User:Spoon!

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1 Central angle in simplices between radii to vertices
2 Oxyanion chart

[edit] Central angle in simplices between radii to vertices

I know that many people have figured this out long ago, but I like to share it anyhow, because I have wondered about it for a long time when I was in high school...

You know how in high school they told you that the angle between two bonds in methane was about $109.47^\circ$ or something like that? Did you ever wonder what that came from? It is $\cos^{-1} (-\frac{1}{3})$ . And I will now show you why:

[edit] Theorem

If circumradii are drawn between the center of an $n$ -simplex and its vertices, the angle between these segments is $\cos^{-1} (-\frac{1}{n})$ .

[edit] Proof

Some formulas from this page:

http://www.math.rutgers.edu/~erowland/polytopes.html#sectionII

The height of a regular $n$ -simplex of side $s$ is

$h_n = \sqrt{ \frac{n+1}{2n} } s$

The apothem (inradius) of a regular $n$ -simplex of side $s$ is

$a_n = \frac{h_n}{n+1}$

The circumradius, which is the difference between the height and the apothem, is:

$R = h_n - a_n = \frac{n}{n+1} h_n = \sqrt{ \frac{n}{2(n+1)} } s$

Now consider any two circumradii. They go to two different vertices, which must be joined by an edge of the $n$ -simplex, forming a triangle. Because we know the lengths of all sides of this triangle, we can find the angle between the circumradii using the law of cosines:

$\cos{\gamma} = \frac{a^2 + b^2 - c^2}{2ab}$

Here, $a = b = R$ , and $c = s$ .

$\begin{align} \cos{\gamma} &=& \frac{R^2 + R^2 - s^2}{2R^2} \\ &=& 1 - \frac{1}{2} \left(\frac{s}{R}\right)^2 \\ &=& 1 - \frac{1}{2} \cdot \frac{2(n+1)}{n} \\ &=& 1 - \frac{n+1}{n} \\ &=& -\frac{1}{n} \\ \gamma &=& \cos^{-1} (-\frac{1}{n}) \\ \end{align}$

The angle we want is between $0$ and $180^\circ$ . Since cosine is one-to-one in that range, the angle is uniquely determined.

Q.E.D.

[edit] Conclusion

This explains, among other things, the angles between hybridized orbitals:

hybridization	dimensions	angle between orbitals
sp	n = 1	$\cos^{-1} (-\frac{1}{1}) = 180^\circ$
sp²	n = 2	$\cos^{-1} (-\frac{1}{2}) = 120^\circ$
sp³	n = 3	$\cos^{-1} (-\frac{1}{3}) \approx 109.47^\circ$

That's all the simplices that can fit in 3 dimensions, folks; but you see the pattern...

[edit] Oxyanion chart

Hybridization	Orbital configuration	III	IV	V	VI	VII	VIII
sp (double)	AX_1.5E₀	B₂O₃ M₂O₃
sp	AX₂E₀ linear	AlO₂^- BO₂^-	CO₂ SiO₂ MO₂	NO₂⁺
sp	AX₁E₁		CO	NO⁺
sp² (double)	AX_2.5E₀		Si₂O₅^2-	N₂O₅ P₂O₅ As₂O₅ M₂O₅
	AX_1.5E₁			N₂O₃ As₂O₃
	AX_0.5E₂			N₂O
sp²	AX₃E₀ trigonal planar	BO₃^3-	CO₃^2- SiO₃^2- SnO₃^2- PbO₃^2-	NO₃^- VO₃^-	SO₃ O₄ SeO₃ MO₃
	AX₂E₁ bent <120		SnO₂^2- PbO₂^2-	NO₂^-	SO₂ O₃ SeO₂
	AX₁E₂				SO O₂
sp³ (double)	AX_3.5E₀		Si₂O₇^6-	P₂O₇^4-	Cr₂O₇^2-	Cl₂O₇ M₂O₇
	AX_2.5E₁
	AX_1.5E₂				S₂O₃^2-
	AX_0.5E₃					Cl₂O
sp³	AX₄E₀ tetrahedral		SiO₄^4-	PO₄^3- AsO₄^3-	SO₄^2- SeO₄^2- TeO₄^2- CrO₄^2- MoO₄^2- WO₄^2-	ClO₄^- BrO₄^- IO₄^- MnO₄^- TcO₄^- ReO₄^-	XeO₄ RuO₄ OsO₄
	AX₃E₁ trigonal pyramidal			PO₃^3- AsO₃^3-	SO₃^2- SeO₃^2- TeO₃^2-	ClO₃^- BrO₃^- IO₃^-	XeO₃
	AX₂E₂ bent <109.5			PO₂^3-	SO₂^2-	ClO₂^- BrO₂^- IO₂^-
	AX₁E₃				O₂^2-	ClO^- BrO^- IO^-

User:Spoon!

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Contents

[edit] Central angle in simplices between radii to vertices

[edit] Theorem

[edit] Proof

[edit] Conclusion

[edit] Oxyanion chart

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