4Classical thermodynamics
II Statistical Physics
4.1 Zeroth and first law
We begin by defining some words.
Definition (Wall). A wall is a rigid boundary that matter cannot cross.
Definition
(Adiabatic wall)
.
Adiabatic walls isolate the system completely from
external influences, i.e. the system is insulated.
Definition
(Diathermal wall)
.
A non-adiabatic wall is called diathermal. Sys-
tems separated by a diathermal wall are said to be in thermal contact.
Definition
(Equilibrium)
.
An isolated system with a time-independent state is
said to be in equilibrium.
Two systems are said to be in equilibrium if when they are put in thermal
contact, then the whole system is in equilibrium.
We will assume that a system in equilibrium can be completely specified by
a few macroscopic variables. For our purposes, we assume our system is a gas,
and We will take these variables to be pressure
p
and volume
V
. Historically,
these systems are of great interest, because people were trying to build good
steam engines.
There will be 4 laws of thermodynamics, which, historically, were discovered
experimentally. Since we are actually mathematicians, not physicists, we will
not perform, or even talk about such experiments. Instead, we will take these
laws as “axioms” of the subject, and try to derive consequences of them. Also,
back in the days, physicists were secretly programmers. So we start counting at
0.
Law
(Zeroth law of thermodynamics)
.
If systems
A
and
B
are individually in
equilibrium with C, then A and B are in equilibrium.
In other words, “equilibrium” is an equivalence relation (since reflexivity and
symmetry are immediate).
In rather concise terms, this allows us to define temperature.
Definition
(Temperature)
.
Temperature is an equivalence class of systems with
respect to the “equilibrium” relation.
More explicitly, the temperature of a system is a quantity (usually a number)
assigned to each system, such that two systems have the same temperature iff
they are in equilibrium. If we assume any system is uniquely specified by the
pressure and volume, then we can write the temperature of a system as
T
(
p, V
).
We have a rather large freedom in defining what the temperature of a system
is, as a number. If
T
is a valid temperature-assigning function, then so is
f
(
T
(
p, V
)) for any injective function
f
whatsoever. We can impose some further
constraints, e.g. require that
T
is a smooth function in
p
and
V
, but we are still
free to pick almost any function we like.
We will later see there is a rather natural temperature scale to adopt, which
is well defined up to a constant, i.e. a choice of units. But for now, we can just
work with an abstract “temperature” function T .
We can now move on in ascending order, and discuss the first law.
Law
(First law of thermodynamics)
.
The amount of work required to change
an isolated system from one state to another is independent of how the work is
done, and depends only on the initial and final states.
From this, we deduce that there is some function of state
E
(
p, V
) such that
the work done is just the change in E,
W = ∆E.
For example, we can pick some reference system (
p
0
, V
0
), and define
E
(
p, V
) to
be the work done required to get us from (p
0
, V
0
) to (p, V ).
What if the system is not isolated? Then in general ∆
E 6
=
W
. We account
for the difference by introducing a new quantity Q, defined by
∆E = Q + W.
it is important to keep in mind which directions these quantities refer to.
Q
is the
heat supplied to the system, and
W
is the work done on the system. Sometimes,
this relation ∆
E
=
Q
+
W
is called the first law instead, but here we take it as
the definition of Q.
It is important to note that
E
is a function of state — it depends only on
p
and
V
. However,
Q
and
W
are not. They are descriptions of how a state
changes to another. If we are just given some fixed state, it doesn’t make sense
to say there is some amount of heat and some amount of work in the state.
For an infinitesimal change, we can write
dE = ¯dQ + ¯dW.
Here we write the
¯d
with a slash to emphasize it is not an exact differential in
any sense, because Q and W aren’t “genuine variables”.
Most of the time, we are interested in studying how objects change. We will
assign different labels to different possible changes.
Definition
(Quasi-static change)
.
A change is quasi-static if it is done so slowly
that the system remains in equilibrium throughout the change.
Definition
(Reversible change)
.
A change is reversible if the time-reversal
process is possible.
For example, consider a box of gases with a frictionless piston. We can very
very slowly compress or expand the gas by moving the piston.
If we take pressure to mean the force per unit area on the piston, then the work
done is given by
¯dW = −p dV.
Now consider the following two reversible paths in pV space:
V
p
A
B
The change in energy
E
is independent of the path, as it is a function of state.
On the other hand, the work done on the gas does – it is
Z
¯dW = −
Z
p dV.
This is path-dependent.
Now if we go around a cycle instead:
V
p
A
B
then we have ∆E = 0. So we must have
I
¯dQ =
I
p dV.
In other words, the heat supplied to the gas is equal to the work done by the
gas. Thus, if we repeat this process many times, then we can convert heat to
work, or vice versa. However, there is some restriction on how much of this we
can perform, and this is given by the second law.