Презентация на тему Basics of thermodynamics & kinetics

DEFINITION

1st Baron Kelvin (1824 -1907)
 Ludwig Eduard Boltzmann (1844 –

1906)

“small part” with ε & all
states of thermostat

with E-ε. Mall(E-ε) = 1 • Mt(E-ε)

k • ln[Mt(E-ε)] ≡ St(E-ε) ≅ St(E) - ε•(dSt/dE)|E

Mt(E-ε) ≅ exp[St(E)/k] • exp[-ε•(dSt/dE)|E/k] conclusions

WHAT IS “TEMPERATURE”?

S ~ ln[M]

exp[-

ε1• (dSt/dE)|E/k] (GIBBS)
and
Probability1(ε1) = exp(-ε1/kBT) (BOLTZMANN)

One has: (dSt/dE)|E = 1/ T

k = kB
______________________________________________________________
ε ⇒ ε-kBT, M ⇒ M × exp(1) ≡ M × 2.72

Яков Григорьевич Синай, 1935

Abel Prize 2014
“…связь между порядком и хаосом…” 1/r3

Joseph Liouville
(1809 - 1882) 

Σj Pj(εj) ≡ 1
Z(T) = Σi exp(-εi/kBT)

partition function
СТАТИСТИЧЕСКАЯ СУММА

Unstable (falls)
stable

?
unstable
?

Along tangent: S-S(E1) = (E-E1)/ T1
i.e., F = E - T1S = const (= F1 = E1 - T1S1)

mechanics:

P(ε) ~ exp(-ε/kBT) // Classic: ε =

εCOORD + εKIN
εKIN = mv2/2 : does not depend on coordinates
Potential energy εCOORD: depends only on coordinates

P(ε) ~ exp(-εCOORD/kBT) • exp(-εKIN/kBT)


Z(T) = ZCOORD(T)•ZKIN(T) ⇒ F(T) = FCOORD(T)+FKIN(T)

========================================================================================================================

Elementary volume: Δ(mv)Δx ≅ ħ ⇒ (Δx)3 ≅(ħ/|mv|)3
= (ħ2/[mkBT])3/2

Δ(mv) ≅ m|v|, and |mv| ≅ (mkBT)1/2

consider further
only potential energy:

E ⇒ ECOORD
M ⇒ MCOORD
S(E) ⇒

SCOORD(ECOORD )
F(E) ⇒ FCOORD , etc.

~ 1
Transition: |ΔF1|= |-ΔS×ΔT| ~ kT*



ΔE-T*ΔS=0

Melting:
NOT 1-s order phase transition

1-s order phase transition

Helix & coil: 1D objects Ice & water: 3D objects

N

N

n

n

ΔFhelix_n = Const + n×f ΔFICE_n = C×n2/3 + n×f
1D interface 3D interface
Mid-transition: f = 0
ΔShelix_n ~ ln(N) positions ΔSICE_n ~ ln(N)
N : very large; n ~ αN, α<<1 (e.g., α~0.001)
Const << ln(N) α2/3⋅N2/3 >> ln(N)
phases mix phases do not mix

t0→#1→ ≈
≈ τ#→ (n/n#) =

τ#→ × exp(+ΔF#/kBT)

Not
“slowly goes”,
but
climbs, falls
and climbs again…

falls

τ#→

(or: ≈the highest rate)

1/TIME = (1/τ#→) × exp(-ΔF1#/kBT) + (1/τ#→) × exp(-ΔF2#/kBT)

PARALLEL REACTIONS:

RATE = 1/ TIME

+ …
#
#
start
_
CONSECUTIVE REACTIONS:
TRANSITION TIME ≅ SUM OF TIMES
(or:

≈ the highest time)

TIME ≈ τ#→ × exp(+ΔF1#/kBT) + τ#→ × exp(+ΔF2#/kBT) + …

steady-state approximation

t0→… → ≈ t0→#1→1 + t1→#2→ 2 + …

start

_

“long barrier”

“downhill”

“long barrier”:

finish

AND OUT OF PATHWAYS OF CONSECUTIVE REACTIONS:

TRANSITION TIME ≅

SUM OF TIMES
(or: ≈the longest time)

# main

finish

finish

start

start

“trap”: on

“trap”: out

main #

~ kBT = Σj Pj(εj) ∙ εj

v2 = (vX2)+(vY2)+(vZ2) Maxwell :

in 3D

James Clerk  

(1831 –1879)

-(3πDη)v [Stokes law], or m(dv/dt) = -(kBT/Ddiff)v

[Einstein-Stokes]
D – diameter;
m ~ D3 ⋅ 1g/cm3 – mass;
η – viscosity

tkinet ≈ 10-13 sec × (D/nm)2
in water

Sir George Gabriel Stokes Albert Einstein
DIFFUSION: (1819-1903) (1879-1995)

During tkinet the molecule moves by Lkinet ~ v•tkinet

Then it restores its kinetic energy mv2/2 ~ kBT from thermal kicks of other molecules, and moves in another random side

CHARACTERISTIC DIFFUSION TIME: nanoseconds

tkinet

≈ 10-13 sec × (D/nm)2 in water

DIFFUSION:
During tkinet the molecule moves by Lkinet ~ v•tkinet

Then it restores its kinetic energy mv2/2 ≈ kBT from thermal kicks
of other molecules, and moves in another
random side

CHARACTERISTIC DIFFUSION
TIME: nanoseconds

The random walk allows the molecule
to diffuse at distance D (~ its diameter)
within ~(D/L kinet)2 steps, i.e., within

tdifft ≈ tkinet•(D/Lkinet)2 = D2/Ddiff

≈ 4•10-10 sec × (D/nm)3 in water

r1

→

Z(T) = Σj exp(-εj/kBT)
Σj Pj(εj) = 1
E(T) = <ε> = Σj εj∙ Pj(εj)
if all εj = ε : #STATES = 1/P, i.e.: S(T) = kB∙ln(1/P)
S(T) = kB = kB∙Σj ln[1/Pj(εj)]∙Pj(εj)
F(T) = E(T) - TS(T) = -kBT ∙ ln[ Z(T)]
STATISTICAL MECHANICS

exp(-εj/kBTth);

E(Tth) = Σj εj Pj(εj,Tth)
S(Tth) = kBΣj ln[1/Pj(εj,Tth)]Pj(εj,Tth)
Tsmall_part = dE(Tth)/dS(Tth) = Tth

STATISTICAL
MECHANICS

T1

i.e.,
F = E - T1S = const (= F1 = E1 - T1S1)

exp(-ε/kBT) Classic: ε = εCOORD +

εKIN
εKIN = mv2/2 : does not depend on coordinates
Potential energy εCOORD: depends only on coordinates

P(ε) ~ exp(-εCOORD/kBT) • exp(-εKIN/kBT)

ZKIN(T) = ΣK exp(-εK/kBT): don’t depend on coord.
ZCOORD(T) = ΣCexp(-εC/kBT): depends on coord.

Z(T) = ZCOORD(T)•ZKIN(T) ⇒ F(T) = FCOORD(T)+FKIN(T)

========================================================================================================================

Elementary volume: Δ(mv)Δx = ħ ⇒ (Δx)3 =(ħ/|mv|)3

ΣCexp(-εC/kBT): depends ONLY

on coordinates

P(εKIN) = exp(-εKIN/kBT) / ZKIN(T)

ZKIN(T) = ΣK exp(-εK/kBT): don’t depend on coord.

T<0: unstable (explodes)
<εKIN> ⇒ ∞ at T<0
due to
P(εKIN) ~ exp(-εKIN/kBT)