statistical thermodynamics wayne m. lawton department of mathematics national university of...
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STATISTICAL THERMODYNAMICS
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected] (65) 874-2749Fax (65) 779-5452
SECOND LAW REVISITED
Lord Kelvin : A transformation whose only final result is to transform into work heat extracted from a source which is at the same temperature throughout is impossible
Rudolph Clausius : A transformation whose only final result is to transfer heat from a body at a given temperature to a body at a higher temperature is impossible (this principle implies the previous one)
Martian Skeptic : What temperature ?
SECOND LAW REVISITED
BA Definition : Body A has higher temperature than body B ( ) if, when we bring them into thermal contact, heat flows from A to B. Body A has the same temperature as B ( ) if, when we bring them into thermal contact, no heat flows from A to B and no heat flows from B to A. ( [A] := {U | U A} )
Enrico Fermi : (Clausius Reformulated) If heat flows by conduction from a body A to another body B, then a transformation whose only final result is to transfer heat from B to A is impossible.
BA
SECOND LAW REVISITED
[C])([A],f [C])([B],f [B])([A],f
C,Bfor and [B]),([A],fQQ
and Clausius, impliesKelvin &Carnot
BA
0. work W ofquantity a perform and
B, to0QQheat ofquantity
asurrender A, fromheat of 0 Q
quantity a absorb toengine reversible a
usecan then weBA If :Carnot Sadi
BA
A
SECOND LAW REVISITED
DA
Definition : Absolute Thermodynamic Temperature
Choose a body D
DA
If then ])D[],A([f)AT(
then 1)AT( AD then ])A[],D([f1)AT(
In a reversible process
0)BT(Q)AT(Q BA
SECOND LAW REVISITED
System : Cylinder that has a movable piston and contains a fixed amount of homogeneous fluid
States (Macroscopic) : Region in positive quadrant of the (V = volume, T = temperature) plane.
Functions (on region) : V, T, p = pressure Paths (in region) : Oriented curves Differential Forms : can be integrated over paths
pdV )W(
? )Q(
WORK HEAT
SECOND LAW REVISITED
Definition : Entropy Function S
T)Q( T)S(V, Define 11)T,V( state a Choose 11
T),(V, to)T,V( joins where 1121
adiabatic is and ,isothermal is 21
and 2
cyle sCarnot' si 21
(by thermal equilibrium and by thermal isolation)
TdS )Q(
FIRST LAW REVISITED
function U energy) (internalan exists There
dU pdV - TdS
dV pV
U dT
T
UTdS
such that
Therefore
FIRST & SECOND LAWS COMBINED
pV
U
T
1VS ;
T
U
T
1TS
Therefore, the basic (but powerful) calculus identity
VS
T
TS
V
Yields (after some tedious but straightforward algebra)
p-TpT
VU
IDEAL GAS LAW
V/gRnT)p(V,
g
n
R
(Chemists) Boyl, Gay-Lussac, Avogardo
amount of gas in moles
ideal gas constant
Kelvin)degree/joules(8.314
ideal gas temperature in Kelvin
(water freezes at 373.16 degrees)
JOULE’S GAS EXPANSION EXPERIMENT
We substitute the expression for p (given by the ideal gas law) to obtain
gTd
gdT
V
nRVU
and observe that the outcome of Joule’s gas expansion experiment
gT0VU
IDEAL GAS LAW
(Physicists)
VTkNT)p(V,
N
_Kelvin) deg / joules1038.1( 23
k
mole) / molecules100225.6( 23
number of molecules of gas
Boltzmann’s constant
GAS THERMODYNAMICS
Experimental Result : (dilute gases)
1)1(NkdT
dU
Vp VT paths adiabaticon constant &1
)V/Vln(Nk)T/Tln()1(NkS 111
Therefore
GAS KINETICS
Monatomic dilute gas, m = molecular mass
2x
2N
Um
2
13m
2
1
xx m22
tA
V
NtF
average kineticenergy / molecule
V2m21)1(N
AF
p
2m2
1)1(kT 3
5
GAS KINETICS
Photon gases3
4momentumcE
Maxwell Equipartition of Energy
0mm
mm)(
21
221121
kT2
1
directioneach in
energy kinetic
freedom of degrees
21
EQUIPARTITION
) N - N lnN!Nln ( Stirling’s formula
m21 p,...,p,p is
Number of ways of partitioning N objects into m bins with relative frequencies (probabilities)
)!Np()!...Np(!NC m1
yields
)p,...,p(HN Cln m1 where
denotes Shannon’s information-theoretic entropy
)p,...,p(H m1
mm11m1 plnpplnp)p,...,p(H
EQUIPARTITION
),EpEp(NE mm11
If the bins correspond to energies, then
and therefore (nearly) C,
is maximized, subject to an energy constraint
distribution
),p,...,p(H m1
)T(Z)kTEexp(p ii by the Gibbs
ClnkS )T(ZlnNkTTSE
dEdST1 Therefore and free energy Maxwell dist. )kT2/(-mexp)prob(x, 2
THIRD LAW
Nernst : The entropy of every system at absolute zero can always be taken equal to zero
inherently quantum mechanical
molecules 01 24
discrete microstates, a quart bottle of air has about
& smicrostate 01 2210
Maxwell’s demon : may he rest in peace
Time’s arrow : probably forward ???
energy 2ln kT nformationi ofbit 1
REFERENCES
H. Baeyer, Warmth Disperses and Time Passes
R. Feynman, Lectures on Physics, Volume 1
E. Fermi, Thermodynamics
V. Ambegaokar, Reasoning about Luck
F. Faurote, The How and Why of the Automobile
REFERENCES
C. Shannon and W. Weaver, The MathematicalTheory of Communication
K. Huang, Statistical Mechanics
N. Hurt and R. Hermann, Quantum StatisticalMechanics and Lie Group Harmonic Analysis
H. S. Green and T. Triffet, Sources of Consciousness,The Biophysical and Computational Basis of Thought