Transcript Diapositiva 1 - Universidad Autonoma de Madrid

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 nonequilibrium 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
only sometimes
• 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 (Nm number of moles)
• V, volume
• E, internal energy
in this case
system is
isolated
Types of thermodynamic variables:
• Extensive: proportional to system size
N, V, E (simple system)
• Intensive: independent of system size
p, T, m (simple system)
Not all variables are independent. The equations of state relate
the variables:
f (p,N,V,T) = 0
For example, for an ideal gas
pV  NkT
No. of particles
Boltzmann constant
1.3805x10-23 J K-1
or
pV  N m RT
No. of moles
Gas constant
8.3143 J K-1 mol-1
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)
in this case
system is
isolated
in this case system
interchanges
energy with
surroundings
Thermodynamic limit:
N
N  , V  ,    
V
1.2 Laws of Thermodynamics
in an isolated
Thermodynamics
is based on three laws
system
proportional to
system size
1. First law of thermodynamics
Energy, E, is a conserved and extensive quantity
dE  W  Q
change in energy
involved in
infinitesimal process
(explicit)
mechanical work
done on the system
SYSTEM
amount of heat
transferred to the
system
(hidden)
1.2 Laws of Thermodynamics
Thermodynamics is based on three laws
inexact differentials
exact differential
1. First
law of thermodynamics
W & Q do not exist
E does exist (it is a
(not state functions)
state function)
Energy,
E, is a conserved and extensive quantity
dE  W  Q
change in energy
involved in
infinitesimal process
(explicit)
mechanical work
done on the system
SYSTEM
amount of heat
transferred to the
system
(hidden)
Thermodynamic (or macroscopic) work
W   xi dXi  x1dX1  x2 dX2  ...
i
independent of
system size are conjugate variables (intensive, extensive)
i
i
x , X 
xi
intensive
variable
Xi
extensive
variable
xidXi
m
-p
-H
...
N
V
M
...
mdN
-pdV -HdM
...
surroundings
 0 dW
0
(explicit)
0
In mechanics:
 dE
where  0
SYSTEM 1  (hidden)
  F  dr  E1  E0
dE  E1  EW
0
0
and F is a conservative force
In fact dE = dWtot= W + Q
Only the part of dWtot 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 dWtot as Q
surroundings
0
0
(explicit)
0
SYSTEM
0
(hidden)
dE  E1  E0
In fact dE = dWtot= W + Q
Only the part of dWtot 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 dWtot as Q
system’s pressure = F / A
F = external force
A
volume change in
slow compression
• mechanical work (through
macroscopic variable V):
gas
W   pdV  0 if dV  0
• heat transfer (through
microscopic variables):
molecules in base of container get
kinetic energy from fire, and
transfer energy to gas through
conduction (molecular collisions)
Q  0
the system
adsorbs heat
from reservoir 1
the system
performs work
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)
Thermodynamic process
A change in the state of the system from one equilibrium state to
another
It can viewed as a trajectory in a
thermodynamic surface defined by
the equation of state
For example, for an ideal gas
f  p, v, T   pv  kT  0
v  V / N  1/ 
specific volume
f  p, v, T 
initial
state
reversible
path
final
state
• 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
• 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
the wall separating the two parts
is slightly non-adiabatic (slow
flow of heat from left to right)
T1 > T2
A quasistatic process
is not necessarily
reversible
• irreversible process
unidirectional process: once it happens, it cannot be reversed
spontaneously
Calculation of work in a process
work done by the
system
The work done on the system on going
from state A to state B is
VB
WAB    pdV
-
VA
One has to know the equation of state
p = p (v,T) of the substance
work done by the
system along the cycle
In a cycle DE = 0 but W  0
Therefore:
Q  W   pdV
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
dE  Q  dQ
DE   dQ  Q
isochoric
dV  0  W  0
isobaric
• Isobaric: no change in pressure
Q  pdV  dE  d  pV   dE  d E  pV   dH
H  E  pV is the enthalpy.
Q   dH  DH
Also:
important in chemistry and
biophysics where most
processes are at constant
pressure (1 atm)
VB
W    pdV   pVB  VA    pDV
VA
• Isothermal: no change in temperature, i.e. dT = 0
For an ideal gas
3
E  NkT  dE  0  Q  W
2
Q  W
(ideal gas)
• Adiabatic: no heat transfer, i.e. Q = 0
dE  dW  DE  W
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 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 cooling
Dp0
DE<0
for an ideal gas
and many other
systems this means
DT<0
work done by
the system
VA
VB
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
S A  SB
In an infinitesimal process
dS  0
The equality holds for reversible processes; if process is
irreversible, the inequality holds
Example of irreversible process
isolated
system
the internal
wall is
removed
ideal
gas
V/2
V/2
expanded
gas
Arrow of time
V
DS can be easily calculated using statistical mechanics
DS  Sfinal  Sinitial  0
entropy of ideal gas in volume V
entropy of ideal gas in volume V/2
The entropy of an ideal gas is

S  S 0  Nk log vT
• entropy before:
3/ 2

V
, v
N
 v 3/ 2 
Sinitial  S0  Nk log T 
2


• entropy after:
Sfinal  S0  Nk logvT
• entropy change:
3/ 2

DS  Sfinal  Sinitial  Nk log 2
The inverse process involves DS<0 and is in principle prohibited
The existence of S is the price to pay
time evolution from
for not following the hidden degreesnon-equilibrium
of freedom.
state
(N,V,E)
It is a genuine thermodynamic (nonmechanical) quantity
• at equilibrium it is a function
S  S ( N ,V , E )
• it is a monotonic function of E
An adiabatic process involves changes
in hidden microscopic variables at fixed
(N,V,E). In such a process
S  S N ,V , E; i 
maximum
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):
equations of state
1  S 
p  S 
m
 S 
  , 
 ,  

T  E  N ,V T  V  N , E T
 N V , E
Since S increases monotonically
E, it can be inverted to give
equationswith
of state
E = E(N,V,S)
 E 
 E 
 E 
T    , p  
 ,m  

 S  N ,V
 V  N ,S
 N V ,S
S  S ( N ,V , E )
entropy representation of thermodynamics
E  E ( N ,V , S )
energy representation of thermodynamics
Equivalent (more utilitarian) statements of 2nd law
Historically they reflect the early understanding
of the problem
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)
Clausius
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)
As a corollary: the most efficient heat engine
operating between two reservoirs at temperatures
T1 and T2 is the Carnot engine
Lord Kelvin
Carnot
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
alternative
statement of
2nd law
dS 
Q

reversible process

irreversible process
where
T
If Q > 0 (heat from environment to system) dS > 0
B
In a finite process from A to B:
DS  
A
Q
T
For reversible processes T-1 is an integrating factor, since DS only
depends on A and B, not on the trajectory
The name entropy was given by Clausius in 1865 to
a state function whose variation is given by dQ/T
along a reversible process
A clearer explanation of entropy was given Clausius
by Boltzmann in terms of probability arguments in
1877 and then by Gibbs a few years later:
Gibbs
S  k  pi log pi
It can be shown that this S
corresponds to the thermodynamic S
i
where pi is the probability ofCONNECTION
the system being in a
microstate i
WITH ORDER
Wahrscheindlichkeit
(probability)
If all microstates are equally probable (as is the case if
More order means less
E = const.) then pi =1/W, where
W isavailable
the number of
states
microstate of the same energy E, and
1
1
1
S  k  log  kW log W  k log W
W
W
i W
Boltzmann
Does S always increase? Yes. But beware of environment...
In general, for an open system:
dS  dSi  dSe  dSenv  0
dSi  0
entropy change due to
internal processes
dSe <> 0
entropy change of
environment
entropy change due to
interaction with
environment
entropy change of system
may be positive or negative
e.g. living beings...
Processes can be discussed profitably using the entropy concept.
B
For a reversible process:
Q
DS  S B  S A  
• If the reversible process is isothermal:
A
T
B
1
Q
DS  S B  S A   Q   Q  TDS
TA
T
S increases if the system absorbs heat, otherwise S decreases
• If the reversible process is adiabatic:
B
DS  S B  S A  
A
Q
T
 0  DS  0
Reversible isothermal processes are isentropic
But in irreversible ones the entropy may change
B
In a finite process:
Q   TdS (depends on the trajectory)
A
In a cycle:
Q   TdS  W  work done by the system in the cycle
Q
Q
heat absorbed
by system
DS = 0
CARNOT CYCLE
Isothermal process.
Heat Q1 is absorbed
DS AB  Q1 /T1
T1
Adiabatic process.
No heat
DS DA  0
Q=-W
T2
Change of
entropy:
Q1 Q2
DS 

0
T1 T2
Adiabatic process.
No heat
DS BC  0
Isothermal process.
Heat Q2 is released
DSCD  Q2 /T2
Q1 Q2
Q1 T1



  1  Q1  Q2  0
T1 T2
Q2 T2
it is impossible to
perform a cycle with
W  0 and Q2 = 0
Efficiency of a Carnot heat engine:
work done by system
W


heat absorbed by system Q1

T1  T2 S 2  S1 
T2

 1
T1 S 2  S1 
T1
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 T2
can be determined with respect to a reference temperature T1