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.