Ch1. Statistical Basis of
Thermodynamics
1.1 The macroscopic state and the
microscopic state
1) Macrostate: a macrostate of a physical
system is specified by macroscopic
variables (N,V,E).
2) Microstate: a microstate of a system is
specified by the positions, velocities, and
internal coordinates of all the molecules
in the system.
For a quantum system, Y(r1,r2,….,rN),
specifies a microstate.
1
Microstate Number W(N,V,E)



For a given macrostate (N,V,E), there are a
large number of possible microstates that can
make the values of macroscopic variables. The
actual number of all possible miscrostate is a
function of macrostate variables.
Consider a system of N identical particles
confined to a space of volume V. N~1023. In
thermodynamic limit: NVbut n=N/V
finite.
Macrostate variables (N, V, E)
 Volume: V
E  ni i
 Total energy:

i
2
Macrostate variables



Volume: V
E   ni i
Total energy:
i
ni – the number of particles with energy i
i - energy of the individual particles
Total energy:
N   ni
i

Microstate: all independent solutions of
Schrodinger equation of the system. N-particle
Schrodinger equation,
 2 2
   

 




U
(
r
)
Y
(
r
,
r
,...,
r
)

E
Y
(
r
,
r
,...,
r

k
k 
1 2
N
1 2
N)

k 1  2m

N
E   Ek
k 1
3
Physical siginificance of
W(N,V,E)

For a given macrostate (N,V,E) of a physical
system, the absolute value of entropy is
given by
S ( N ,V , E )  k ln W( N ,V , E )
Where k=1.38x10-23 J/K – Boltzman constant


Consider two system A1 and A2 being
separately in equilibrium.
When allow two systems exchanging heat
by thermal contact, the whole system has
E(0)=E1+E2=const. macrostate (N,V, E(0))
4
Problem 1.2

Assume that the entropy S and the statistical
number W of a physical system are related
through an arbitrary function S=f(W). Show
that the additive characters of S and the
multiplicative character of W necessarily
required that the function f(W) to be the form of
f(W) = k ln(W)
• Solution: Consider two spatially separated systems A and B
A
B
5
1.3 Future contact between statistics
and thermodynamics

Consider energy change between two subsystems A1 and A2, both systems can change
their volumes while keeping the total volume
the constant.
A1
(N1,V1,E1)
A2
(N2,V2,E2)
Energy change
Volume variable
No mass change
E(0) = E1+E2=const
V(0) = V1+V2=const
N(0) = N1+N2=const
6
1.3 Future contact between statistics
and thermodynamics –cont.

Initial states

System A1: (N1,V1, E1),
S1(N1,V1,E1)=k lnW1(N1,V1,E1)
System A2: (N2,V2, E2),
S2(N2,V2,E2)=k lnW2(N1,V1,E1)
Thermal contact process
A1
(N1,V1,E1)
A2
(N2,V2,E2)
E(0) = E1+E2=const, E1, E2 changeable
V(0) = V1+V2=const, V1, V2 changeable
N(0) = N1+N2=const, N1, N2 changeable
W(0) (N1,V1,E1; N2,V2,E2)= W1(N1,V1,E1)+W2(N2,V2,E2)
7
1.3 Future contact between statistics
and thermodynamics –cont.

Thermal equilibrium state (N1*,V1*,E1*)
 S ( 0 ) 
  ln W1 
  ln W 2 




 0  
 

N

N

N
1  N 1 N 1*
1  N 1 N 1*
2  N 2  N 2*



m1m2
 S ( 0 ) 
  ln W1 
  ln W 2 




 0  
 

V

V

V
1 V 1V 1*
1
2


V 1V 1* 
V 2V 2*
P1=P2
 S ( 0) 
  ln W1 
  ln W 2 




 0  
 

E

E

E
1  E1 E1*
2
 1  E1 E1*


 E 2 E 2*
T1=T2
8
Summary-how to derive thermodynamics
from a statistical beginning?
1) Start from the macrostate (N,V,E) of the given system;
2) Determine the number of all possible microstate
accessible to the system, W(N,V,E).
3) Calculate the entropy of the system in that macrostate
S ( N ,V , E )  k ln W( N ,V , E )
4) Determine system’s parameters, T,P, m
1
 S 
   ;
 E  N ,V T
P
 S 

  ;
 V  N , E T
m
 S 

 
T
 N V , E
5) Determine the other parameters in thermodynamics
Helmhohz free energy: A= E-T S
Gibbs free energy: G = A + PV = mN
Enthalpy: H = E + PV
9
Determine heat capacity
6) Determine heat capacity Cv and Cp;
 S 
 E 
Cv  T     
 T  N ,V  T  N ,V
 S 
 H 
Cp  T    


T

T
  N ,P 
 N ,P
10
1.4 Classical ideal gas





L
Model:
N particles of nonatomic molecules
Free, nonrelativistic particles
Confined in a cubic box of side L (V=L3)
L
Wavefunction and energy of each particleL
2
ˆ
p
Hˆ 
,
2m

p x  i ,
x
p y  i

,
y
p z  i

z
p 2  p x2  p y2  p z2   2 2
11
1.4 Classical ideal gas-cont.

L
Hamiltonian of each particle


ˆ
HY x , t    Y x , t ,

2  2
2
2  



 2  2 Y  x , t    Y  x , t 
2

2m  x
y
z 

L
L
Separation of variables

Y  x   1  x 2  y 3  z 
 2   x2   y2   z2

Boundary conditions: Y(x) vanishes on the
boundary,
12
1.4 Classical ideal gas-cont.

L
Boundary conditions: Y(x)
vanishes on the boundary
L
1 x   2  y   3 z   0
on x  0, L ; y  0, L; z  0, L
1 x   sin k x x, k x  nx

L
2  y   sin k y y, k y  n y
3 z   sin k z z, k z  nz

L

h2
2
2
2
 nx , n y , nz  
n

n

n
x
y
z
8m L2
nx , n y , nz  1, 2, 3....
L

L
13

Microstate of one particle

L
Boundary conditions: Y(x)
vanishes on the boundary
L
1 x   2  y   3 z   0
on x  0, L ; y  0, L; z  0, L
1 x   sin k x x, k x  nx

L
2  y   sin k y y, k y  n y
3 z   sin k z z, k z  nz

L

L
L

h2
2
2
2
 nx , n y , nz  
n

n

n
x
y
z
8m L2
nx , n y , nz  1, 2, 3....

One microstate is a
combination of (nx,ny,,nz)
14
The number of microstate of one
particle W(1,,V)

The number of distinct microstates for a
particle with energy e is the number of
independent solutions of (nx,ny,nz), satisfying
8m V 2 / 3
nx  n y  nz 
 *
2
h
2

2
2
The number W(1,,V) is
the volume in the shell
of a 3 sphere. The
volume of in (nx,ny,nz)
space id 1.
nz
ny
nx
15
Microstates of N particles

L
The total energy is

h2
2
2
2
E    i nx , n y , nz   
n

n

n
ix
iy
iz
2
8
m
L
i 1
i 1
N
N
3N
  r
r 1
h2
2
where  r 
n
r
8m L2
L

L
• One microstate with a given energy E is a solution of
(n1,n2,……n3N) of
n1  n2  ...... n3 N
2
2
2
8m L2
 2 E  E*
h
3N-dimension sphere
with radius sqrt(E*)
16
The number of microstate of N
particles W(N,E,V)

The volume of 3N-sphere with radius R=sqrt(E*)
N ,VE  

 3N / 2
3N / 2!
R
3N
1
 
 2
3N
 3N / 2
3N / 2!
E *3 N / 2
(Appendix C)
The number W(N,E,V) is the
volume in the shell of a 3N-sphere.
 N , VE 
W( N , V , E ) 
, with   E
E
 V  4m E 3 / 2  3
ln WN ,V , E   ln N ,VE   N ln  3 
  N
h
3
N
  2
 
n3
n2
n1
17
Entropy and thermodynamic
properties of an ideal gas
 V  4m E 3 / 2  3
S ( N ,V , E )  k ln W  Nk ln  3 
   NK
 h  3N   2
• Determine temperature
1  S 
3
1
    Nk
T  E  N ,V 2
E
3  3

E  N  kT   n RT 
2  2

• Determine specific heat
3
3
 S 
 E 
Cv  T 
 
  Nk  nR
2
 T  N ,V  T  N ,V 2
5
5
 S 
  ( E  PV ) 
Cp  T
 
  Nk  nR
T
2
 T  N , p 
N, p 2
18
State equation of an ideal gas
• Determine pressure
P  S 
1

  Nk
T  V  N , E
V
PV  NkT
• Specific heat ratio
Cp
5


Cv 3
19
1.5 The entropy of mixing ideal gases
• Consider the mixing of two ideal gases 1 and 2, which are
initially at the same temperature T. The temperature of the
mixing would keep as the same.
N1,V1,T
N2,V2,T
mixing
N1,V,T
N2,V,T
• Before mixing
• After mixing
20
P1-11
Four moles of nitrogen and one mole of oxygen at P=1
atm and T=300K are mixed together to form air at the
same pressure and temperature. Calculate the entropy of
the mixing per mole of the air formed.
21