SB Lecture 2

Question 1

What is the definition of symmetry?

Answer 1

Symmetry is the property of remaining invariant under certain changes (as of orientation in space, for example).

Question 2

Give three reasons why symmetry is important in chemistry

Answer 2

1. Using symmetry significantly reduces the complexity of mathematical solutions. It simplifies the description of molecules, which in turn simplifies the mathematical treatment of processes. (E.g., spectroscopy; quantum mechanics)
2. Symmetry is necessary to describe the structure of crystalline materials, which are comprised of regular, repeating arrangements of atoms.
3. Symmetry underpins the specific reactivity of many molecules in nature. (E.g., reactivity between chiral species – v. important in pharmaceuticals!)

Question 3

What is the definition of symmetry operation?

What is an example of a symmetry operation?

Answer 3

A symmetry operation is an action that leaves the molecule apparently unchanged.

A rotation is an example of a symmetry operation.

The dashed line represents the axis of rotation.
We can rotate the H2O molecule by (multiples of) 180° about this axis without apparent change to its position in space.

Question 4

What is the definition of a symmetry element?

Answer 4

A symmetry element is the point, line or plane in the molecule about which the symmetry operation occurs.

Question 5

What are the 5 symmetry elements needed to fully describe the symmetry of a molecule?

Answer 5

All 5 are not always present!

Question 6

Describe the symmetry element “Identity, E”

Answer 6

• This is the simplest operation: nothing moves!
• All objects have symmetry element E.

Question 7

Explain 2-Fold proper rotation C₂

Answer 7

• This is the simplest example of an n-fold rotation.
• Symmetry Operation: rotation
• Symmetry Element: C₂ axis (a line)
• Water (H₂O): C₂ rotation (360º/2 =180º)
  
  • ​The water molecule may be rotated through any angle about the bisector of the H-O-H bond angle but only a rotation of (a multiple of) 180° (C₂) leaves its position unchanged in space

Question 8

Describe 3-Fold proper rotation, C₃

Answer 8

• A rotation by 120° is one-third of a complete circle (360/3)° = C₃
• 2 x 120° = 240° is two-thirds of a circle 2(360/3)° = C₃
• Unchanged if you rotate 120º about the axis
• Does not have C₂
• Ammonia NH₃: C₃ and C₃² rotation
• A 3-fold rotation is possible about and the C₃ axis.
  
  • There are 2 rotations associated with this axis: C₃¹ and C₃²

Question 9

Describe 4-Fold proper rotation, C₄

Answer 9

• Rotation by 90° = (360/4)° is C₄ (more precisely, C₄¹)
• Rotation by 2 x 90°= 180° is C₂ not C₄²!!
• Rotation by 3 x 90° = 270° is C₄³
• Iodine pentafluoride IF₅: C₄ , C₂ and C₄³ rotation
• The C₄ axis and C₂ axis share the same position.

Question 10

Can rotation axes have different positions?

Answer 10

YES!

Higher symmetry molecules can have multiple rotation axes.

Tetrachloroplatinate [PtCl₄]^2-

Question 11

What is the principal axis?

Answer 11

• The principal axis is the highest order axis of rotation (largest n).
• It is possible to have multiple principal axes!!
• Axes that are perpendicular to the principal axis are labeled with primes (‘ ) and (“ ).

Question 12

What are the 3 possible subscripts for a mirror plane?

What is the symmetry element, the symmetry operation?

Answer 12

3 Possible subscripts for a σ plane:

v (vertical), h (horizontal), d (dihedral)

Symmetry element: mirror plane

Symmetry operation: reflection

Question 13

What is σ_v?

Describe water, H₂O with respect to σ_v and σ_v’ mirror plane

Answer 13

σ_v (v = vertical) is a mirror plane collinear with a unique principal rotation axis

Water, H₂O: σ_v and σ_v’ mirror plane

• There are two vertical mirror planes in H₂O
• They are perpendicular to one another
• One cute through all the atoms
• One cuts the O atom and reflects the H atoms into one another

Question 14

Describe σ_h mirror plane (Boric Acid)

Answer 14

Boric Acid, B(OH)₃: σ_h mirror plane

• σ_h (h= horizontal) is a mirror plane perpendicular to the principal rotation axis or axes
• In this configuration, there is a C3 axis through the boron atom…you are looking straight down it in this image.
• The only mirror plane present is perpemndicular to this axis. Therefore, it is called a horizontal mirror plane.

Note: The terms “horizontal” and “vertical” have nothing to do with the orientation of the molecules in space!! They only refer to orientation w.r.t. principal axis.