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A New Thermodynamic Variable: Entropy
Michael Fowler 7/10/08
Introduction
The word “entropy” is sometimes used in everyday life as a synonym for chaos, for example: the
entropy in my room increases as the semester goes on.
But it’s also been used to describe the
approach to an imagined final state of the universe when everything reaches the same
temperature: the entropy is supposed to increase to a maximum, then nothing will ever happen
again.
This was called the Heat Death of the Universe, and may still be what’s believed, except
that now everything will also be flying further and further apart.
So what, exactly, is entropy, where did this word come from?
In fact, it was coined by Rudolph
Clausius in 1865, a few years after he stated the laws of thermodynamics introduced in the last
lecture.
His aim was to express both laws in a quantitative fashion
.
Of course, the first law—the conservation of total energy including heat energy—is easy to
express quantitatively: one only needs to find the equivalence factor between heat units and
energy units, calories to joules, since all the other types of energy (kinetic, potential, electrical,
etc.) are already in joules, add it all up to get the total and that will remain constant. (When
Clausius did this work, the unit wasn’t called a Joule, and the different types of energy had other
names, but those are merely notational developments.)
The second law, that heat only flows from a warmer body to a colder one, does have quantitative
consequences: the efficiency of any reversible engine has to equal that of the Carnot cycle, and
any nonreversible engine has less efficiency.
But how is the “amount of irreversibility” to be
measured?
Does it correspond to some thermodynamic parameter?
The answer turns out to be
yes
: there is a parameter Clausius labeled entropy that doesn’t change in a reversible process, but
always increases in an irreversible one.
Heat Changes along Different Paths from
a
to
c
are Different!
To get a clue about what stays the same in a reversible cycle, let’s review the Carnot cycle once
more.
We know, of course, one thing that doesn’t change: the internal energy of the gas is the
same at the end of the cycle as it was at the beginning, but that’s just the first law.
Carnot
himself thought that something else besides total energy was conserved: the heat, or caloric fluid,
as he called it.
But we know better: in a Carnot cycle, the heat leaving the gas on the return
cycle is
less
than that entering earlier, by just the amount of work performed.
In other words, the
total amount of “heat” in the gas is
not
conserved, so talking about how much heat there is in the
gas is meaningless.
To make this explicit, instead of cycling, let’s track the gas from one point in the (
P
,
V
) plane to
another, and begin by connecting the two points with the first half of a Carnot cycle, from
a
to
c
: