Essays on Science for the Common Good

(Vol. Two) 

XXIV The Concept of Surface area and Volume

The concepts of surface area and volume are important concepts in both the physical (nonliving) world and the biological (living) world.  Now, combine those attributes into a single relationship, the surface area to volume ratio and you have a whole new phenomenon that has enormous ramifications.  Let’s look at the relationship in the physical world first.

Let’s say you have a chunk of coal about the size of a large rock that you, want to burn and all you have to light it is a single match. What chance do you think you have to light it?  Now you break that chunk of coal into a few smaller chunks.  Can you light one? Perhaps with the aid of some fuel such as charcoal lighter fluid and paper “starters” you may succeed.  Now pulverize those chunks into fine powder and light the match… You better have a “long reach” or you may take a trip to the ER at a local hospital. Why the difference?  Let’s look at the mathematics of what you happens to the SA:V ratio when you break it up into smaller pieces.  But first let’s consider what you need to burn– anything. From science we know that three conditions must be met. We need:

        • a combustible material (i.e. coal)

        • a supply of oxygen.

        • enough heat to ignite the substance (bring it to its kindling temperature)

For now let’s focus on the oxygen supple.

The formula for calculating the surface area of a cube, as an example, is length x width × number of sides (6) or SA = 6s² and let’s compute that for a 1 cm cube:

SA = 1x1x 6 (6 x 1²) = 6cm³                                   and for a 2 cm cube:

SA = 2 x2x6 (6 x 2²⁾ = 24 cm³                and for a 3cm³ cube

SA = 3×3×6 (6×3²⁾ = 54 cm³

Now let’s figure the volume for each of these. The formula is length x width x height (l³⁾

        V for a 1 cm³ = 1x1x1 (1³) = 1cm³

        V for a 2cm³ = 2x2x2 (2³) = 8 cm³

        V for a 3 cm³ = 3x3x3 (3³) = 27 cm³

Comparing:       SA:V

1cm³                =   6:1

2cm³ = 24: 8 =    3: I

3cm³ = 54: 27 = 2: 1

Now   6:1 vs. 2:1 =3:1 = 3

which means that a l cm³ has three times as much surface are as a 3cm3 which means it has three times as much contact with oxygen in the air which also means that a fine powder’s                                                                                                                                             contact with oxygen compared to a large rock size chunk of coal is enormous, more like humongous when you consider it has 10^umteenth  pieces.

This principle, of course, applies to all combustible materials and even extends to other scenarios besides burning.  It applies to baby powder as well as gunpowder, and to highway salt as well as fine fertilizer pellets.

    Now let’s consider the concept of increased surface area in living systems. In our lungs we have tiny microscopic air sacs called alveoli, millions of them that increase the surface area of our lungs for the exchange of both oxygen and carbon dioxide, oxygen into and carbon dioxide out of the lungs. Remember from past essays that CO₂ is a product of cellular respiration and is toxic to living cells. The alveoli are richly supplied with capillaries where CO₂ from body cells is exchanged for O₂ from outside thebody.  See the diagram below. Red indicates O₂ rich blood and blue represents high CO₂ blood. The whole process is called diffusion.

Obviously, the millions of alveoli inside our lungs increase the surface area and enable an animal our size or larger to carry on life’s activities without an enormous set of lungs.  Next, let’s consider a unique group of animals that are adapted for breathing in the water and on land, amphibians.  Many of my students were surprised to find such small lungs during dissection—until they learned that frogs breathe subcutaneously, through their blood vessel rich skin.  Think of the size of the surface area, basically all of the body exposed to water. 

Your small intestine contains mullions of microscopic finger-like projections called villi which allows digested food to be absorbed by their rich supply of blood in, you guessed it, the capillaries This includes active transport, which requires energy (from ATP breakdown) and passive transport which does not require energy (diffusion and facilitated diffusion).

Our brains are convoluted (wrinkled) on the surface to increase the surface area. There are other examples of the extremely important concept of increased area of concentration. Let’s consider cell reproduction next.

Why do cells reproduce? Obviously one reason is to replace worn out or dead cells.  However, a lesser known reason is to increase cell efficiency.  As cells grow their surface area to volume ratios goes down. Once again, compare the SA: ratio of the 3 cm³ to the l cube. Remember all substances enter and leave the cell through the cell membrane (surface area). So as cells grow they become less efficient at bringing in nutrients and getting rid of toxic substances. Cell reproduction (mitosis and cytokinesis) replaces large cells by smaller more efficient cells.

I hope you have a greater understanding of, and appreciation for, the concept of increased surface area in our daily lives.

References

Mader, S. Biology, eighth edition (2004) McGraw-Hill, New York, NY