Schrodinger's Wave Equation

Schrodinger's wave equation :- Let the electron meves in three dimensional axis ie. x, y and z axis. To, explain the movement of electron through all the axis, Schrodinger's explain that electron meves with a wave Function Lot, the wave Fuction along the three axis. along x-axis is 4x, Y-axis is 4y and along Z-axis is Yz. along Schrodinger develops a mathematical wave equation to describe the swave motion of e araind the nudeus. The movement of e Stationary wave having obey all the condition of a wave Feenction. Ψ = A. sin 2πα (1) Wheren wave Function of an e crontasis moe Position in which e moves. sull Fi Atini pridet wavelengthis s/m 122.2 Differentiating eq (1) win.tx (mi) asitizen Eq (6) on (7) is called Schrodinger wave equatèon For an e in three dimensional axis. When e moves around the nucleus, then potential energy (V) = - ze dy + day dya + 12 + 872m dza 62 (E+ze²) = 0 For H. atom, Z=1. d2 + d24+ d24 224 da2 dy dez + 872m 2 This equation is known as Fon hydregen atom. Schrodingers weave eq Significans of 4 and 42 :- (1) The wavefunction & must be single valued, Finite and contineous. Multi valued wave Funt (Infinite wave) Function (Discontineous) Well behaved wave Fent (2.) In all wave function, the square of wave function (42) is the property that give physical significance (3.) The probablity of Finding and that in overcall space e is maximeem in an must be equal to 1. orbital too Eq^ (6) on (7) is called Schrodinger wave equatèon For an e in three dimensional axis. When e moves around the nucleus, then potential energy (V) = -ze? N Her d24 + day dya + d2 dza + 872m h2 (E+ze²) = 0 +Ze. For H. atom Z=1. d2 + d24+ d24 224 da dy2 22 + 872m This equation is known as Fon hydregen atom. Schrederger's Lucive eq Significans of 4 and 42 :- (1) The wavefunction & must be single valued, finite and contineous (Multi valued wave Fent (Infinite wa Function wave) (Discontineous) Well behaved wave fent (2.) In all wave function, the square of wave function (42) is the property that give physical significance" (3.) The probablity of Finding e is maximem in an and that in overcall space must be equal to 1. too orbital 4. For e wave, we represent the probablity of finding e in overall space where as it represent in light wave. intensity of light Quantum Number :- Atem → Shell Subshell Orbital spin These are the identification no. or Inder no. which gives the complete information or address of all e in an atom. The Quantum no identity the distance of e From andeus (radius), energy of e in a shell, type of axentation spin of e around its own axis. There are four quantum no. namely, (i) Arincipal Q.no. (i) Azimuthal/subsidiary/secondary / Angular ano. (ii) Magnetic Magnetic Q.no (v.) Spin Q.no. The First three & no. are intere related to each othere. (1) Principal Q. No It is rrepresented by the letter 'n' It is derived from radial part of schreedinger wave eq". The value of n are 1,2,3,4. It represents main energy Level on [shell] to which are an electron belonge in an atom. It determine the Shell, and that can be accomodate in a maxm no. of e = 202. Azimuthal Quantum No. :- It is represented by the letter 'L'. It is derived From angular part of schrecelinger's wave eq?. The value of I ranges From o to (n-1). It's value depends upon the principal quantum no.1 therefore, it is called subsidiary/ secondary quantum No. It represents the subshell to which an electron belongs The marm no. of electron that can be accomodate in a subshell 2C2L+2). 20 Value of I Sub shell May noirFe2(2+1) Magnetic Quantum No. It is represented by the letter 'm' It is derived From angular part of schrodinger's wave equation. zero. Indida to SqANG threugh The value of m ranges From 1 to th It represents the e- Orbital of a sub-shell to which an belengs in an atom. It describe orientation of orbitalis n=1 (K.Shell)1=0 (s-subshell) mas orbital)a n=2&shell)1=0 (4) 1-1 (P sub-shell) m=-1,0,1, orchital) (P, Py, Pa n=3 (M shell) 1=0 (8) 1=1) sub-shell) m3,-2,-1,0,1,2,3 (-orbital) (dry dyz, den chiry-dyedon n=4 (N-shell) L=0) 1=1 (P) 2=2(d) 1-3 (3-subshell) m=-3,-2,-1,0,1,2,3 (F-orbital FX, Fys, F23, fryz, FX (4222) Fy (2x2). Spin Quantum no. - It is denoted by the letter 's'. The value of & are +1/2 and -1/2. It represents the spin of e anaend their own axis. If the electron spin in clockwise direction, it is represented by uparrow (1) and its value is +2/2. If the electron spen in anticlockwise din, it is represented by dewn arrow (+) and its value is -1/2. $=11/2012 Shape of Orbital :- → The three dimensional orientation of e around the nucleuw of an atom is called Orbital. The region in three dimensional space where probability of Finding e is, maxm in an atom, is called orbital. S-Ochital When L=0 (s=sub-shell), then m=0. →S-sub-shell has only spherically symmetrical. one orientation or orbital, which is → At centree part of the orbital, nucleus is located. → The probablity of Finding e inside the s-orcbital is 100%. → From the value of Prenupal Quantum no. (n), the size of different sorbetal increase with the velocity of n. → Graphically, the shapeof 23, 25 and 3s orbital can be represented as, Node → Expect 18, in other s-orbital, there is a region where probablity of finding e is zera, called radial node/node. b) P-Orbital :- Where L= 1 (P-sub-shell) then m=-1, byte (3) →P-sub-shell has three orientation/cubital such as Px, Py, Pz →P-orbital has dumbled shape which consists of two lobes directiseed along the ares Uppolobe (UL) tuel To Nadal point Lower Lobe (LL). Nodal Point "It is a point where two lobes joint and probablity of Finding e is zero ↳ Graphically, the shape of Pr, Py and Pe Orbital can be represented as No. of radial node = 0-1-1 The no. of radial nede for 2P, 3P, 4P orchital are zerre, one and two respectively. Nede (2P) 3P) Intido- (c.) d-Orbital When 1-2 (d-subshell), then m=-2,-1,0,1,2, 4d-Subshell thas five orientation/orbital such as day, dyz, den, day, dez 4d-orbital has a double dumbled shape which consists of Four lobes directed along the axes on in between the axel dry dyz, dex, da²-y² orbital have Clove leaf shape Structure dza orbital has baby soother on daighnut shape (d) F-arbital When 1:3 (J-subshell) then m=-3,-2,-1,6,11,1213. F. Subshell has seven orientation / orbital having triple. concentric or complex shape. dumbled shape or Cubic Set F2, Fy, fz, Fryz, Fx(y²-z²). Fy (22²), F2x²y²) Physical Significance of Quantum no. - (1.) Principal Quantum ne. am electron in a shell. describe the energy and position of (2) Azimuthal G. no. describe the shape of e cloud. It also tells that the energy level/shall consists of silb-shelt. →The magnitude of angular mementum of an e in a subshell is given by Orbital angular mementum. (3) Magnetic &. no. describe the Orientation of ecloud in spos It tells that stub-shell consists of a no. of orbital. 4. Spin Qine. describe the spin of e araerd its own axis. Spin angular mementum of spining e is S(STL) SC Distribution of electron in different sub-shell of an abr Electrons are Filled in different subshell of an atom in accordance with different principleie 1. Autbus Principle. 2. Pauli's Exclusion Principle. 3. Hund's rule of maximeem meeltiplecity. (4) 4. Stable electronic configuration/ Exchange energy rule. (1) Aufbau's Principle Aufbaus is a It state that, u german word which mean "Building up". Electron's are filled in differcent subshell of an atom in the increasing order of their energy." Subshell with laun energy is filled up first in prefencape to other sub-shellsanotimpice losisu19 The energy of different subshell can be complamed by using (1) rule. 118 Increasing conder of energy of different sub-shell in an atom is given by 15 225 220 2 3 5 2 3P 445 < 3d < 4P 255 2 4d 25P < 65 < uf < 5 d 2 G P < 7 3 4 5 F & Gd <7P. Br 35 252 252 206 352 3p6us² 3d ups 2.88, 18, 18, 32, 32 Pauli's Exclusion Principle:- No two electron in an atom can have all, the Four Quantum no. alice." Two electreon in an atom can't have all the Four quantum no. exactly same. If two electrons are predene in same snell (value same), same suh-shell (I valice same) and same orbital (m value same) then they miut have different spin quantiem no.(s) le. if an electron spin in clockwise din", then other most spin in anticlockwise din. According to this principle, the maximum capacity of an orbifal to accomodate electron is two. This principle is called as Exclusion principló es it excludes the probablity of two electron in an atom-capital. containing more then Significance :- 1. Maximum capacity of an atoon orbital to accomodate = 2. 2. Maximum capacity of a sub-shell to accomodate e = 2(211)