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
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