PHYSICS-15-10- 11th (PQRS) SOLUTION

https://docs.google.com/document/d/1qFKa1T194bQy70WF3Bfv3crqw6oCQV2R/edit?usp=sharing&ouid=109474854956598892099&rtpof=true&sd=true a nd Rutherfo rd a tomic mode ls a nd their l imita tions; Na ture of electromagnetic radiation, photoelectric effect; spectrum of hydrogen atom, Bohr model of hydrogen atom - its postulates, derivation of the relations for energy of the electron and radii of the different orbits, limitations of Bohr's model; dual nature of matter, de-Broglie's relationship, Heisenberg uncertainty principle. Elementary ideas of quantum mechanics, quantum mechanical model of atom, its important features, Ψ and Ψ2, concept of atomic orbitals as one electron wave functions; Variation of Ψ and Ψ2 with r for 1s and 2s orbitals; various quantum numbers (principal, angular momentum and magnetic quantum numbers) and their significance; shapes of s, p and d - orbitals, electron spin and spin quantum number; rules for filling electrons in orbitals - aufbau principle, Pauli's exclusion principle and Hund's rule, electronic configuration of elements, extra stability of half-filled and completely filled orbitals. RUTHERFORD MODEL OF THE ATOM Rutherford model of the atom was given by projecting alpha particles which have very high energy. The results of Rutherford’s experiment were quite unexpected. Most of the α-particles passed through the gold foil undeflected. A small fraction of the alpha particles was deflected by small angles. A very few alpha particles bounced back The alpha-particle - scattering experiment proposed the nuclear model of atom, according to which An atom consists of nucleus at centre The positive charge is due to proton The total mass is centered at nucleus. Electron move around the nucleus in circular path Number of electrons in an atom is equal to the number of protons in it. ELECTROMAGNETIC RADIATIONS Electromagnetic radiations are the waves which are produced by acceleration of an electric charge and posses electric and magnetic fields perpendicular to each other and also perpendicular to path of propagation in space. It does not require a medium for propagation and has velocity equal to 3.0 × 108m sec-1. Electromagnetic waves are characterised by the following five characteristics : Wave length It is defined as the CHAPTER INCLUDES Rutherford Model of the atom Electromagnetic radiations Bohr's atomic model Hydrogen spectrum Dual nature of matter Heisenberg uncertainity principle Quantum numbers Electronic configuration of an atom Aufbau principle Hund's rule distance between two neighbouring crests or troughs. It is denoted by λ . Vibrating Source Energy Pauli's exc lusion principle It is generally measured in Angstrom unit (Å) or nanometers (n) 1Å = 10-10 m 1nm = 10-9m Frequency It is no. of waves which pass through a particular point in one second. Its units are cycle per second (cps), Hertz or s-1. Large units are K.Hz = 103 Hz. M Hz = 106 Hz. Velocity It is distance travelled by a wave in one second. Its units are ms-1 or cm s-1. Amplitude It is defined as the height of crest or depth of trough of a wave. The amplitude of a wave determine the intensity of radiation. It is expressed in the units of length. Wave Number It is defined as the no. of wavelengths per centimeter. Its units are cm-1 or m-1. v = 1 λ BOHR'S ATOMIC MODEL The model is based on the quantum theory of radiation and the classical law of physics. Postulate The path of electron is circular. The force of attraction between nucleus and electron is equal to centrifugal force of the moving electron. Electron can revolve only in those orbits whose angular momentum is an integral multiple of mvr = nh . (m = mass of electron, v = velocity of electron, r = radius of orbit) 2π Electron remains in stationary orbit where it does not lose energy. h 2π . i.e., Each stationary orbit is with definite amount of energy (E) and E1 < E2 < E3 Similarly (E2 – E1) > (E3 – E2) > (E4 – E3). The Energy of Electron Total energy (E) = K.E. + P.E. 2π2Z2me4 En = − n2h2 where , n = 1, 2, 3 ....... E = energy of electron in nth orbit Z = nuclear charge e = charge of electron m = mass of electron h = planck’s constant Z2 i.e., En = E1 × n2 for H-like atom i.e., E = − 21.79 × 10−19 Z2 n2 J/atom = − 13.6 Z 2 n2 eV per atom 313.6 × Z2 = n2 kcal/mol = − 1312 n2 kJ/mol Radii of Orbits n2h2 r = 4π2mkZe 2 By putting value of h, π, m, e and k n2 r = 0.529 × Å Z r = 0.529 × n2 × −10 Z r = 0.529 × 10−8 n2 cm Z For H-like atoms. Thus rn = r1 × n2 Velocity of Electron v = 2.188 × 108 × Z cm/sec n HYDROGEN SPECTRUM 0 –52.5 –82.0 –146 n = ∝ 5th shell (n = 5) 4th shell (n = 4) Paschen series 3rd shell (n = 3) –328 Balmer series 2nd shell (n = 2) Lyman series (ultraviolet) Lyman series Paschen series (infrared) Balmer series (visible) –1312 1st shell (n = 1) (b) When energy is supplied, the electron moves to higher energy shells depending on the amount of energy absorbed, when it comes back it emits the energy. For e.g. : If electron comes back from energy level having energy E2 to energy level having energy E1, then energy of emitted radiation is given as ΔE = E2 – E1 = hν. Thus, different spectral lines in the spectra of atoms corresponds to different transition of electrons from higher energy level to lower energy levels. Spectral Series n1 n2 Region (i) Lyman Series 1 2, 3, 4... UV (ii) Balmer Series 2 3, 4 .... Visible (iii) Paschen Series 3 4, 5, 6... IR (iv) Brackett Series 4 5, 6, 7 .... IR (v) Pfund Series 5 6, 7, .... IR (vi) Humphrey Series 6 7, 8 ..... Far IR Rydberg Equation The wavelength (λ), wave number ( ν ) for the electromagnetic radiation can be calculated by Rydberg equation. ν = 1 = R × Z2 ⎡ 1 − 1 ⎤  λ ⎢n2 n2 ⎥ ⎣⎢ 1 2 ⎥⎦ Z = Atomic number R = 109743 cm–1 - Rydberg constant n2 = higher orbit n1 = lower orbit Total number of spectral lines (i) n(n − 1) 2 → when electron jumps from nth level to ground level. (ii) (n2 − n1)(n2 − n1 + 1) 2 → when electron returns from n2 to n1. DUAL NATURE OF MATTER : de Broglie Equation Louis de Broglie proposed that the material particles are also associated with wave nature, just as radiations. The wavelength of the wave associated with a particle mass 'm' moving with velocity 'v' as . Derivation : The energy of photon (E) of frequency 'ν' is given by E = hν …(i) According to Einstein's equation, E = mc2 …(ii) From equation (i) and (ii), mc2 = hν 2 c ⎛Θ ν = c ⎞ ∴ mc ∴ λ = = h ⋅ ⎜ ⎟ ⎝ ⎠ h mc Thus by replacing c, the velocity of the photon by v, the velocity of particle, . Number of revolution per sec by an electron in a shell may be given as = de Broglie’s equation & K.E. Let K.E. of the particle of mass ‘m’ is E E = 1 mv 2 2 2Em = m2v 2 Velocity = 2πr v 2πr = mv = P λ = h = P Number of waves in an orbital = 2πr = 2πr × mv mvr × 2π or λ h h nh mvr = angular momentum = 2π (according to Bohr Model) Thus = nh × 2π = n 2π h ∴ Number of waves in an orbital = n. HEISENBERG UNCERTAINTY PRINCIPLE For small particles such as electrons, it is impossible to determine simultaneously its position and velocity at a given instant with absolute certainty. The principle states that, "it is impossible to measure simultaneously both the position and velocity or momentum of a microscopic particle with absolute accuracy or certainty. Mathematically, . where Δx = uncertainty in position Δv = uncertainty in velocity h = planck's constant m = mass of the particle Significance of uncertainty principle is that it rules out existence of definite paths or trajectories. The effect of Heisenberg uncertainty principle is significant only for motion of microscopic objects and is negligible for that of macroscopic objects. QUANTUM NUMBERS The set of four integers required to define the state of electron in an atom are called quantum numbers. The set of quantum numbers are Principle quantum number (n) Azimuthal quantum number (l) Magnetic quantum number (m) Spin quantum number (s) Principle quantum number, (n), relates to the amplitude (i.e. size) of an electron wave and also the total energy of the electron. It has integral values of 1, 2, 3, 4 ... etc., also denoted as K, L, M, N etc. Azimuthal quantum number, (l), tells us about the subenergy shell of electron. For each main energy shell there can be ‘n’ number of subenergy shells. These subenergy shells are designated by different values of l. For each value of n, l can have values from 0, 1, 2, 3 n – 1. Magnetic quantum number, (m), explains the behaviour of an electron in the external magnetic field or in other words it tells us about orbitals of the electrons. The values of m gives the number of orbitals associated with a particular sub shell in shell. For each value of l, m can have values from –l to +l including zero. e.g., when l = 1, m = -1, 0, +1; l = 2, m = -2, -1, 0, +1, +2 Spin quantum number, (s), gives an idea about the electron spinning on its axis. Each spinning electron can have two values of + 1/2 (clockwise spin) and – 1/2 (anticlockwise spin). ELECTRONIC CONFIGURATION OF AN ATOM The filling of electron in an atom to make electronic configuration is governed by the following three rules Aufbau principle Hund’s rule Pauli’s Exclusion principle Aufbau principle According to this rule electrons are added progressively to the various orbitals in the order of increasing energy starting with the orbit of lowest energy. The compilation of increasing energy in orbit has been done on the basis of two simple guidelines Atomic orbitals fill in the order of increasing (n + l) values. Higher the (n + l) value for an orbital higher will be the energy. If two orbitals have the same value for (n + l), the one which has lower value of n will be of lower energy and would be filled first. Hund’s rule of maximum multiplicity This rule determines the arrangement of electron in the orbitals of same energy e.g. within px, py and pz or amongst dxy, dyz, dxz, dx2-y2 and dz2. It states that pairing of electron in any orbital p,d or f shall take place only when all the orbitals of that sub shell contain one electron each. Pauli’s Exclusion Principle According to this principle “no two electrons of an atom can have the same set of all the four quantum numbers”. In other words An orbital cannot have more than two electrons If an orbital has two electrons, they must have opposite spin (paired electrons). ❑ ❑ ❑