Chapter 10: Q32E (page 468)

By the “vector” technique of example 10.1 , show that the angles between all lobes of the hybrid ${sp}^{3}$ states are $109.5 ° .$ .

Short Answer

Expert verified

The Angle between all lobes of hybrid ${sp}^{3}$ states are $109.5 °$

Step by step solution

Finding Four States that participate in hybrid.

$ψ_{1} = ψ_{2 s} - ψ_{2 p z} ψ_{2} = ψ_{2 s} + \frac{1}{3} ψ_{2 p z} - \frac{\sqrt{8}}{3} ψ_{2 p x} ψ_{3} = ψ_{2 s} + \frac{1}{3} ψ_{2 p z} - \frac{\sqrt{2}}{3} ψ_{2 p x} - \frac{\sqrt{6}}{3} ψ_{2 p x} ψ_{4} = ψ_{3} = ψ_{2 s} + \frac{1}{3} ψ_{2 p z} - \frac{\sqrt{2}}{3} ψ_{2 p x} - \frac{\sqrt{6}}{3} ψ_{2 p x}$

Now We will Use Dot product to Find The Angle.

A.B=AB Cos $θ$ (Here A and B are vectors).

To Calculate the Dot Product.

Ignoring the 2s state contribution, to calculate the length of each state.

$|ψ_{1}| = {(- 1)}^{2} = 1 {|ψ_{1}|}^{2} = (\frac{1}{3}) + {(\frac{\sqrt{8}}{3})}^{2} = 1 ψ_{3 or 4} l^{2} = {(\frac{1}{3})}^{2} + {(\frac{\sqrt{2}}{3})}^{2} + {(\frac{\sqrt{6}}{3})}^{2} = 1 For Dot Products Calculation . . ψ_{1 .} ψ_{2} = (0, 0, 1) . (\frac{- \sqrt{8}}{3}, 0, \frac{1}{3}) = \frac{- 1}{3} ψ_{2} . ψ_{3 or 4} = (0, 0, - 1) . (\frac{\sqrt{2}}{3} \pm (\frac{\sqrt{6}}{3}, \frac{1}{3}) = \frac{- 1}{3} ψ_{2} . ψ_{3 or 4} = (\frac{\sqrt{2}}{3} - (\frac{\sqrt{6}}{3}, \frac{1}{3}) . (\frac{\sqrt{2}}{3}, (\frac{\sqrt{6}}{3}, \frac{1}{3}) = \frac{2}{9} - \frac{6}{9} + \frac{1}{9} = \frac{- 1}{3} In all cases the Dot Product is \frac{- 1}{3} and the length is always 1 . So, Cos θ = \frac{AB}{|AB|} = \frac{- 1}{3} θ = \cos^{- 1} (\frac{- 1}{3}) = 109.5 ° . . Hence, The Angle between all lobes of hybrid {sp}^{3} states are 109.5 °$

For Dot Products Calculation..

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By the “vector” technique of example 10.1 , show that the angles between all lobes of the hybrid ${sp}^{3}$ states are $109.5 ° .$ .

Short Answer

Step by step solution

Finding Four States that participate in hybrid.

To Calculate the Dot Product.

One App. One Place for Learning.

Most popular questions from this chapter

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By the “vector” technique of example 10.1 , show that the angles between all lobes of the hybridsp3states are 109.5°..

Short Answer

Step by step solution

Finding Four States that participate in hybrid.

To Calculate the Dot Product.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Physics of Motion

Electricity and Magnetism

Conservation of Energy and Momentum

Measurements

Force

Study anywhere. Anytime. Across all devices.

By the “vector” technique of example 10.1 , show that the angles between all lobes of the hybrid ${sp}^{3}$ states are $109.5 ° .$ .