Surface/Volume: How Geometry Explains Why Grain Elevators Explode, Hummingbirds Hover, and Asteroids are Colder than Ice

187 Pages·2023·8.462 MB·English

by  Alan E. Rubin

Alan E. Rubin
Surface/Volume
How Geometry Explains Why
Grain Elevators Explode,
Hummingbirds Hover, and
Asteroids are Colder than Ice
Surface/Volume
Alan E. Rubin
Surface/Volume
How Geometry Explains Why Grain
Elevators Explode, Hummingbirds Hover,
and Asteroids are Colder than Ice
Alan E. Rubin
Department of Earth, Planetary, and Space
Sciences
University of California
Los Angeles, CA, USA
ISBN 978-3-031-23748-5 ISBN 978-3-031-23749-2 (eBook)
https://doi.org/10.1007/978-3-031-23749-2
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A nanosphere is very small
It is a very tiny ball
Its atoms—a surficial field
But little volume is concealed.
The planet Earth is quite immense
The heat inside is quite intense
It takes so very long to cool
The S/V ratio makes this rule.
For Geometers
and those who move in their circles
Introduction
Hardy tourists can hike up the K¯ılauea volcano on the big island of Hawai’i. It is
an 8.5-km round trip from the Visitor Center along Crater Rim Trail and around
the K¯ılauea Iki Trail loop. The terrain is rugged with deep cracks and razor-sharp
lava. K¯ılauea is probably the most active volcano in the world, erupting, on average,
every two to three years. A series of magma chambers beneath K¯ılauea (at depths
of 3 km, 30 km, and 90 km) provides the molten rock. During eruptions, low-viscosity
basaltic lava pours from volcanic vents and flows downslope. At night, visitors can
see a river of thin, slowly undulating red-orange stripes surrounding dark amoeboid
slabs of congealed lava.
Whether K¯ılauea is erupting or not, once a month, if skies are clear, the full moon
bathes the volcano in pale white light. The Moon’s nearside is covered with basaltic
lava plains similar in composition to basalts from K¯ılauea. But the mantle of the
Moon is cold; major volcanic activity on the Moon ended about 2½ billion years
ago.
Why do volcanoes erupt every day on Earth, but not on the Moon?
It is a matter of size and geometry. The radius of the Earth is 3.7 × that of the
Moon. The surface area of the Earth (continents and oceans combined) is about 13 ×
that of the Moon; the volume of the Earth is 49 × that of the Moon. The difference
in volume far exceeds the difference in surface area (49 vs. 13). This is because
volume increases as the third power of radius (r × r × r) while surface area increases
only as the second power (r × r). This geometric constraint ensures that the (surface-
area)/volume ratio of the Earth is less than 30% that of the Moon. The Earth therefore
has a lot of volume (in which heat can be generated), but comparatively little surface
area from which interior heat can escape. Temperatures in the Earth’s lithosphere can
exceed 1300ºC; hence, the active volcanoes. The Moon has a much smaller volume
and a relatively high amount of surface area; this geometric property allowed most
of the Moon’s internal heat to escape into space early in Solar System history.
The surface-area/volume ratio (commonly simplified as surface/volume)
constrains much of the physical world. Small objects lose heat faster than large ones.
This general property applies not only to planets and moons but also to mammals and
birds. Small animals lose heat quickly. To avoid shivering or freezing to death, small
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x Introduction
vertebrates must maintain a high metabolic rate. While an elephant’s heart may beat
25 times a minute, a shrew’s heart beats more than 600 times a minute. To go about
its daily activities, a house mouse (two-to-three times the size of a shrew) requires
about 40 times as much energy per unit mass as an elephant.
Surface/volume ratios also influence the structures of plants. Leaves from decid-
uous trees are broad and flat with high surface/volume ratios. The large surface area
captures a lot of sunlight; the thin leaf structure assures that the Sun’s rays do not
have far to travel before entering the chloroplasts to facilitate photosynthesis.
The rates of chemical processes are also governed by geometry. Example 1—
Dissolution: Small soluble substances (e.g., salt crystals) have high surface/volume
ratios and dissolve readily. A large fraction of the molecules in the crystals is near
the surface and can be easily dislodged when soluble crystals are immersed in
water. Example 2—Evaporation: The amount of liquid that evaporates from an open
container is a direct function of the surface area of the exposed liquid. As an illus-
tration, let’s put two open-top containers atop the dry scrub in California’s Mojave
Desert; each container holds 1 cubic meter of water. The first container is a cube, 1
m on a side; the exposed surface is 1 m2. The second container is shaped like a giant
sheet pan: 10 m × 10 m × 1 cm; the exposed surface is 100 m2. Water from the
second container evaporates 100 times faster than from the cube (even though both
containers initially held the same amount of water).
Nanoparticles have extremely high surface/volume ratios, these tiny objects are
nearly all surface with little enclosed volume. A particle 1 nm (a billionth of a
meter) across has a surface/volume ratio eight million times greater than that of a
pea. Because surface forces predominate in nanometer-size metal particles, these
particles melt at much lower temperatures than centimeter-size metal nuggets.
Because mass, like volume, is a three-dimensional property, many physical struc-
tures are constrained by their surface/mass ratios. Architect Frank Lloyd Wright’s
dream of a mile-high skyscraper in Chicago will never materialize. It would be too
difficult for such a structure to support its own weight, even though the building was
designed to narrow toward the top.
The effective strength of muscles is also constrained by the surface/mass ratio.
Raw muscle strength is directly related to a muscle’s cross-sectional area, increasing
as the square of length. Muscle mass, however, increases as the cube of length. The
result is that large birds have such heavy muscles they lack the power to hover, but
diminutive hummingbirds have no trouble at all.
Scientists can deduce basic principles from the geometric constraints imposed
on physical structures: Small bodies lose heat faster than large bodies. Chemical
reactions proceed more rapidly in small objects. Large bodies tend to be so heavy
that steps must be taken to reduce the weight—this accounts for the high proportions
of (low-density) spongy bone in dinosaurs and the multiple-mirror designs of many
modern giant optical telescopes.
Introduction xi
The concept of the surface/volume ratio is of enormous importance in under-
standing the physical world. It explains how rapidly a warm object will lose heat
to its surroundings—whether the object is a planet or a gerbil. It governs how effi-
ciently sunlight can penetrate leaves and facilitate photosynthesis. It accounts for the
numerous folds in the human brain, explains why our small intestine is all coiled up
and why krill have finely branched appendages. It shows why we are much more
likely to get frostbite on our toes and fingers than our legs and arms. And it warns
us to be cautious around grain elevators.
Other Books by Alan E. Rubin
Disturbing the Solar System: Impacts, Close Encounters, and Coming Attractions.
Princeton University Press, 2002, Princeton, New Jersey. 376 pp.
Son of Man: A Personal Memoir by Yehoshuah ben Yahweh, Translated and Edited
by Arthur Melton and Monica Wheatley. A novel. 2015, CreateSpace. 147 pp.
With Chi Ma: Meteorite Mineralogy. Cambridge University Press, 2021, Cambridge.
404 pp.
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