  = =  (Bohr's radius) (b) 2 n 1 r r n = 4. 2 n 0 e v 2 nh =  where n v , is the velocity of electron in H-atom in its nth orbit. Also, 1 n v v n = where 1 v , is the velocity of the electron in its first orbit. 5. n c v n =  where 2 0 e 2 ch  =  is fine structure constant