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Atomic Structure And Quantum Numbers

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Published in: Chemistry | IIT JEE Mains | Physics
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Atomic Structure Quantum Numbers

Akhilesh K / Lucknow

4 years of teaching experience

Qualification: M.Sc (NIT Rourkela - 2019)

Teaches: All Subjects, English, Mathematics, Science, Chemistry, Physics, Algebra, IIT JEE Mains, AIPMT, NEET

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  1. Atomic Structure Atomic structure: Overview of Bohr's atomic model, Schrodinger wave equation, Interpretation of wave function (radial and angular), Hydrogen like atom, Concept of atomic orbitals. Books: 1. Atkins ' Physical Chemistry by Peter Atkins and Julio de Paula 2. Inorganic Chemistry by Gary L. Miessler and Donald A. Tarr
  2. Atoms ' Two regions. ' Nucleus- protons and neutrons. ( Occupies very small space ' Electron cloud- region where you might find an electron. [Occupies large (most of the) 1 fm 1 100,000 fm space] Name Electron Nucleus Proton Symbol Subatomic particles Discovered by, year J. J. Thomson, 1897 E. Rutherford, 1911 H.G.J. Moseley, 1913
  3. Atomic Structure Bohr's Model (planetary model)-1914 Assumes electron as particle Got Nobel Prize in 1922 for foundational contribution to understand atomic structure. Bohr atomic model Of a nitrogen atom neutron 0 2011 Encyclopaedia Britannica, Inc. electron orbits electron proton Schrödinger Model (electron cloud model)- 1926 Assumes electron as wave Got Nobel Prize in 1933 for his Schrödinger wave equation for H-like atoms. 1 fm 1 Å = 100,000 fm
  4. Bohr's Model r = E n2h2/nge2Z E = -Z2e4g/8E02n2h2 or, E = -13.6/n2 ev E = -Ze2/8neor = reduced mass of electron-nucleus combination Z = Charge of nucleus or atomic number e = electronic charge h Planck's constant n = principal quantum number 5 4 3 2 1 Bohr atomic model of a nitrogen atom neutron @ 2011 Encyclopedia Britannica, Inc. 0.00 ev 0.54 0.85 ev -1.51 ev 3.4 ev -13.6 ev electron orbits electron
  5. Drawbacks of Bohr model The model only worked for hydrogen-like atoms. Could not explain why the intensity of the spectra lines were NOT all equal. The existence of fine and hyperfine structure in spectral lines. The Zeeman effect - changes in spectral lines due to external magnetic fields. Wave nature of electron was not considered in Bohr's model Information gained from Bohr's model It provide foundation to understand atomic structure. The total energy of electron in H-atom it predicts is exactly agree with these obtained from Schrodinger equation (discussed later).
  6. Photoelectric Effect (1905) Photoelectrons radiation Metal Quantum revolution A beam of light is not a wave propagating through space, but rather a collection of discrete wave packets (photons), each with energy h v Emission of electrons from metals when exposed to (ultraviolet) radiation proves particle nature of electron.
  7. Photoelectric Effect The kinetic energy of the emitted electron Ek = h v— Ø= h( v— vo) v is the frequency of incident radiation G) is the work function (minimum energy required to remove an electron from the metal surface ) = hvo , vo is the threshold frequency and for radiation with v > vo, electron emitted. Photoelectron (e-) EK(e-) Free, stationary electron Bound electron
  8. de Broglie equation (1924) (Wave-particle duality) All the moving particles have wave properties. So electrons have dual character. The wave length of a particle moving with a velocity 'v' is given by: = h/mv The wave length of a macroscopic object is insignificant to measure. However, for an electron, the wave length is measurable.
  9. Experimental proof of de Broglie hypothesis Davisson—Germer experiment (1923-1927) confirms the de Broglie hypothesis (wave-particle duality). They discovered the wave nature of electron by getting a diffraction pattern from a Ni crystal by bombarding a beam of electron. Electron gun hot filament to Mrelease electrons +54 V Accelerating o electrode Theory -1.67Åfor54V Experiment Pathlength difference d sine 1.65Å for constructive interference Not bad for a three ear old idea! Nickel lattice spacing d = 2,15Å Second Series December, 1927 THE Vol. 30, 6 1924 de Broglie's hypothesis 0 Nickel stal 1927 Davisson. Germer experiment Electron scattering peak at 500 1929 Nobel Prize for de Broglie PHYSICAL REVIEW DIFFRACTION OF ELECTRONS BY A CRYSTAL OF NICKEL BY C. DAVISSON AND L. H. GERMER ABSTRACT The intens•it" of scattering of a homogeneous beam of electrons of adjustable speed incident upon a single crystal of nickel has been measured as a function of direction. The crystal is cut parallel to a set of its {111} -planes and bombardment is at normal incidence. The distribution in latitude and azimuth has been determined for such scattered electrons as have lost little or none of their incident energy,
  10. Heisenberg's uncertainty principle (1927) The more precisely we can define the position of an electron the less certainly we are able to define its velocity, and vice versa. Ax , APx > h/47t
  11. Heisenberg's uncertainty principle (Concept of orbital) The energy of spectral line is measured with great accuracy, and hence the energy of electron. Precision in energy Precision in momentum So there is large uncertainty in the location of electron. Hence orbits are replaced by orbitals (the probability of finding an electron in a given space). Orbitals are described by wave functions. n 5 4 3 2 1 0.00 —0.54 —0.85 -1.51 Paschen series IR -3.4 Balmer series Visible Lyman series UV -13.6 An energy level diagram for hydrogen.
  12. Schrödinger's equation The Schrödinger equation describes the wave properties of an electron in terms of position, mass, total and potential energy. The simplest form of Schrodinger equation: H = Hamilton operator E = Total energy of electron W = wave function of electron —h2/81t2m V2 + V H potential energy, (V = — Ze2/4a€ r) V [-h2/8a2m V2 W = E W
  13. Wave function (W) (PSI) 1. A wavefunction is a mathematical function (like sinx, ex). Like any mathematical function it can have large value at some place, small in other and zero elsewhere. It can be real or complex 2. A wavefunction contains all information about the system 4. If the wavefunction is large at a point in space, the particle has a large probability at that point 5. The more rapidly a wavefunction changes from place to place, higher the K.E. of the particle it describes
  14. Wave function (W) (PSI) How to determine the observables from wavefunction (W) By performing a set of well defined mathematically operations on W. These mathematical operations are called operators (operator) (function)= (constant factor) (same function) ax dx ax —ae (operator) (function)= (eigenvalue) (eigenfunction) 14
  15. Understanding the Schrödinger equation (Hamiltonian operator) (Eigenfunction) = (Eigenvalue) (Eigenfunction) Eigenfunction is the wave function of an electron corresponding to the energy E. Eigenfunction is different for each eigenvalue. By solving Schrödinger equation one can find the wave functions (eigenfuncctions) and the corresponding allowed energies (eigenvalues). (operator corresponding to an observable) (W) = (value of observable) (W) Hamiltonian operator is used to find out the total energy
  16. Understanding the Schrödinger equation How to extract the information about the location of electron from the wave function W? The probability of finding the particle in an infinitesimal volume, d V, about a given point is proportional to IW12 d V. Normalization the probability of existence of the particle in the entire space should be 1. vv*dx = 1
  17. Identifying an eigenfunction Q]. Show that eax is an eigenfunction of the operator d/dx, and find the corresponding eigenvalue. Q2. Show that eax2 is not an eigenfunction of d/dx. d/dx(eax2) = 2ax (eax2) This is not an eigenvalue eventhough the same function occurs on the right, because the function is now multiplied by a variable factor (2ax), not a constant factor.
  18. Solutions of Schrödinger equation There are many solutions to the above equation. However, the acceptable solutions must satisfy following conditions. 1. V must be single valued. 2. V must be continuous. 3. Vmust be finite. For an atom several wave functions (WI, W 2, W 3) will satisfy these conditions and each of these has a corresponding energy (El, E2, E3) These wave functions are called as orbitals analogy to orbits in Bohr's theory. Each orbital is described uniquely by a set of three quantum numbers, n, l, ml.
  19. Quantum Numbers There are four quantum numbers. Orbital is defined by three quantum numbers (n, l, ml). An electron is defined by four quantum numbers (n, l, ml, ms). Principal quantum number (n): Determines the total energy of an electron. E = -1/n2 (Z2e4W8e02h2) or, E = 1/n2 (-13.6) ev Has value 1, 2, 3
  20. Quantum Numbers Angular momentum quantum number (l): Describes angular dependency of the wave function (shape of orbital) and rotational kinetic energy (angular momentum). Angular momentum + 1) h 1=0, 1, 2, 3, Q: why the value of 'l' is always less than 'n Magnetic quantum number (ml); Describe orientation of orbital in space. It is the z-component of angular momentum. (so it never be larger than the 'l' value) It may have +ve z-component or —ve z-component So the value ranges from —l, O, +1
  21. Quantum Numbers Spin quantum number (ms); Describes orientation of electron spin in a magnetic field. The value of either +1/2 (in the direction of the field) or —1/2 (opposed to it). Arises due to spinning of electron.
  22. Radial and Angular part of wave function W In 3D, W may be expressed either in Cartesian coordinates (x,y,z) or in spherical coordinates r = represents the distance from the nucleus. 0 = is the angle from z-axis, it varies from 0-71. = is the angle from x-axis, it varies from 0-211. W(x,y,z) can be converted to W using: x = r sino cos() Y = r sino sin() z = r cos0 z
  23. Radial and Angular part of wave function W W may be separated into radial component and two angular components. Combining the two angular components L.....-r.....J I V radial function angular function
  24. Radial and Angular part of wave function W The radial function, R(r) : The radial function is determined by the quantum numbers n and l. The radial probability function, 47tr2R2 is derived from the radial part of the wave function. It describes the probability of finding the electron at a given distance from the nucleus. Is Distance from nucleus (r)
  25. Radial and Angular part of wave function W The angular functions, ) Ø(+) : The angular function is determined by the quantum numbers I and ml. The angular functions, ) Ø(+), describes shape of the orbitals and their orientation in space.
  26. Hydrogen atom Hydrogen has special significance 'No approximation is required in solution of Schrödinger equation 'Can get expression for energy levels For H atom the Schrödinger wave equation can be written as -(h2/2m V2 + Ze2/4n€or 26
  27. Solution of Schrödinger wave equation for Hydrogen atom Solution of the Schrödinger wave equation (shown in spherical coordinate) may be described as a product of two functions. where R n (r) is called the radial part of y, and Y (0 +) its angular l,ml , part. The wavefunction of the electron in the hydrogenic atom is called an atomic orbital. An orbital is a one-electron wavefunction. 27
  28. Solution of the Schrödinger wave equation: Some examples Is atomic orbital (n = 1, I = 0, ml = 0 Is — Radial part Z = effective nuclear charge = Bohr radius a r = distance from the nucleus n = principal quantum number Angular part
  29. Is atomic orbital (n = 1 , O, 11'11 = O 1= For H atom 1, so Is Y is a constant and does not depend on 0 and , s-orbitals are spherically symmetrical.
  30. Note: Z is the atomic number and z is the z-direction/component In polar coordinate the above equation becomes cose 2P atomic orbitals (n = 2, 1, +1, (Since, z = r cos0) 1 z 3 - 206 a Radial part (R n l) 2 Angular part (Y
  31. 2P atomic orbitals (n = 2 For H atom Z = 1, so 3 1 1 3 206 ao cose The angular variation of wave function depend on cos 0. The probability density is proportional to cos20. The probability density has maximum value along an arbitary axis (z-axis) on either side of the nucleus ( at 0= 0 and 1800)
  32. p-orbitals are given below: — 2U6 ao 2px 2P atomic orbitals Similarly the wave functions (real part of wave functions) of other two (Since, x/r = sin0cos+) (Since, y/r = sinesino sin OcosØ sin sin" 2py 2pz
  33. 3d atomic orbitals (n 1 1 15 2 xy xy' xz' yz' x2-y2' z Similarly the d d xz' yz 1 1 15 2 x2—y 2 2 1 5 2 2z2— y 2 2 3dyz Since the angular part contains two or more variables, so these orbitals have shapes in two axial directions or more. 3dx2_ 2
  34. Nodal surface(s) of atomic orbitals Nodal surface is a surface with zero electron density. node ONE CYCLE Nodes appear naturally as a result of the wave nature of the electron. At the nodal surface the wave function changes its sign. At nodal surface, W = 0 either R(r) = 0 Radial node Node Is 2s z Angular node x 2pz
  35. Nodal surface(s) of atomic orbitals If Y (0,+) = 0, angular nodes result. Angular nodes are planar or conical. Number of angular nodes Orbital s- orbital p-orbital d-orbital f-orbital No. of angular nodes 1 2 3 Total number nodal surface = If R(r) = 0, radial nodes or spherical nodes result. Number of radial nodes = n—l -1 Orbital Radial Orbital Radial Orbital Is 2s nodes 1 2 3p 4p nodes 1 2 4d Radial nodes 1 2
  36. Q. Describe the nodal surfaces for a 3dxy orbital, whose angular wave function is 1 4 1 15 2 xy x z xz -plane z -plane 3d Ans. In nodal surfaces, Y = 0 Y = 0, when either x = 0 or y = 0 So the nodal surfaces are x = 0 (yz plane) and y = 0 (xz plane)
  37. Sign of atomic orbital Sign of lobes does not mean electronic charge. It represents the phase/direction of wave function. Atomic orbital along positive axial direction has a positive sign and negative axial direction has a negative sign. post tavt xz x + ve phase ve phase ONE 3 2 toe teo CYCLE
  38. Sign of atomic orbital The angular part of s-orbitals contain constant value, no variables. So there is no change in sign. The angular part of p-orbital contains either x or y or z. Since the x, y, or z direction may be positive or negative, so the wave function along positive axis has a positive sign and along negative axis has negative sign. The angular part of d d d contain product of two component. So the product xy' xz' yz of signs gives the sign of the lobe. the angular part contains x2—y2. so the wave function is positive along x- For d X2-y2' direction (positive and negative) and negative along y direction. For dz2 , the angular function contains 2z2—x2 Y2 so the wave function is positive along z direction (both positive and negative) and negative in xy-plane.

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