Comparing Surface Area and Volume
chemical reactions  hot air balloons
Chemical reations
The smaller an object is, the greater its surface area to volume
ratio is. This is important for chemical reactions, since more surface
area between the reactants will mean a faster reaction.
Let's compare the two simplest Rubik's Cubes in the picture above,
and assume that each small block is 2 cm on a side. The 3 x 3 x 3 cube
will then be 6 cm across, while the 2 x 2 cube will be 4 cm on a side.
3 x 3 x 3 cube:
surface area = 6 faces x 6 cm x 6cm = 216 cm^{2}
volume = 6 cm x 6 cm x 6cm = 216 cm^{3}
ratio of surface area:volume = 1:1
2 x 2 x 2 cube:
surface area = 6 faces x 4 cm x 4 cm = 96 cm^{2}
volume = 4 cm x 4 cm x 4cm = 64 cm^{3}
ratio of surface area:volume = 3:2
So in a reaction with particles similar to these Rubik's cubes, the
smaller particles, 2/3 the size of the larger ones, would react 50%
faster (all else being equal).
If we had one litre of particles, if it was just one solid cubic lump
it would have a surface area of 600 cm^{2}, but if it was ground
up into cubic lumps one millimeter on a side we would have a surface
area of 0.06 cm^{2} per lump, but with a million of these in
a litre the total surface area would be 60,000 cm^{2}, which
would produce a much faster reaction.
Note that a faster reaction isn't always what we want. If a reaction
occurs too quickly it can be very dangerous. In that case we would want
to have larger particles, to slow the reaction down. We take advantage of this when we put a large log on a log fire before going to bed. The bigger the log, the slower it'll burn.
Hot air balloons
Going the other way, the larger an object is, the lower its surface
area to volume ratio is. This is important for hot air balloons,
since less surface area per volume means less comparative weight of
the balloon for the hot air it contains, which means more lifting power
for the weight.
If we have a cubical balloon 1 metre across we'll have 1 m^{3}
of hot air for (roughly) 6 m^{2} of balloon (1:6 ratio). For
a cubic balloon 2 m on a side, we'll have 8 m^{3} of hot air
for 24 m^{2} of balloon surface area – twice as much lifting
power (1:3 ratio). A cubeshaped balloon just 10 cm on a side will contain
1 L (0.001 m^{3}) of hot air and have a surface area of 0.06
m^{2} – much less lifting power (1:60 ratio).
When Mythbusters made a helium balloon out of lead foil (thereby disproving
the idea that a lead balloon will invariably sink) they had to make
it reasonably large – three metres across – because their first attempt
didn't have enough lift (volume of helium) for the amount of lead
weight (surface area).
