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Ch1. Statistical Basis of Thermodynamics
1.1 The macroscopic state and the microscopic state1) 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, (r1,r2,….,rN), specifies a microstate.
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Microstate Number (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: NVbut n=N/V finite.
Macrostate variables (N, V, E) Volume: V Total energy:
iiinE
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Macrostate variables
i
inN
i
iinE Volume: V Total energy:
ni – the number of particles with energy i
i - energy of the individual particles Total energy: Microstate: all independent solutions of
Schrodinger equation of the system. N-particle Schrodinger equation,
),...,,(),...,,()(2 2121
1
22
NN
N
kkk rrrErrrrU
m
1k
kEE
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Physical siginificance of (N,V,E)
For a given macrostate (N,V,E) of a physical system, the absolute value of entropy is given by
),,(ln),,( EVNkEVNS
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))
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Problem 1.2
Assume that the entropy S and the statistical number of a physical system are related through an arbitrary function S=f(). Show that the additive characters of S and the multiplicative character of necessarily required that the function f() to be the form of
f() = k ln()
A B
• Solution: Consider two spatially separated systems A and B
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1.3 Future contact between statistics and thermodynamics Consider energy change between two sub-
systems 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 changeVolume variableNo mass change
E(0) = E1+E2=const
V(0) = V1+V2=const
N(0) = N1+N2=const
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1.3 Future contact between statistics and thermodynamics –cont. Initial states A1
(N1,V1,E1)A2 (N2,V2,E2)
System A1: (N1,V1, E1), S1(N1,V1,E1)=k ln1(N1,V1,E1)
System A2: (N2,V2, E2), S2(N2,V2,E2)=k ln2(N1,V1,E1)
E(0) = E1+E2=const, E1, E2 changeable
V(0) = V1+V2=const, V1, V2 changeable
N(0) = N1+N2=const, N1, N2 changeable
Thermal contact process
(0) (N1,V1,E1; N2,V2,E2)= 1(N1,V1,E1)+2(N2,V2,E2)
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1.3 Future contact between statistics and thermodynamics –cont.
Thermal equilibrium state (N1*,V1*,E1*)
*222
2
*111
1
*111
)0( lnln0
NNNNNNNNN
S
*222
2
*111
1
*111
)0( lnln0
EEEEEEEEE
S
*222
2
*111
1
*111
)0( lnln0
VVVVVVVVV
S
2
P1=P2
T1=T2
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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, (N,V,E).3) Calculate the entropy of the system in that macrostate
TN
S
EV
,
;;1
,, T
P
V
S
TE
S
ENVN
),,(ln),,( EVNkEVNS 4) Determine system’s parameters, T,P,
5) Determine the other parameters in thermodynamics
Helmhohz free energy: A= E-T SGibbs free energy: G = A + PV = NEnthalpy: H = E + PV
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Determine heat capacity
6) Determine heat capacity Cv and Cp;
PNPN T
H
T
STCp
,,
VNVN T
E
T
STCv
,,
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1.4 Classical ideal gas
Model: N particles of nonatomic molecules Free, nonrelativistic particles Confined in a cubic box of side L (V=L3)
222222
2
,,
,2
ˆˆ
zyx
zyx
pppp
zip
yip
xip
m
pH
L
L
L
Wavefunction and energy of each particle
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1.4 Classical ideal gas-cont.
txtxzyxm
txtxH
,,2
,,,ˆ
2
2
2
2
2
22
L
L
L Hamiltonian of each particle
Separation of variables
2222
321
zyx
zyxx
Boundary conditions: (x) vanishes on the boundary,
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1.4 Classical ideal gas-cont.
L
nkzkz
Lnkyky
Lnkxkx
LzLyLxon
zyx
zzz
yyy
xxx
,sin
,sin
,sin
,0;,0;,0
0
3
2
1
321
L
L
L Boundary conditions: (x) vanishes on the boundary
....3,2,1,,
8,, 222
2
2
zyx
zyxzyx
nnn
nnnmL
hnnn
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Microstate of one particle
L
nkzkz
Lnkyky
Lnkxkx
LzLyLxon
zyx
zzz
yyy
xxx
,sin
,sin
,sin
,0;,0;,0
0
3
2
1
321
L
L
L Boundary conditions: (x) vanishes on the boundary
....3,2,1,,
8,, 222
2
2
zyx
zyxzyx
nnn
nnnmL
hnnn
One microstate is a combination of (nx,ny,,nz)
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The number of microstate of one particle (1,,V)
The number of distinct microstates for a particle with energy e is the number of independent solutions of (nx,ny,nz), satisfying
*8
2
3/2222
h
mVnnn zyx
The number (1,,V) is the volume in the shell of a 3 sphere. The volume of in (nx,ny,nz) space id 1. nx
ny
nz
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Microstates of N particles
2
2
23
1
1
222
2
2
1
8
8,,
rr
N
rr
N
iiziyix
N
izyxi
nmL
hwhere
nnnmL
hnnnE
The total energy is
• One microstate with a given energy E is a solution of (n1,n2,……n3N) of
L
L
L
*8
......2
22
322
21 EE
h
mLnnn N 3N-dimension
sphere with radius sqrt(E*)
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The number of microstate of N particles (N,,V)
Ewith
E
VENEVN
,,
),,(
2/3
2/333
2/3
*!2/32
1
!2/3, N
NNN
N
EN
RN
VEN
The volume of 3N-sphere with radius R=sqrt(E*)
The number (N,E,V) is the volume in the shell of a 3N-sphere.
n1
n2
n3
(Appendix C)
NN
mE
h
VNVENEVN
2
3
3
4ln,ln,,ln
2/3
3
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Entropy and thermodynamic properties of an ideal gas
ENk
E
S
T VN
1
2
31
,
RTnkTNE
2
3
2
3
NKN
mE
h
VNkkEVNS
2
3
3
4lnln),,(
2/3
3
• Determine temperature
• Determine specific heat
nRNkT
PVE
T
STC
nRNkT
E
T
STC
pNpNp
VNVNv
2
5
2
5)(
2
3
2
3
,,
,,
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State equation of an ideal gas
VNk
V
S
T
P
EN
1
,
NkTPV
• Determine pressure
• Specific heat ratio
3
5
v
p
C
C
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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.
• Before mixing
N1,V1,T N2,V2,Tmixing N1,V,T
N2,V,T
• After mixing
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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.