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lecture%204 - Number of Orbitals and Orientation in Space :...

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Unformatted text preview: Number of Orbitals and Orientation in Space : Each orbital in a particular subshell is specified by m l (the magnetic quantum #). In a given subshell, possible values for m l are: - l, (1 - l ), (2 - l ) ... 0 ... ( l- 2), ( l - 1 ), l Therefore 2 l + 1 orbitals in a particular subshell. Each of the orbitals within a particular subshell contributes a portion of the total angular momentum of that subshell. Examples of Electron Probability Density Calculations Each orbital is defined by the appropriate combination of quantum #s. If n = 1, l = 0, m l = 0 it is an s-orbital. Such an orbital is radially symmetric so the angular wavefunction Y does not depend on the spherical polar coordinates (therefore a constant). (Refer to eq 17 & 19.) 2 ( r ) = [ R(r)Y ] 2 = e-2r/ a(0) / a 3 Max value when r = 0 (nucleus). Decreases to insignificant level at r = 250 pm. However we know that electron is not in the nucleus. Discrepancy is rectified by the radial distribution function. Orbitals specified by l = 0, m l = 0 are s-orbitals. The radial wavefunction component of is R(r) . This component has max value when r = 0. It goes to zero as r increases. Radial Distribution Function Gives the probability of the presence of an electron at any position within a spherical shell of radius r and thickness r . Does not depend on spherical polar coordinates, so only the R(r) portion of the (, , r ) wavefunction is needed . P ( r ) = r 2 [ R ( r )] 2 For the s-orbital: R(r) = 2e-r/ a(0) /( a ) 3/2 Another way to express this function is below [ ( , , r ) is the wavefunction for the ground state of the H atom] P(r) = 4 r 2 [ ( r )] 2 Only includes the R(r) portion of . Goes to zero at r = 0. Peak for 1 s orbital corresponds to: r = a = 53 pm. Difference Between Probability Density Predicted by 2 and Predicted by Radial Distribution Function [(, , r )] 2 gives the probability of the electron being in a volume V at a site specified by r and the polar coordinates. P(r) gives the probability of the electron being in a spherical shell of radius r and thickness r. [ (, , r 29 ] 2 is max at the nucleus ( r = 0) then decreases as r increases. P(r) is zero at the nucleus because the volume of the spherical shell becomes zero there (the 4 r 2 in front of 2 in the equation on previous pg goes to zero). A plot of P(r) as a function of r gives the most realistic portrayal of the electron density in each energy level. For an s-orbital in the n = 1 Energy Level: Both l and m l are zero. P(r) is max at r = 53 pm ( Bohr radius, a ) then decreases with further increase in r (because 2 goes to zero). Orbital Wavefunctions in Subshells Specified by l = 1 If n > 2 then l can be > 1. The l = 1 subshell contains three p-orbitals designated by m l = -1, 0 or +1, or by the names: p x , p y or p z ....
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This note was uploaded on 04/24/2008 for the course CHEM 6A taught by Professor Pomeroy during the Fall '08 term at UCSD.

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lecture%204 - Number of Orbitals and Orientation in Space :...

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