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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.