Surface Area

LxW πr2 ½ bh

Suppose you were given 100ft of string and wanted to construct a geometric shape with

the greatest possible area. What shape should you create?

Suppose you were given 100ft of string and wanted to construct a geometric shape with

the greatest possible area. What shape should you create?

25ft

25ft

25ft

25ft 20ft 20ft

30ft

30ft

In all three cases the perimeter = 100ft

C = 100ft

Suppose you were given 100ft of string and wanted to construct a geometric shape with

the greatest possible area. What shape should you create?

25ft

25ft

25ft

25ft 20ft 20ft

30ft

30ft

In all three cases the perimeter = 100ft

C = 100ft

A = LxW= 25ft2 = 625ft2 A = LxW= 20ft2 x30ft2 = 600ft2 C = 2πr; r= 100/2π = 15.9ftA = πr2 = π15.92 = 794ft2

Three Objects of Equal Perimeter (12 units) but clearly the square has the largest surface area! You

can’t cut a piece off of a rectangle and attach it turning it into a 3x3 square.

Square vs Circle where the perimeter/circumference is 12 units. The circle is clearly bigger.

Perimeter is not directly related to surface area!

Square vs Circle where the perimeter/circumference is 12 units. The circle is clearly bigger.

Perimeter is not directly related to surface area!

How do you find the surface area of a cylinder?

If you unravel a cylinder you have two circles (top and bottom) and a rectangle whose length is the circumference of the cirlces

πr2

πr2

2πrh

πr2 + πr2 + 2πrh = 2πr2 + 2πrh = 2πr(r+h)

HOMEWORK!Mathematically demonstrate

that the area of a heart can be found using the formula:

(π + 4)r2. Show all steps!

HOMEWORK!Mathematically demonstrate

that the area of a heart can be found using the formula:

(π + 4)r2. Show all steps!

HINT!

Surface Area to Volume Lab!

Image modified from Hewitt: Conceptual Physics

What happens if we double the Linear Dimensions of a 1g Object?

Length

Surface Area

Mass

Volume

Density

SA:V

SA:M

Length

Surface Area

Mass

Volume

Density

SA:V

SA:M

What happens if we double the Linear Dimensions of a 1g Object?

Length 4

Surface Area 96

Mass 8

Volume 64

Density 1/8

SA:V 3/2

SA:M 12/1

Length 2

Surface Area 24

Mass 1

Volume 8

Density 1/8

SA:V 3

SA:M 24/1

What happens if we double the Linear Dimensions of a 1g Object?

Length 4

Surface Area 96

Mass 8

Volume 64

Density 1/8

SA:V 3/2

SA:M 12/1

Length 2

Surface Area 24

Mass 1

Volume 8

Density 1/8

SA:V 3

SA:M 24/1• Doubling the linear dimensions decreases the SA:V ratio.• Doubling the linear dimensions decreases the SA:M ratio.• The volume and mass go up by a factor of 3 (23) or 8x as much• Surface Area increases with the square and mass and volume

with the cube of the increase in linear dimensions.• Volume and Mass increase faster than surface area when we

scale an object up.

What happens if we quadruple the Linear Dimensions of an object?

• The volume increases by a factor of?

• The Surface Area increases by a factor of?

• The mass increases by a factor of?

What happens if we quadruple the Linear Dimensions of an object?

• The volume increases by a factor of?• 43 or 64x VB = 64VA

• The Surface Area increases by a factor of? • 42 or 16x SAB = 16SAA

• The mass increases by a factor of?• 43 or 64x mB = 64mA B

A

Remember, we are quadrupling the linear dimensions, not increasing the volume by 4x as much! As we see, volume and

mass increase much faster than area when scaling up!

Surface Area to Volume Ratios and Biology

• Cells must be small because they rely on the process of diffusion to transport needed material in and out of the cell walls.

• When the surface area to volume ratio is high diffusion is more efficient and the cell is happy.

• If the surface area to volume ratio is low then we have a high volume/mass of material with a comparatively smaller surface area for nutrient transfer. The Cell is sad. Big Cell with Low SA:V

SAD CELL!

Small Cell with High SA:V

HAPPY CELL!

Plant Cells > Animals cells but have larger Vacuoles

Surface Area to Volume Ratios and Biology

• Unicellular Organisms: small with high SA:V ratio so diffusion works well. Once a cell gets too big it must DIVIDE! (and conquer the world!)

• Multicellular Organisms: are large and overcome a low SA:V ratio having lungs (gas exchange organs) and a circulatory system (blood) that helps transfer needed materials into and out of the organism.

Model Cell Diffusion with Agar

Surface Area and ChemistryZn + 2HCl ZnCl2 +H2 (DEMO)

ZINC CUBE METAL

H+

Cl -

H+

H+H+

H+

Cl -

Cl -

Cl -

Cl -

We are going to put a block of Zinc into a cup full of Hydrochloric Acid and model a single replacement reaction.

Surface Area and ChemistryZn + 2HCl ZnCl2 +H2 (DEMO)

ZINC CUBE METAL

H+

Cl -

H+

H+

H+

H+

Cl -

Cl -

Cl -

Cl -

Hydrogen Gas is leaving

Cl -

Cl -

Cl -

H+

H+H+

The zinc atoms on the surface of the Zinc react with the Chloride ions forming ZnCl2

Cl -Cl - Zn+2

Surface Area and ChemistryZn + 2HCl ZnCl2 +H2 (DEMO)

ZINC CUBE METAL

The Rate of any Chemical Reaction will be proportional to the Surface area of the reactants.

Supposing our Zinc cube is 2in by 2in by 2in what is its total surface area?

2”

2”

2”

Surface Area and ChemistryZn + 2HCl ZnCl2 +H2 (DEMO)

ZINC CUBE METAL

The Rate of any Chemical Reaction will be proportional to the Surface area of the reactants.

Supposing our Zinc cube is 2in by 2in by 2in what is its total surface area?

A Cube has 6 sides or faces and the surface area of each one is LxW so

SA = 6(LxW) = 24in22”

2”

2”

Surface Area and ChemistryZn + 2HCl ZnCl2 +H2 (DEMO)

ZINC CUBE METAL

What happens to surface area if we cut the cube in half?

2”2”

2”ZINC CUBE METAL

ZINC CUBE METAL

SA = 24in2

Surface Area and ChemistryZn + 2HCl ZnCl2 +H2 (DEMO)

ZINC CUBE METAL

What happens to surface area if we cut the cube in half?

2”2”

2”ZINC CUBE METAL

ZINC CUBE METAL

SA = 24in2

We have two extra sides or faces. Thus the SA goes up.

SA=???

Surface Area and ChemistryZn + 2HCl ZnCl2 +H2 (DEMO)

ZINC CUBE METAL

What happens to surface area if we cut the cube in half?

2”2”

2”ZINC CUBE METAL

ZINC CUBE METAL

SA = 24in2

We have two extra sides or faces. Thus the SA goes up.

SA=32in2

Surface Area and ChemistryReaction Rate is Surface Area!

ZINC CUBE METAL

Continuing the Process Results in more and more Surface Area Thus, a crushed substance or powder should react more quickly

than a larger version of itself. Mass and Volume remain constant but SA increases by crushing!

2”2”

2”ZINC CUBE METAL

ZINC CUBE METAL

ZINC CUBE

METAL

ZINC CUBE

METAL

ZINC CUBE

METAL

ZINC CUBE

METAL

Whoosh Bottle & CombustionAlcohol is flammable. In the whoosh bottle demonstration we use 92% Isopropyl alcohol.

QUESTION: Why is this reaction so spectacular whereas a cup of alcohol will simply light on fire?

Whoosh Bottle & CombustionAlcohol is flammable. In the whoosh bottle demonstration we use 92% Isopropyl alcohol.

QUESTION: Why is this reaction so spectacular whereas a cup of alcohol will simply light on fire?

ANSWER: A puddle of alcohol will evaporate into gas quicker than a puddle of water because alcohol has a low vapor pressure. Crushing a substance increases the surface area but those particles may still touch one another or be piled on one another reducing the surface area available for a reaction! Gas particles are very small and spread out leading to maximal surface area for a reaction.

I swirl the jug for a bit so the alcohol evaporates into a gas and has more surface area for the reaction. The same combustion reaction of lighting a cup of alcohol occurs, but this one occurs at a much faster rate due to increased surface area!

2 C3H7OH (l) + 9 O2 (g) ----> 6 CO2 (g) + 8 H2O (g)

Extensions of Surface Area: A puddle will evaporate much faster than a cup with the same amount of water. Granular sugar will dissolve much faster in water than sugar cubes. Small wood or kindling will burn much faster than a large log. Table salt will dissolve and dissociate in water much quicker than rock salt. Why? Rusting, Evaporation, Dissolving, Burning and any Chemical Reaction are all

proportional to the surface area of the object

DEMOS:Burning wood vs wood Dust!Burning or rusting Steel (solid) vs steel wool vs steel powderCornstarch burned as a powder vs blown.

THINK ABOUT IT:Why will steel wool combust better than steel powder?

Extensions of Surface Area: A puddle will evaporate much faster than a cup with the same amount of water. Granular sugar will dissolve much faster in water than sugar cubes. Small wood or kindling will burn much faster than a large log. Table salt will dissolve and dissociate in water much quicker than rock salt. Why? Rusting, Evaporation, Dissolving, Burning and any Chemical Reaction are all

proportional to the surface area of the object

DEMOS:Burning wood vs. wood Dust!Burning or rusting Steel (solid) vs. steel wool vs. steel powderCornstarch burned as a powder vs. blown.

THINK ABOUT IT:Why will steel wool combust better than steel powder?In order for combustion to occur you need O2. As a powder the steel particles are all on top of one another. There isn’t a lot of oxygen between them. Steel wool is “frizzy” and there is a lot of room and oxygen between the thin strands of metal.

Video/Demo: combustion of steel wool in an oxygen rich environment.