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Lecture 4
•
Equilibrium and noequilibrium processes.
•
Adiabatic, isotermic, isobaric and isochoric processes.
•
Connection
quantities.
•
Helmholtz free energy
Energy G.
•
Thermodynamic potentials and Heat capacity.
•
The laws of thermodynamics.
•
Thermodynamic functions for the canonical ensemble.
•
Partition functions.
•
Alternative expression for the partition function.
•
Density of states.
•
A system of harmonic oscillators.
between
statistical
and
thermodynamic
F, Enthalpy H and Gibbs Free
1
Equilibrium and noequilibrium process
The typical example of nonreversible ( in statistical sense)
process is the relaxation process.
The process is reversible if during every it’s moment the
system is in equilibrium state and the process can go any
direction. The reversible processes are usually connected
with some variations of external conditions and the energy
of the system.
The variations have to be so slow that the system can
reach equilibrium. Very slow process can be defined as
quasystatic. The meanings «slow» depends on process
time and have to be compared with the relaxation time.
Adiabatic processes can be defined as the processes at
the
constant
material and temperature conditions.
Isothermal, isobar and isochoric processes are going at a
constant temperature, pressure and volume respectively2
Connection between Statistical and
Thermodynamic Quantities
We have seen that for a system in equilibrium =(E,x,Ni),
where E is the energy; x denote the set of external
parameters describing the system; and the Ni are the
numbers of molecules of the several chemical species
present. If the conditions are changed slightly, but
reversibly in such a way that the resulting system is also in
equilibrium, we have
  
  
  
d  
 dx   
 dE   
 dN i 
 E 
  x 
i  N i 
=
dE


1

 X  dx 

1
i dN i


(4.1)
i
3
We may write this result as
dE  d   X dx   i dNi
(4.2)
Let us first consider a simple example with the number of
particles fixed and the volume as the only external
parameter;
dN i  0; xv  V ; X V  
(4.3)
Then from (4.2)
dE  d  dV
(4.4)
We see that the change in internal energy consists from
two parts. The term d represents the change in internal
energy when the external parameters are kept constant.
This is just what is meant by heat. Thus
4
DQ  d
(4.5)
is the quantity of heat added to the system in a reversible
process. The symbol D is used instead of d because DQ is
not an exact differential-that is, Q is not a state function.
The term - dV is the change in internal energy caused by
the change in external parameters; this is what we mean
by mechanical work, and
DW  dV
(4.6)
is the work done on the system in the volume change dV.
By elementary mechanics the work done must be given
by -pdV. Therefore
 p
(4.7)
5
where p is the pressure. We see that (4.2) is equivalent to
the equation
dE  DQ  DW
(4.8)
which is the First Law of Thermodynamics.
The statement that dS=DQ/T is a perfect (exact) differential in a
reversible process is a statement of the Second Law of
Thermodynamics.
That is, DQ/T is a differential of state function, entirely
defined by the state of the system. Now from (4.5) we
know that
DQ  d
6
Specific Heat
d  DQ / 
(4.9)
is a perfect differential, as  is a state function. We note
that both 1/T and 1/ are integrating factors for DQ. As
we know that =kT, thus
S  k
(4.10)
as the connection between the usual thermodynamic
entropy S and the entropy  as we prefer to define it for
use in statistical mechanics.
The specific heat at constant volume Cv, and the one at
constant pressure, Cp, would be given by
 S 
 E 
  
CV  T 






T

T


 N ,V

 N ,V

 N ,V
7
 S 
  ( E  PV ) 
  
Cp  T    





T
 T  N , P 
N ,P
   N , P
We defined E as a function of  and V. Other quantities of
interest are then obtained from E:
 E 
 E 
dE = d  pdV  
d




 dV
   V
 V  
whence
and
E 

   V
  kT  

 E 
p

 V  
(4.11)
(4.12)
(4.13)
The independent variables  and V are often quite
inconvenient and it is more convenient to work with , p or
, V for example.
8
To do this we introduce auxiliary functions called thermodynamic potentials:
F, H, G.
Helmholtz Free Energy F
F(V, ) is defined as
Now
 E 
F  E    E    
   V
 F 
 F 
dF  dE  d  d   pdV  d    dV    d
 V  
   V
From (4.15)
(4.14)
(4.15)
 F 
 p

 V  
(4.16)
 F 
  

   V
(4.17)
and
9
Therefore if V,  are the independent variables it is natural to introduce
F, from which p, are readily calculated
Enthalpy H
H(,p) is defined by
Now
 E 
H  E  pV  E  V 

 V  
dH=dE  pdV  Vdp=d  Vdp 
 H 
 H 
 dp
=
 d  
   p
 p 
whence
and
(4.18)
 H 
 

  
(4.19)
(4.20)
p
 H 
V 

 p  
(4.21)
10
Gibbs Free Energy G
G(,p) is defined by
 E 
 E 
G  E    pV  E   
 V

   V
 V 
(4.22)
Now
dG  dE - d  d  pdV  Vdp  d  Vdp 
whence
and
 G 
 G 
 dp

 d  
   p
 p 
 G 
   
   p
 G
V  
 p

(4.23)
(4.24)




(4.25)
11
The Helmholtz free energy of a body has the property that
the work done on the body in a reversible process at
constant temperature is the change of its Helmholtz free
energy.
This easily shown: in a reversible process
DW  dE - dQ  dE -d  d ( E   )  dF
(4.26)
Note that -dF is the maximum work, which can be, done by
the system in a change at constant temperature.
In the case of one component system with the volume as
only one external parameter we can write the main
thermodynamic equation for quasi-static processes
dE  d  pdV  dN
(4.27)
12
Thermodynamic potentials
Energy
E(,V,N)
dE=-pdV+dN
Entropy
(E,V,N)
d=dE+pdV-dN
Enthalpy
H(,p,N)=E+pV
dH=d+Vdp+dN
Helmholtz Free
Energy
F(,V,N)=E-
dF=-d-pdV+dN
Gibbs
Thermodynamic
Potential
G(,p,N)=F+pV=N
dG=-d+Vdp+dN
Great potential
(,V,)=F-N=-pV
d=-d-pdV-Nd
13
Recapitulation of thermodynamic laws
Zero law -postulated the existence of equilibrium states.
All parts of closed equilibrium system are in the state of
internal equilibrium and heat equilibrium between each
other, that means one general characteristic from all
subsystems is taking place (temperature principle).
the law of energy continuity. The energy can
be transformed to the system by the heat. It impossible to
make any work without the energy. (The perpetual mobile
of the 1st order is impossible).
First law-
14
the entropy of the close system is
increasing. It can be defined also through the Clausius
principle: as the irreversible process of the transforming
the heat from the hot body to the cold one.
Second law-
As the principle of Kelvin (Tomson) the second law read: It
is impossible to build the cycle machine that can work by
absorption of the heat from the thermostat with any other
changes in the system (the perpetual mobile of the 2nd
order can not be created).
Third Law - (Nernst-Plank heat theorem) - the entropy
of the system is going to zero if the absolute temperature
is also tends to zero.
15
Thermodynamic Functions for the Canonical
Ensemble
Let us define the entropy of the canonical ensemble with
mean energy <E> as being equal to the entropy of a
microcanonical ensemble with energy <E>.
This corresponds to the thermodynamic situation because in
thermodynamics the entropy is fixed by the energy
independently of whether the system is isolated or in
contact with a heat bath.
The entropy for the microcanonical ensemble is equal to
ln where  is the volume of phase space corresponding
to energies between E0 and E0+E. As we have seen, the
precise value of E is unimportant and we may choose it
equal to the range of reasonable probable values of the
16
energy in the canonical ensemble.
Let us first write  in terms of E. If (E) denotes the
volume of phase space corresponding to energies less
than E we have
  ( E ) 
  
(4.28)
 E
 E  E
We now estimate E, the range of reasonable probable
values for the canonical ensemble. Let p(E)dE be the
canonical ensemble probability that the system will have
energy in the range dE at E. Then,
p( E )dE   ( E )d( E )
(4.29)
where (E) is the occupancy probability of a unit volume of
phase space at energy E. p(E) is distributed according to
the Gauss distribution. The function is normalized and this
means that we may estimate the breadth E of the
17
distribution peak by
p( E )E  1
i.e., by
(4.30)
  ( E ) 
 E  1
 E  E
 (  E )
(4.31)
Substituting the E given by this equation in the
expression for , we obtain, using (3.39)
E
 ( E )  Ae
  1 /  (  E ) 
E
A1e kT

E
A1e kT

kT
(4.32)
so that
  ln    ln A  E / kT
(4.33)
We have
18
ln A  ( E   ) / 
(4.34)
But we recall the Helmoholtz free energy FE-, whence
Ae
and
F / kT
( E )  e( F  E ) / kT
(4.35)
(4.36)
We have further, by the normalization of 
( F  E )/ kT

d


e
d  1


(4.37)
and
e  F / kT   e  E ( p,q )/ kT d
(4.38)
19
The partition function
If we define the partition function as
Z  e
Z 
 E ( p,q )/ kT
d
 E / kT
e

(classical)
(4.39)
(quantum)
(4.40)
i
i
we have
F  -kT ln Z
(4.41)
The other thermodynamic functions can be calculated from
the partition function, using thermodynamic potentials.
20
Alternative expression for the partition function.
Density of states.
In most physical cases the energy levels accessible to a
system are degenerate, i.e. one has a group of states, gr
in number, all belonging to the same energy value Er . In
such a case it would be more appropriate to write the
partition function
Z (V , T )   gr e
 E r / kT
(4.42)
r
the corresponding expression for Pr , the probability that
the system be in any of the states with energy Er , would
be
21
gr exp(  Er )
Pr 
 gr exp(  Er )
(4.43)
r
Clearly, the gr states with a common energy Er are all
equally likely to occur. As a result the probability of a
system having energy Er becomes directly proportional to
the multiplicity gr of this level; gr thus plays the role of
"weight factor" for the level Er. The actual probability is
then determined by both the weight factor gr and the
Boltzmann factor exp(-Er/kT) of the level, as we indeed
have in (4.43).
Now in view of the largess of the number of particles
constituting a given system and the largess of the volume
to which these particles are confined, the consecutive
energy values Er of the system must be extremely close to
one another.
22
Accordingly, there lie, within any reasonable interval of
energy (E,E+dE), a very large number of energy levels.
One may then regard E as a continues variable and write
P(E)dE for the probability that the given system, as a
member of the canonical ensemble, may have its energy in
the specified range.
Clearly, the product of the relevant single-state probability
and the number of energy states lying in the specified
range will give this. Denoting the latter by g(E)dE, where
g(E) stands for the density of states of the system around
the energy value E, we have
P( E )dE  exp(  E ) g ( E )dE
(4.44)
23
which on normalization becomes
P( E )dE 
exp(  E ) g ( E )dE

(4.45)
 exp( E ) g( E )dE
0
The denominator is clearly another expression for the
partition function of the system:

Z  e
 E / kT
g ( E )dE
(4.46)
0
The expression for <f> any average value of physical
quantity f may be written in this case as
24

 f ( Er )gr e E / kT  f ( E )e E / kT g( E )dE
0
 f  r


 gr e E / kT
 e E / kT g( E )dE
r
r
r
(4.47)
0
Let us consider the relation (4.46) If we regard =1/kT as a
complex variable, then the partition function Z() is just
Laplace transform of the density of states g(E).

Z   e  E / kT g ( E )dE
0
The integral is, of course convergent over the positive half
plane of  (because g(E)0 for all E and lim g( E )exp(E )  0
E 
for all >0).
We can, therefore, write g(E) as the inverse Laplace
transform of Z()
25
 '  i
E
1
g( E ) 
e Z (  )d (  ' > 0)

2i  ' i
1
=
2

e
(  '  i ") E
Z (  'i ")d "
(4.48)

the path of integration runs parallel to, and to the right of,
the imaginary axis, i.e. along the straight line Re = >0.
26
A system of harmonic oscillators
We shall now study, as an example, a system of N,
practically independent, harmonic oscillators. We start
with the specialized situation when the oscillators can be
treated classically. The Hamiltonian of any one of them
(assumed to be one-dimensional) may then be written as
1
1 2
2 2
H (qi , pi )  M qi 
pi
2
2M
of course, the index i will run from 1 to N. For the
partition function, we readily obtain
(4.49)
single-oscillator
 

1
1
1

Z1    exp    M 2 q 2 
p 2 dqdp 
h   
2M
2


1/ 2
=
1  2 


2 
h  M 
 2M

 
1/ 2




1

(4.50)
27
where   h / 2
The partition function of the N-oscillator system would then
be
Z N  Z1N  (  )  N
(4.51)
The Helmholtz free energy of the system is now given by
  
F   kT ln Z N  NkT ln

(4.52)
 kT 
whence we obtain for other thermodynamic quantities
  
(4.53)
  kT ln 
P0
 kT 
    
S  Nk ln 
  1
  kT  
(4.54)
(4.55)
E  NkT
(4.56)
C p  CV  Nk
(4.57)
28
We note that the mean energy per oscillator is in complete accord with
the equipartition theorem, namely 2
1
,
2
for
kT E we have here two
independent quadratic terms in the single oscillator Hamiltonian.
We may determine the density of states, g(E), of this
system from the expression (4.51) for its partition
function. We have, in view of (4.48),
Z  Z N  (  )  N
N
g( E ) 
  i
1
(  )
1
N
1
e E

d

N
2i   i 
('>0),
that is
 1
E N 1


N
( ) ( N  1)!


g (E)  
0



for
E0
(4.58)
for
E0
29
To test this correctness, we may calculate the entropy of
the system with the help of this formula. Taking N>>1 and
making use the Stirling approximation, we get
  E  
S ( N , E )  k ln g ( E )  Nk ln
  1
  N  
(4.59)
which yields for the temperature of the system
1
E
 S
T 
 
 E  N
Nk
(4.60)
Eliminating E between these two relations, we obtain
precisely our earlier result for the functions S(N,T). This
indeed assure us of the inner consistency of our approach:
more so, it gives us confidence to accept (4.58) as the
30
correct expression for the density of states of this system.
We now take up the quantum-mechanical situation,
according to which the energy eigenvalues of a one-
dimensional harmonic oscillator are given by
 n  (n  21 )
(4.61)
n=0,1,2,...
Accordingly, we have for the single-oscillator partition
function
Z1 (  ) 

e
  ( n  ) 
1
2
n0
 21 


e

 2 sinh
 
1 e
1
2

 
1
(4.62)
The N-oscillator partition function is then given by
Z N (  )   Z1 (  )
N


e
 ( N / 2 ) 
 2 sinh
1
2

 

N
1 e

   N
(4.63)
31
For the Helmholtz free energy of the system, we have
F  NkT ln[2 sinh( 21  )]  N [ 21   kT{1  e  }]
(4.64)
whence we obtain for other thermodynamic quantities
F/N
(4.65)
P0
(4.66)
S  Nk

1
2


 coth 21    ln 2 sinh 21  
 

= Nk  
 ln(1  e   ) 
1
e

 

E  21 N coth( 21  )  N  21   
e
 1

(4.67)
(4.68)
32
C p  Cv  Nk ( 12  ) 2 cosech 2  12  

e
 Nk (  ) 2 
(e
 1) 2
(4.69)
33