ones, the faster ones slow down (losing heat) and the slower ones speed
up (gaining heat). Let’s assume that 1 unit of heat is transferred from a
small amount of water at temperature T 1 to an ice cube at temperature T 2.
Since water is warmer than ice, T 2 T 1.
Heat accounting is very similar to checkbook accounting; units gained
are viewed as positive (we add deposits in our checkbook), and units lost
are viewed as negative (we subtract checks or withdrawals, and subtract-
ing a positive number yields the same result as adding a negative one). So
the contribution to the change in entropy from the loss of the heat unit
from the small amount of water is 1/T 1. The contribution to the change
in entropy from the gain of the heat unit by the ice cube is 1/T 2. The to-
tal change in entropy from this heat transaction is 1/T 1 1/T 2 ; this ex-
pression is positive since T 2
T 1. As the water cools and the ice melts,
each one of these heat transactions changes the entropy by a positive
amount, and so the entropy of the system increases.
Once the system has reached equilibrium, with all the cubes melted and
the system at a uniform temperature, no more heat transactions can take
place and the glass of water is at maximum entropy. The glass of water is
a microcosm of what is happening in the universe. For the most part,
warm things are cooling and cool things are warming, entropy is increas-
ing, and we are headed toward a dim and distant future where everything
is at the same temperature, no more heat transactions can take place, and
things are really, really dull because nothing can happen. This is the so-
called heat death of the universe.
At least entropy doesn’t always increase everywhere at every time; the
second law only requires entropy to increase in reversible procedures,
and, fortunately, a lot of the really interesting procedures do not fall into
that category. The freezing of ice cubes, or the birth of a child, requires a
local decrease in entropy—but it is always at the expense of the increase
in entropy in the universe as a whole, because the universe must supply
heat to run the refrigerator to freeze the ice cubes, and in order to produce
the child it requires a lot of entropy in the form of material and energy.
Local decreases in entropy take place for a variety of reasons, not just
because you need to use electricity to run your refrigerator to make the
ice cubes. Gravity, of which there’s a lot lurking around the universe, con-
tributes to local decreases in entropy that help power our existence. A
cloud of hydrogen gas, when viewed strictly from the standpoint of its
thermodynamic properties, is a high-entropy system. What the thermo-
dynamic viewpoint fails to take into consideration is the role that gravity
plays in causing a local entropy decrease. The cloud, if it is large enough,
collapses under its own gravitation until its mass is dense enough to
190 How Math Explains the World