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

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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.
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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.) Ψ 2 ( r ) = [ R(r)Y ] 2 = e -2r/ a(0) / π a 0 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.
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Orbitals specified by l = 0, m l = 0 are s -orbitals. The radial wavefunction component of Ψ is R(r) . This component has max value when r = 0. It goes to zero as r increases.
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Radial Distribution Function Gives the probability of the presence of an electron at any position within a spherical shell of radius r and thickness δ r . Does not depend on spherical polar coordinates, so only the R(r) portion of the Ψ ( θ, φ, r ) wavefunction is needed . P ( r ) = r 2 [ R ( r )] 2 For the s -orbital: R(r) = 2e -r/ a(0) /( a 0 ) 3/2 Another way to express this function is below [ Ψ ( θ,φ , r ) is the wavefunction for the ground state of the H atom] P(r) = 4 π r 2 [ Ψ( r )] 2
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Only includes the R(r) portion of Ψ . Goes to zero at r = 0. Peak for 1 s orbital corresponds to: r = a 0 = 53 pm.
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Difference Between Probability Density Predicted by Ψ 2 and Predicted by Radial Distribution Function [Ψ( θ , φ , r )] 2 gives the probability of the electron being in a volume δ 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 δ r. [ Ψ( θ , φ , 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 π r 2 in front of Ψ 2 in the equation on previous pg goes to zero).
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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 0 ) then decreases with further increase in r (because Ψ 2 goes to zero).
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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 . All of these functions go to zero at r = 0 because the R(r) portion of Ψ( θ, φ, r ) goes to zero (Table 1.2) .
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