Changing Surface Area and Volume with Changing Shape - NCLT

Sample Student Activity
Changing Surface Area and Volume
with Changing Shape
When you are cold and not in a position to put on more clothes, you keep
your arms close to the rest of your body. If you were really cold, you may sit
hugging your knees to your chin. Most land mammals, lizards, and snakes
"curl up" to make their bodies as compact as possible in a cold environment.
Bears, for example, can become almost spherical. Conversely, when animals
are hot, they tend to stretch out.
An individual animal can control its shape-up to a point. That point is
determined by the animal's anatomy. In ecology, there is an anatomical
rule-of-thumb, or general guideline, called Bergmann's Rule. It applies to
"warm-blooded" animal species that have populations that live in both warm
climates and cold climates. It simply states that there is an inverse
relationship between average ambient temperature and animal size.
Examples of mammals include white-tailed deer, which are larger in Canada
than in the Florida Keys. The common wolf is 20% larger in northern
Canada than in northern Mexico. At latitudes greater than 30 o , both south
and north, mountain lions are larger than those closer to the equator. Flying
squirrels in the Arctic are more than 50% larger than those in Central
America. The smaller species have a proportionally larger surface area per
unit volume compared to their larger counterparts. Therefore, the smaller
animal has exterior surface for the same volume available to dissipate heat
in the warmer climates.
Scientists have discovered that this kind of surface effects also has a
dramatic impact on nanoscale materials when they become so small. The
melting point of substances is normally a constant characteristic (as long as
pressure is kept constant). In fact, it can be used to identify substances. In
Shown at the same scale: Emperor
penguins (left) in Antarctica are bigger
than Fairy penguins (right) in Australia
and New Zealand.
metals, however, scientists have observed large decreases in melting points as the size of metal particles
decreases in the nanoscale. The melting point for gold that we can hold in our hands is 1064 o C. If this piece of
gold is broken into smaller and smaller pieces, the melting point of each is still 1064 o C—until the pieces get
smaller than 1 µm. Then the melting point starts to decrease, very little at first. It is not until the gold particle
gets to 20 nm that the melting point really drops, when the amount of surface area per unit volume increases by
a million times. It has been measured for particles as small as 2 nm to be 350 o C, and by extrapolation, it would
be 200 o at 1 nm. Another property change that is related to increase in surface area is chemical reactivity. The
surface area per volume increase due to the nano-size of the catalytic materials within the catalytic converter
volume of 1000 - 2000 cm 3 is impressive: it is the area of about 10 football fields!
Size is an ambiguous word, although we generally understand it to mean volume. This activity deals with the
quantities: surface area and volume. As an object decreases in size, obviously its volume (V) decreases, and so
does its surface area (SA), but not at the same rate. A major reason that the nanoscale is special involves the
scaling down of the surface area to volume ratio: SVR. You will explore this ratio for geometric objects.
Try This
You will need: 32 identical sticks and 27 linking cubes.
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Sample Student Activity
A. Ideal 2-D Objects
___1. Make predications: 1) Sketch a rectangle that you think has the smallest possible ratio of perimeter to
area (P/A). 2) Sketch a rectangle that has about the same A as the rectangle you sketched in step 1, but
has a larger ratio of P/A.
___2. Obtain 32 sticks. Divide the sticks into 4 groups of 8 sticks. You will make four polygons, as described
below. For each polygon, use one group of sticks, making sure to use all 8 sticks.
a) a rectangle with the minimum perimeter to area ratio (P/A)
b) a rectangle with the maximum P/A
c) a polygon with the minimum P/A
d) a polygon with the maximum P/A
___3. Sketch each of the four polygons you made. Discuss with your teacher which measurement units to use.
B. Properties of 3-D Objects
___1. Make a predication: Read steps 1 and 2 of the Procedure below. Predict a
shape for the minimum and the maximum ratios of surface area to
volume (SVR).
___2. Each group should obtain at least 24 linking cubes. Divide the linking
cubes into 3 groups of 8 linking cubes. You will make two or three
shapes, as described below. For each shape, use one group of linking
cubes, making sure to use all 8.
a) Make a shape with the minimum SVR.
b) Make a shape with the maximum SVR.
Is there a different shape with the same maximum SA/V? If so, create it.
___3. Record a description of each shape. Don't forget units for SA and V.
___4. Consider SVR for the two cubes illustrated on the right. Which do you think has a
larger SVR? Make a prediction. 2) Imagine more cubes of different sizes. Predict
the shape of a graph of SA vs. L (the length of an edge) and SVR vs. L
for the cubes. Sketch your predicted graph.
___5. Create more cubes to explore the relationships between
V vs. L, SA vs. L, SVR vs. L, SVR vs. 1/L.
Before making the cubes, predict the shape of the graph for each of
these relationships.
Questions
1. How did your results compare with your predictions? Were any of the results unexpected? Explain.
2. Consider how you could make P/A for a rectangle larger and larger. What is the limit?
3. Which (SA or V) increases faster with L? How do you know?
5. If you had obtained data for the same variables but for a different shape, let's say a sphere rather than a
cube, how would the graphs differ?
4. For the SVR vs. L graph, is there a minimum ratio of SA to V? Is there a maximum ratio of SA to V?
6. The graph created SVR vs. L leaves something to be desired with respect to illustrating the geometric
relationship for very small L. Create a log-log graph of the data.
7. A sucrose molecule has L ~ 0.4 nm. For the SVR vs. L graph you created, indicate approximately where a
sucrose molecule should be plotted.
Information from the World Wide Web
1. An interactive cube building simulation for creating larger block of cube is available at the National Center for
Learning and Teaching (NCLT) in Nanoscale Science and Engineering’s NanoEd Resource Portal://nanoed.org
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