1.The Statistical Basis of Thermodynamics
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Transcript of 1.The Statistical Basis of Thermodynamics
1. The Statistical Basis of Thermodynamics
1. The Macroscopic & the Microscopic States
2. Contact between Statistics & Thermodynamics:
Physical Significance of the Number (N,V,E)
3. Further Contact between Statistics & Thermodynamics
4. The Classical Ideal Gas
5. The Entropy of Mixing & the Gibbs Paradox
6. The “Correct” Enumeration of the Microstates
1.1. The Macroscopic & the Microscopic States
System of N identical particles in volume V, with
, ,N
N V finiteV
(Thermodynamic limit )
E.g., Non-interacting particles:
1
N
ii
N n
1
N
i ii
E n
i = single particle energies ni = # of p’cles with energy i
A macrostate is specified by parameters ( N, V, E, ... ).
Postulate of equal a priori probabilities:All microstates satisfying the macrostate parameters are equally likely to occur.
, , ,N V E = # of all microstates that give rise to the macrostate (extensive) parameters N, V, E, ... .
Let
1.2. Contact between Statistics & Thermodynamics: Physical Significance of the Number (N,V,E)
Consider 2 systems A1 & A2 in thermal contact with each other,
i.e., partition is fixed, impermeable but heat conducting.
( Nj , Vj & E(0) = E1 + E2 are fixed )
A1
( N1 , V1 , E1 )
A2
( N2 , V2 , E2 )
01 2
0 01 1 2 2 E E E
E E E
Equilibrium is achieved if E1 ( with E2 = E(0) E1 ) maximizes (0) :
0
1
0E
1 1 2 2
2 2 1 11 2
E EE E
E E
2 2 2 2
1 2
E E
E E
(0) denotes properties of the composite system
1 1 2 22 2 1 1
1 2
0E E
E EE E
1 1 2 2
1 1 1 2 2 2
1 1E E
E E E E
1 1 2 2
1 2
ln lnE E
E E
ln E
E
Let 2 systems are in thermal equilibrium if they have the same .
Thermodynamics :,
1
N V
S
E T
Planck :
lnS k Boltzmann :
lnS k
1
k T
k = Boltzmann constant
3rd law
0th law ( thermal eqm.)
1.3. Further Contact between Statistics & Thermodynamics
For an impermeable but movable & heat conducting partition,
Nj , V(0) = V1 +V2 & E(0) = E1 + E2 are fixed.
Equilibrium is achieved, i.e., (0) is maximized, if
0
1
0E
0
1
0V
and
i.e., both system have the same values of &,
ln
N EV
1st law: dE T dS P dV dN
,N E
SP T
V
P
k T
chemical potential
~ mech. eqm.
For a permeable, movable & heat conducting partition,
N(0) = N1 + N2 , V(0) = V1 +V2 & E(0) = E1 + E2 are fixed.
Equilibrium is achieved, i.e., (0) is maximized, if
0
1
0E
0
1
0V
i.e., Both system have the same values of , , & ,
ln
V EN
1st law: dE T dS P dV dN
,V E
ST
N
k T
0
1
0N
~ chemical eqm.
Summary
Connection between statistical mechanics & thermodynamics is
lnS k
Once is written in terms of the independent thermodynamical variables,
all other thermodynamic quantities can be obtained via the Maxwell relations.
UInternal Energy
SV , X
P, YT
HEnthalpy
GGibbs free energy
FHelmholtz free energy
V
UT
S
Mnemonics for the Maxwell Relations
P
GS
T
U
S P
V T
G
T V
P S
dU TdS P dV Y dX 1
z x y
x y z
y z x
dH TdS Vd P Y dX
varvarvarF W
2
S V
PT
V S
U
S V
Good Physicists Have Studied Under Very Fine Teachers
= U ( P) V Y X
= F ( P) V Y X = H TS
= U TS
= U(V,S,X)
1.4. The Classical Ideal Gas
Non-interacting, classical ( distinguishable), point particles:
, , NN E V V
lnS k N V const
,N E
S P
V T
P V n R T
Nk
V
Cf n R
kN
A
R
N 231.38 10 /J K
23
8.31 /
6.02 10 /
J mol K
mol
58.62 10 /eV K
const here means indep. of V.
Quantum (Obeying Schrodinger Eq) Free Particles
Let these particles be confined within a cube of edge L.
Dirichlet boundary conditions: 0 at walls ( where x,y,z = 0,L ).
sin sin sinyx znn n
A x y zL L L
1,2,3, ; , ,in i x y z
Neumann boundary conditions: n 0 at walls.
cos cos cosyx znn n
A x y zL L L
,1,2,3,0in
22 2 2
2 2 2, ,2 2x y z x y z
kn n n n n n
m m L
1-particle energy :
2
2 2 22/38 x y z
hn n n
m V 3V L
2
2 2 22/38 x y z
hn n n
m V
* 2 2 2x y zn n n
2*
2/38
h
m V
i.e.
Let
( * is a positive integer )
1, ,N E V # of { nx, ny, nz } satisfying2/3
* 2 2 22
8x y z
m Vn n n
h
, ,N E V # of { nix, niy, niz } satisfying
For N non-interacting particles
2
2 2 22/3
18
N
i x i y i zi
hE n n n
m V
2
*2/38
hE
m V
* 2 2 2
1
N
i x i y i zi
E n n n
3
2
1
N
rr
n
2/3
* 2 2 22
1
8N
i x i y i zi
m V EE n n n
h
2/3, , ,N E V N V E
2/3, , ,N E V N V E 2/3, , ,S N E V S N V E
For reversible adiabatic processes, S & N are kept constant.
2/3V E const
2/31/3
,
20
3 N S
EdV V dE
V
,
2
3N S
E E
V V
,N S
EP
V
2
3
EP
V
,f N S
Valid for both classical & quantum statistics
(adiabatic processes)
5/3PV const
Better behaved quantity is ( N,E,V),
defined as the # of lattice points with non-negative coordinates & lying within
the volume bounded by the surface of a sphere, centered at the origin, and
with radius
Counting States: Distinguishable Particles
State labels { nix, niy, niz } form a lattice in the 3N-D n-space.
( N,E,V) = # of lattice points with non-negative coordinates & lying on the
surface of a sphere, centered at the origin, and with radius
*R
fluctuates wildly even for small E changes unless N >>1.
2/3*
2
8m V ER
h
* * * 3/21
1 41,
8 3N
As R , the lattice points become a continuum.
* 3/2
6
Better approximations:
Number of points on the x-y, y-z, z-x planes is
Since these points are shared by 2 neighboring sectors, the
volume integral counts each as half a point.
* * 3/2 *1
3
6 8
Dirichlet B.C.(exclude all nj = 0 points )
* * 3/2 *1
3
6 8
Neumann B.C.(include all nj = 0 points )
*13
4
( Density of states in n-space is 1. )
Volume of an n-D sphere of radius R is/2
!2
nn
sphV Rn
( see App.C )
Volume of points with non-negative coordinates
1
2
n
sphV V
( Take non-negative-half of every dimension )
3 3 /2
* * 3 /21,
32 !2
N NNN E E
N
2/3
*2
8, , ,
mV EN V E N E
h
3 /2
3
2
3!
2
NN m EVNh
! 1n n
3 /2
3
2, ,
3!
2
NN m EVN V E
Nh
Stirling’s formula: ln ! lnn n n n for n >>1
3/2
3
3ln , , ln 2 ln !
2
V NN V E N m E
h
3/2
3
4 3ln , , ln
3 2
V m E NN V E N
h N
3
23 3 3ln ! ln
2 2 2
N N NN
Let (N,V,E) = # of states lying between E ½ & E+ ½ .
, ,, ,
N V EN V E
E
3, ,
2
NN V E
E
3/2
3
4 3ln , , ln
3 2
V m E NN V E N
h N
ln , ,N V E ln , ,N V E
3/2
3
4 3ln , , ln
3 2
V m E NN V E N
h N
3/2
3
4 3, , ln ln
3 2
V m E N kS N V E k N k
h N
2
2/3
3 2, , exp 1
4 3
h N SE S V N
m V N k
3
2E N k T
3
2n R T ,N V
ET
S
2
3E
N k
,
V
N V
EC
T
3
2N k
3
2n R
,N S
EP
V
2
3E
V
2
3P V E N k T n R T
,
P
N P
HC
T
,N P
E PV
T
VC n R 5
2n R
5
3P
V
C
C
3/2
3
4 3, , ln
3 2
V m E N kS N V E N k
h N
3
2E N k T
Isothermal processes ( N, T = const ) : E const
,N E
SS V
V
N kV
V lnN k V
ln lnf i f iS S N k V V ln f
i
VN k
V
Adiabatic processes ( N, S = const ) :
3/2V E const 3/2V T const
P V n R T 5/2P V const P V
1V T
Alternatively, ,N SdE P dV
2
3
EdV
V also leads to 3/2V E const
5
3
1.5. The Entropy of Mixing & the Gibbs Paradox
3/2
3
4 3ln
3 2
V m E N kS N k
h N
This S is not extensive, i.e., , , , ,S N V E S N V E
Mixing of 2 ideal gases 1 & 2 (at fixed T ) :
3
3ln
2before itotal i i
i i i
VS S k N
2
3 4ln 1 ln
2 3
m EN k V N k
h N
3/2
2
2 3ln
2
m k TS N k V
h
3
2E N k T
3
3ln
2aftertotal mixed i
i i
VS S k N
ii
V V
3
3ln
2
VN k
22
mkT
Thermal wavelength
Entropy of mixing of gases :
after beforetotal totalmixing
S S S ln 0ii i
Vk N
V
Gibb’s paradox :
For the mixing of different parts of the same gas in equilibrium (Ni / Vi = N / V ,
i = ), the formula still applies & we also have S > 0, which is
unacceptable.
i
i
N N
V V
3
3ln
2before itotal i i
i i i
VS S k N
3
3ln
2aftertotal i
i i
VS k N
Irreversible process: S > 0 is expected.
For the mixing of different parts of the same gas in eqm., ,ii
i
N N
V V
lnimixi i
VS k N
V ln lni i
i
k N N k N N
ln ln 0mixed i i ii
S k N N S k N N
Thus, Gibbs’ paradox is resolved using Gibbs’ recipe :
3/2
3
4 3, , l 1n
3 2
V m ES N V E N k N
h NNk
Sackur-Tetrode eq.
S is now extensive, i.e., , , , ,S N V E S N V E
ln ln 0mixed i i i ii
S k N N N S k N N N or
ln ln !N N N N
ln !S S k N
3
3ln
2before itotal i
i
VS k N
3
3ln
2aftertotal
VS k N
lnii i
Nk N
N
Revised Formulae
3/2
3
4 3, , l 1n
3 2
V m ES N V E N k N
h NNk
2/323 2
, , e2
p4 3
x 13
h N SNE S V N
m V N k
extensive
In general, relations derived using the previous definition of S
are not modified if they do not involve explicit expression of S.
,V S
E
N
2 21
33
E S
N N k
intensive
Gibbs’ recipe is cancelled by removing all terms in red.
,N V
ET
S
2
3
E
N k
,N S
EP
V
2
3
E
V
2
3
ET
N k
2
3
EP
V
2 2
33
E
N
E E S
N N N k 1
PE STN
V
3/2
3
4 3l 1n
3 2
V m ES N k N k
h NN
3
3ln
21k
N
VN
3
2 2
3 3
3 21 ln 1
2 3 N
Vk T
3lnk TV
N
21
3
2
3
E S
N N k
3ln 1k TN
NV
A G P V N P V N N kT
21
3
2
3
E S
N N k
22
mkT
1.6. The “Correct” Enumeration of the Microstates
Elementary particles are all indistinguishable.
In the distribution of N particles such that ni particles occupy the i state,
!
!Di
i
N
n
for distinguishable particles
1 for indistinguishable particles
In the classical (high T ) limit, 0in i !D N
!D
N
Gibbs’ recipe corresponds to