Thermodynamics versus Statistical Mechanics

bronzekerplunkΜηχανική

27 Οκτ 2013 (πριν από 3 χρόνια και 5 μήνες)

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Thermodynamics versus Statistical Mechanics

1.
Both disciplines are very general, and look for description of
macroscopic (many
-
body) systems in equilibrium

2.
There are extensions (not rigorously founded yet) to non
-
equilibrium processes in both

3.
But thermodynamics does not give definite quantitative answers
about properties of materials, only relations between properties

4.
Statistical Mechanics gives predictions for material properties

5.
Thermodynamics provides a
framework

and a
language

to discuss
macroscopic bodies without resorting to microscopic behaviour

6.
Thermodynamics is not strictly necessary, as it can be inferred
from Statistical Mechanics

1. Review of Thermodynamic and Statistical Mechanics

This is a short review

1.1. Thermodynamic variables

We will discuss a simple system:



one component (pure) system



no electric charge or electric or magnetic polarisation



bulk (i.e. far from any surface)

The system will be characterised macroscopically

by 3 variables:



N
,
number of particles

(
N
m

number of moles
)



V
,
volume



E
,
internal energy

only sometimes

in this case
system is
isolated

Types of thermodynamic variables:



Extensive
: proportional to system size



Intensive
: independent of system size

Not all variables are independent. The equations of state relate
the variables:


f (p,N,V,T) = 0

For example, for an ideal gas

or

N, V, E

(simple system)

p, T,
m

(simple system)

Boltzmann constant

1.3805
x
10
-
23

J K
-
1

No. of particles

Gas constant

8.3143 J K
-
1

mol
-
1

No. of moles

Any 3 variables can do. Some may be more convenient than
others. For example, experimentally it is more useful to consider
T
, instead of
E

(which cannot be measured easily)

Thermodynamic limit:

in this case
system is
isolated

in this case system
interchanges
energy with
surroundings

1.2 Laws of Thermodynamics

Thermodynamics is based on three laws

1.
First law of thermodynamics

SYSTEM

Energy,
E
, is a
conserved

and
extensive

quantity

(hidden)

(explicit)

change in energy
involved in
infinitesimal process

mechanical work
done
on

the system

amount of heat
transferred
to

the
system

proportional to
system size

in an isolated
system

1.2 Laws of Thermodynamics

Thermodynamics is based on three laws

1.
First law of thermodynamics

SYSTEM

Energy,
E
, is a
conserved

and
extensive

quantity

(hidden)

(explicit)

change in energy
involved in
infinitesimal process

mechanical work
done
on

the system

amount of heat
transferred
to

the
system

inexact differentials

W

&
Q

do not exist
(not state functions)

exact differential

E

does exist (it is a
state function
)

Thermodynamic (or macroscopic) work

are conjugate variables (intensive, extensive)

x
i

intensive
variable

m

-
p

-
H

...

X
i

extensive
variable

N

V

M

...

x
i
dX
i

m
dN

-
pdV

-
HdM

...

independent of
system size

(explicit)

(hidden)

SYSTEM

surroundings

In fact
dE = dW
tot
=

W +

Q

Only the part of
dW
tot

related to
macroscopic variables

can be
computed (since we can identify a displacement). The part related
to
microscopic variables

cannot be computed macroscopically
and is separated out from
dW
tot

as

Q

In mechanics:


where



and
F

is a conservative force

(explicit)

(hidden)

SYSTEM

surroundings

In fact
dE = dW
tot
=

W +

Q

Only the part of
dW
tot

related to
macroscopic variables

can be
computed (since we can identify a displacement). The part related
to
microscopic variables

cannot be computed macroscopically
and is separated out from
dW
tot

as

Q

A

system’s pressure =
F / A

F =
external force

volume change in
slow compression



mechanical work (through
macroscopic

variable
V
):



heat transfer (through
microscopic

variables):

molecules in base of container get
kinetic energy from fire, and
transfer energy to gas through
conduction (molecular collisions)

gas

if

the system
performs work

the system
adsorbs heat
from reservoir 1

the system
transfers heat to
reservoir 2

HEAT
ENGINE

Equilibrium state

A state where there is no change in the variables of the system

(only statistical mechanics gives a meaningful, statistical definition)

A change in the state of the system from one equilibrium state to
another

Thermodynamic process

specific volume

It can viewed as a trajectory in a
thermodynamic surface defined by
the equation of state

For example, for an ideal gas

initial
state

final
state

reversible
path



quasistatic process

a process that takes place so slowly
that equilibrium can be assumed at all
times. No perfect quasistatic processes
exist in the real world



irreversible process

unidirectional process: once it happens, it cannot be reversed
spontaneously



reversible process

a process such that variables can be reversed and the system would
follow the same path back, with no change in system or
surroundings. The system is always very close to equilibrium

A quasistatic process
is not necessarily
reversible

the wall separating the two parts
is slightly non
-
adiabatic (slow
flow of heat from left to right)

T
1

> T
2

Calculation of work in a process

The work done on the system on going
from state A to state B is

One has to know the equation of state
p = p (v,T)

of the substance

In a cycle
D
E = 0

but

-

work done by the
system

work done by the
system along the cycle

Therefore:

the heat adsorbed by the system is
equal to the work done by the system
on the environment

Types of processes



Isochoric
: there is no volume change



Isobaric
: no change in pressure

is the
enthalpy.

Also:

important in chemistry and
biophysics where most
processes are at constant
pressure (1 atm)

isobaric

isochoric



Isothermal
: no change in temperature, i.e.
dT = 0


For an ideal gas


(
ideal gas)


Adiabatic cooling

If the system
expands

adiabatically
W<0

and
E
decreases

(for an ideal gas and many systems this means
T
decreases: the gas
gets cooler)



Adiabatic heating

If the system
contracts

adiabatically
W>0

and
E
increases

(for an ideal gas this means
T
increases: the gas gets hotter)




Adiabatic
: no heat transfer, i.e.


Q = 0

Adiabatic cooling

work done by
the system

D
p
0

D
E<0

for an ideal gas
and many other
systems this means
D
T<0

V
A

V
B

2.
Second law of thermodynamics

There is an extensive quantity,
S
, called entropy, which is a state
function and with the property that

In an isolated system (E=const.), an adiabatic process from
state A to B is such that

In an infinitesimal process

The equality holds for reversible processes; if process is
irreversible, the inequality holds

D
S

can be easily calculated using statistical mechanics

the internal
wall is
removed

ideal
gas

expanded
gas

Example of irreversible process

entropy of ideal gas in volume V

entropy of ideal gas in volume V/2

V/2

V

V/2

Arrow of time

isolated
system

The entropy of an ideal gas is



entropy before:



entropy after:



entropy change:

The inverse process involves
D
S<0

and is in principle prohibited



at equilibrium it is a function




it is a monotonic function of

E

The existence of
S

is the price to pay
for not following the hidden degrees
of freedom.

It is a
genuine thermodynamic

(non
-
mechanical)
quantity

An adiabatic process involves changes
in hidden microscopic variables at fixed
(N,V,E).
In such a process

maximum

(N,V,E)

time evolution from
non
-
equilibrium
state

S

is a
thermodynamic potential
: all thermodynamic quantities can
be derived from it (much in the same way as in mechanics, where
the force is derived from the energy):

Since

S

increases monotonically with
E
, it can be inverted to give
E = E(N,V,S)

entropy

representation of thermodynamics

energy

representation of thermodynamics

equations of state

equations of state

Equivalent

(more utilitarian) statements of 2
nd

law

Kelvin
: There exists no thermodynamic process
whose sole effect is to extract heat from a system
and to convert it entirely into work


(the system releases some heat)

As a corollary
: the most efficient heat engine
operating between two reservoirs at temperatures
T
1

and
T
2

is the Carnot engine

Clausius
: No process exists in which
the sole effect is that heat flows from
a reservoir at a given temperature to
a reservoir at a higher temperature

(work has to be done on the system)

Clausius

Lord Kelvin

Carnot

Historically they reflect the early understanding
of the problem

S

is connected to the energy transfer through hidden degrees of
freedom, i.e. to

Q
. In a process the entropy change of the system is

where

reversible process

irreversible process

If

Q > 0

(heat from environment to system)
dS > 0

In a finite process from
A

to
B
:

For reversible processes

T
-
1

is an integrating factor, since
D
S

only
depends on
A

and
B
, not on the trajectory

alternative
statement of
2
nd

law

The name entropy was given by Clausius in 1865 to
a state function whose variation is given by

dQ/T
along a reversible process

where
p
i

is the probability of the system being in a
microstate
i

If all microstates are equally probable (as is the case if
E

= const.) then
p
i

=1/
W
, where
W

is the number of
microstate of the same energy
E
, and

It can be shown that this
S

corresponds to the thermodynamic
S

by Boltzmann in terms of probability arguments in
1877 and then by Gibbs a few years later:

CONNECTION
WITH ORDER

More order means less
states available

W
ahrscheindlichkeit

(probability)

Gibbs

A clearer explanation of entropy was given

Clausius

Boltzmann

Does S always increase? Yes. But beware of environment...

In general, for an open system:

entropy change due to
internal processes

entropy change due to
interaction with
environment

>

<

entropy change of
environment

entropy change of system

may be
positive

or
negative

e.g. living beings...

Processes

can be discussed profitably using the entropy concept.

For a reversible process:



If the reversible process is
isothermal
:

S

increases if the system absorbs heat, otherwise
S

decreases

Reversible isothermal processes are
isentropic

But in irreversible ones the entropy may change



If the reversible process is
adiabatic
:

In a finite process:

(depends on the trajectory)

In a cycle:

work done by the system in the cycle

Q

heat absorbed
by system

Q

D
S = 0

CARNOT CYCLE

Q=
-
W

Isothermal process.
Heat Q
1

is absorbed


Adiabatic process.
No heat


Isothermal process.
Heat Q
2

is released


Change of
entropy:

T
1

T
2

Adiabatic process.
No heat


it is impossible to
perform a cycle with
W 0 and Q
2

= 0

Efficiency of a Carnot heat engine:

Carnot theorem:

The efficiency of a cyclic Carnot heat engine only depends on the
operating temperatures (not on material)

By measuring the efficiency of a real engine, a temperature T
2

can be determined with respect to a reference temperature T
1