Derivation of the orbital analysis equation and its physical significance

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Discussion Overview

The discussion revolves around the derivation of an equation relating bond angles to the s-character and p-character of hybridized orbitals, specifically in the context of ##sp^3d## hybridization. Participants explore the theoretical implications of this equation and its connection to established rules in chemistry, such as Bent's rule and Dragos rule.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents the equation $$cos\theta = \frac{s}{1-s} = \frac{p-1}{p}$$, asserting its use in demonstrating the s-character of axial orbitals in ##sp^3d## hybridization and its relation to Bent's rule.
  • Another participant questions the validity of the equation, noting that for s > 0.5, cos(θ) would exceed 1, which raises concerns about the equation's applicability.
  • A subsequent post corrects a typo in the equation to $$cos\theta = \frac{s}{s-1}$$ but still faces criticism regarding its validity for s > 0.5.
  • One participant argues that no hybridized orbital can possess more than 50% s-character, referencing the characteristics of sp hybridization as a basis for their claim.
  • Another participant acknowledges that the equation may hold for "standard" non-fractional hybridizations, suggesting that the equation might not be widely recognized or applicable in general cases.
  • One participant expresses uncertainty about their understanding of the problem, indicating potential confusion regarding the equation's formulation.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity of the equation, particularly in relation to the s-character exceeding 50%. There is no consensus on the equation's derivation or its general applicability.

Contextual Notes

Participants note limitations regarding the equation's applicability to fractional hybridizations and its potential lack of recognition in standard discussions of hybridization.

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TL;DR
unable to prove/derive theoretically the relation between bond angle and s and p character of a hybridized orbital.
$$cos\theta = \frac {s}{1-s} = \frac{p-1}{p}$$
in this equation ##\theta## is the bond angle and ##s## and ##p## are the fractional s-character of the orbital and p-character of the orbital.
This is equation is used rigorously in showing that the s-character of the axial orbitals in a ##sp^3d## hybridized orbital is 0 and hence it is also used to prove bents rule which states that the most electronegative element in a ##sp^3d## hybridized orbital takes the axial positions of the trigonal bipyramid that is formed. But I am unable to find a formal proof/derivation of this equation on the internet or in my textbook.
Is this relation between the bond angle and the s/p character very obvious that it does not need a proof/derivation?
(also I am assuming there might be an experimental proof for this formula but I am not looking for that rather a theoretical proof of this equation will be nice)
 
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Can't say this relation makes sense, for s>0.5 cos(θ) would be larger than 1.
 
Borek said:
Can't say this relation makes sense, for s>0.5 cos(θ) would be larger than 1.
I am so sorry I made a typo it is actually $$cos\theta = \frac{s}{s-1}$$
 
Still fails, now for s>0.5.
 
Borek said:
Still fails, now for s>0.5.
I thought it was clear no hybridized orbital can have an s character more than 50%(0.5 fractional) as sp hybridized orbital has the least number of orbitals(2) combining and has 50% s-character in both orbitals. Hence the equation does not fail.
also, this very fact was used in Dragos rule...
 
OK, for "standard", non fractional hybridizations it will hold.

Apparently I know a bit too much to properly understand the problem as defined here.
 
Borek said:
OK, for "standard", non fractional hybridizations it will hold.

Apparently I know a bit too much to properly understand the problem as defined here.
maybe this particular equation isn't very popular(because it isn't very general?)
 
I don't remember being taught it. Doesn't mean much.

On the second thought I could get s and p reversed. My bad.
 

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