Transcript Ch.1.3-1.4
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 (r 1 ,r 2 ,….,r N ), 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~10 23 . In thermodynamic limit: N finite.
V but n=N/V Macrostate variables (N, V, E) Volume: V Total energy:
E
i n i
i
2
Macrostate variables
Volume: V Total energy: n i i
E
i n i
i
– the number of particles with energy - energy of the individual particles i Total energy:
N
n i i
Microstate: all independent solutions of Schrodinger equation of the system. N-particle Schrodinger equation,
k N
1 2 2
m
2
k
U
(
r k
) Y (
r
1 ,
r
2 ,...,
r N
)
E
Y (
r
1 ,
r
2 ,...,
r N
)
E
k
1
E k
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) =E 1 +E 2 =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( multiplicative character of W W ). Show that the additive characters of S and the 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 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 change Volume variable No mass change E (0) V (0) = E 1 +E 2 =const = V 1 +V 2 =const N (0) = N 1 +N 2 =const 6
1.3 Future contact between statistics and thermodynamics –cont.
Initial states A1 (N1,V1,E1) System A1: (N1,V1, E1), S1(N1,V1,E1)=k ln W 1 (N1,V1,E1) System A2: (N2,V2, E2), S2(N2,V2,E2)=k ln W 2 (N1,V1,E1) Thermal contact process E V (0) (0) = E 1 +E 2 =const, E1, E2 changeable = V 1 +V 2 =const, V1, V2 changeable N (0) W (0) = N 1 +N 2 =const, N1, N2 changeable (N1,V1,E1; N2,V2,E2)= W 1 (N1,V1,E1)+ W 2 (N2,V2,E2) A2 (N2,V2,E2) 7
1.3 Future contact between statistics and thermodynamics –cont.
Thermal equilibrium state (N1*,V1*,E1*)
S
( 0 )
N
1
N
1
N
1 * 0 ln W 1
N
1
N
1
N
1 * ln
N
W 2 2
N
2
N
2 * m1m 2
S
( 0 )
V
1
V
1
V
1 * 0 ln W 1
V
1
V
1
V
1 * ln W 2
V
2
V
2
V
2 *
S
( 0 )
E
1
E
1
E
1 * 0 ln W 1
E
1
E
1
E
1 * ln W 2
E
2
E
2
E
2 * P1=P2 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
S
E N
,
V
1 ;
T
S
V N
,
E
P
;
T
S
N
V
,
E
m
T
5) Determine the other parameters in thermodynamics Helmhohz free energy: A= E-T S Gibbs free energy: G = A + PV = m N Enthalpy: H = E + PV 9
Determine heat capacity 6) Determine heat capacity Cv and Cp;
Cv
T
S
T
N
,
V
E
T
N
,
V Cp
T
S
T N
,
P
H
T N
,
P
10
1.4 Classical ideal gas Model: N particles of nonatomic molecules Free, nonrelativistic particles Confined in a cubic box of side L (V=L 3 )
p x
2
i
2
m
,
x
,
p y
i
y
,
p
2
p x
2
p
2
y
p z
2 2 2
p z
i
z
L L 11
1.4 Classical ideal gas-cont.
Hamiltonian of each particle Y
x
, Y , 2 2
m
2
x
2 2
y
2 2
z
2 Y
x
Y
x
Separation of variables L Y 1
x
2
y
3
z
2
x
2
y
2
z
2 Boundary conditions: Y (x) vanishes on the boundary, L L 12
1.4 Classical ideal gas-cont.
Boundary conditions: Y (x) vanishes on the boundary
on
1 1 2 3
x
0 , 2
L
; sin sin sin
y k x x
,
k y y
,
k z z
, 3 0 ,
L
;
z
0 0 ,
L k x k k z y
n x L
n y
L n z L
L L L
n x
,
n y
,
n z
n x
,
n y
,
n z h
2 8
mL
2 1 , 2 , 3 ....
n x
2
n y
2
n z
2 13
Microstate of one particle Boundary conditions: Y (x) vanishes on the boundary
on
1 1 2 3
x
0 , 2
L
; sin sin sin
y k x x
,
k y y
,
k z z
, 3 0 ,
L
;
z
0 0 ,
L k x k k z y
n x L
n y
L n z L
n x
,
n y
,
n z n x
,
n y
,
n z
h
2 8
mL
2 1 , 2 , 3 ....
n x
L 2
n y
2 L L
n z
2 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
n x
2
n y
2
n z
2 8
mV
2 / 3
h
2 * The number space id 1.
W (1, ,V) is the volume in the shell of a 3 sphere. The volume of in (nx,ny,nz) n x n z n y 15
Microstates of N particles The total energy is
E
i N
1
i
n x
,
n y
,
n z
i N
1
h
2 8
mL
2
n ix
2
n iy
2
n iz
2
r
3
N
1
r where
r
h
2 8
mL
2
n r
2 L L L • One microstate with a given energy E is a solution of (n1,n2,……n3N) of
n
1 2
n
2 2 ......
n
3
N
2 8
mL
2
h
2
E
E
* 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
3
N
3
N
/ / 2 2 !
R
3
N
1 2 3
N
3
N
3
N
/ / 2 2 !
E
* 3
N
/ 2 (Appendix C) The number W (N,E,V) is the volume in the shell of a 3N-sphere. n 3 W (
N
,
V
,
E
)
N
,
VE
E
, ln W
N
,
V
,
E
ln
N
,
VE
N
ln
V h
3
with
E
4
mE
3
N
3 / 2 3 2
N
n 1 n 2 17
Entropy and thermodynamic properties of an ideal gas
S
(
N
,
V
,
E
)
k
ln W
Nk
ln
V h
3 • Determine temperature 1
T
S
E N
,
V
3 2
Nk
1
E
4 3
mE N
3 / 2
E
N
3 2
kT
n
3 2
RT
3 2
NK
• Determine specific heat
C v
T
S
T N
,
V
E
T N
,
V C p
T
S
T N
,
p
3 2
Nk
3 2
nR
(
E
T PV
)
N
,
p
5 2
Nk
5 2
nR
18
State equation of an ideal gas • Determine pressure
P
T
S
V N
,
E
1
Nk V
• Specific heat ratio
C p C v
5 3
PV
NkT
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.
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