*Michael Fowler 7/10/08*

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.

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*:

Evidently, heat *Q _{H}*
has been supplied to the gas—but this does

This is a perfectly well defined reversible route, ending at
the same place, but with quite a *different
amount of heat supplied*!

So we cannot say that a gas at a given (*P*, *V*) contains a definite
amount of heat. It does of course have a
definite internal energy, but that energy can be increased by adding a mix of
external work and supplied heat, and the two different routes from *a* to *c*
have the same total energy supplied to the gas, but with more heat and less
work along the top route.

However, notice that one thing (besides total internal
energy) *is* the same over the two
routes in the diagrams above:

*the** ratio of the heat supplied to the temperature at which is was
delivered*:

_{}

(This equation was derived in the last lecture.)

Of course, we’ve chosen two particular reversible routes
from *a* to *c*: each is one stretch of isotherm and
one of adiabat. What about more
complicated routes?

Let’s begin by cutting a corner in the previous route:

Suppose we follow the path *aefgc* instead of *adc*, where *ef* is an isotherm
and *fg* an
adiabat. Notice that *efgd* is a little
Carnot cycle. Evidently, then,

_{}

and

_{}

from which

_{}

But we can now cut corners on the corners: any zigzag route
from *a* to *c*, with the zigs
isotherms and the zags adiabats,
in other words, any reversible route, can be constructed by adding little
Carnot cycles to the original route. * In fact, any path you can draw in the plane
from a to c can be approximated arbitrarily well by a
reversible route made up of little bits of isotherms and adiabats.
*

We can just apply our “cutting a corner” argument again and again, to find

_{}

where the *i* labels intermediate isothermal
changes.

We can therefore define a ** new state variable**,

_{}

where the integral is understood to
be along *any* reversible path—they all
give the same result. So knowing _{} at a single point in
the plane, we can find its value everywhere, that is, for any equilibrium
state.

This means an ideal gas has four state variables, or
thermodynamic parameters: *P*, *V*, *T*,
*S*. Any two of them define the state (for
a given mass of gas), but all four have useful roles in describing gas behavior.
If the gas is *not* ideal,
things get a lot more complicated: it might have different phases (solid,
liquid, gas) and mixtures of gases could undergo chemical reactions. Classical thermodynamics proved an invaluable
tool for analyzing these more complicated systems: new state variables were
introduced corresponding to concentrations of different reactants, etc. The methods were so successful that a hundred
years ago some eminent scientists believed thermodynamics to be the basic
science, atomic theories were irrelevant.

In fact, for the ideal gas, we can find the entropy
difference between two states exactly!
Recall that the internal energy of a monatomic gas, the total kinetic
energy of the molecules, is _{} for *n* moles at temperature *T*.
For a diatomic gas like oxygen, the molecules also have rotational
energy because they’re spinning, and the total internal energy is then _{} (this is
well-confirmed experimentally). The
standard way to write this is

_{}

because if the gas is held at
constant volume, all ingoing heat becomes internal energy. Note that *C _{V}*
is for the amount of gas in question, it is

Suppose now we add heat *dQ*, but allow volume variation *dV* at the same
time. Then, for *n* moles of gas,

_{}

so

_{}

and doing the integrals

_{}

So the entropy change depends only on the final *T* and *V*, in other words, the final*
P* and *V* since we always have *PV* = *nRT*. This is just restating what we’ve already
established:

** S is a state variable**,

the (macroscopic) state of the
ideal gas is fully determined by *P*, *V* (or *P*, *T*), and therefore so
is the entropy *S*.

**This means we can get
to the final state any way we want, the
entropy change is the same, we don’t have to go by a reversible route! **

Irreversible processes are easy to find—just hold something
hot. When heat *Q* flows from a body at *T _{H}* to one
at

_{}

since the temperatures must be
different for heat to flow at all. There
is *no energy loss* in this heat
exchange, but there *is* a loss of *useful* energy: we could have inserted a
small heat engine between the two bodies, and extracted mechanical work from
the heat flow, but once it’s flowed, that opportunity is gone. This is often called a *decrease in availability* of the energy. To get work out of that energy now, we would have
to have it flow to an even colder body by way of a heat engine.

Note that the increase in entropy is for *the two bodies considered as a single system*. The hot body does of course lose
entropy. Similarly, when a heat engine
has less than Carnot efficiency, because some heat is leaking to the
environment, there is an overall increase in entropy of the *engine plus the environment*. When a reversible engine goes through a
complete cycle, its change in entropy is zero, as we’ve discussed,* and* the change in entropy of its
environment, that is, the two reservoirs (hot and cold) taken together is *also* zero: entropy is simply transferred
from one to the other.

**The bottom line is
that entropy change is a measure of
reversibility:**

**for a reversible process, total entropy change (system + environment) Δ***S*= 0,

**for****an irreversible process, total entropy***increases*, Δ*S*> 0.

Now consider the following scenario: we have a box of volume
2*V*, which is two cubes of volume *V* having a face in common, and that face
is a thin partition in the box.

At the beginning, *n*
moles of ideal gas are in the left-hand half of the box, a vacuum in the other
half. Now, suddenly, the partition is
removed. What happens? The molecules will fly into the vacuum, and in short
order fill the whole box. Obviously, they’ll never go back: this is
irreversible.

*Question*: What happens to the temperature of the
gas during this expansion?

*Answer*:
Nothing! There’s no mechanism for the molecules to lose speed as they
fly into the new space. (*Note*: we don’t need a molecular model to
see this—an ideal gas expanding against nothing does no work.)

*Question*: What happens to the pressure? Can you explain this?

*Question*: What is the entropy change? It is *nR*ln2.

But no heat flowed in on this route!

If we had followed a *reversible*
route, for example moving slowly along an isothermal, letting the partition
gradually retreat to one end of the box like a piston in a cylinder, we would
have had to supply heat.

That supplied energy would have all been used in pushing the
cylinder, the gas itself ending up in the same state as just removing the
partition. But we would have had a
nonzero_{}.

Just removing the partition quickly there is *no* heat transfer, and **this action doesn’t correspond to any path in the P, V plane**, since
each point in that plane represents a gas in equilibrium at that

Clausius realized that the entropy measured both something to do with heat content, but also how “spread out” a system was. This is clear for the ideal gas: the entropy change formula has two terms, one depending on the temperature difference, the other the volume difference. We shall return to this in the next lecture, when we examine entropy in the kinetic theory.

Notice the argument above only tells us entropy *difference* between two points, it’s a bit like potential energy. Actually, though,
there *is* a natural base point: a
system at absolute zero temperature has zero entropy. This is sometimes called the *Third Law of Thermodynamics*, or Nernst’s
Postulate, and can only be really understood with quantum mechanics. We don’t need it much for what we’re doing
here, we only work with entropy differences, but it makes things convenient
because we can now write *S*(*P*, *V*)
without ambiguity.