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Презентация на тему Basics of thermodynamics & kinetics

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2. THERMODYNAMISC&STATISTICAL PHYSICS
3. WHAT IS “TEMPERATURE”?EXPERIMENTAL DEFINITION := t,oC + 273.15oEXPERIMENTAL DEFINITION
4. Benoît Paul Émile Clapeyron (1799 – 1864)William
5. THEORYClosedsystem:energy E = constCONSIDER: 1 state
6. COMPARE:Probability1(ε1) = Mt(E-ε1) / M(E) =
7. Josiah Willard Gibbs (1839 –1903)
8. (dSth/dE) = 1/ TP1(ε1) ~ exp(-ε1/kBT)Pj(εj) =
9. Unstable (explodes, v → inf.)
10. Separation of potential and kinetic energiesin classic
11. IN THERMAL EQUILIBRIUM:TCOORD = TKIN = TouterWe
12. TRANSITIONS:THERMODYNAMICS
13. gradual transition“all-or-none” (or 1st order) phase transitioncoexistence& jump-liketransitioncoexistence(ΔE/kT*)(ΔT/T*) ~ 1Transition: |ΔF1|= |-ΔS×ΔT| ~ kT*ΔE-T*ΔS=0
14. Second order phase transitionchangeRecently observed in proteins;rare case
15. LANDAU: Helix-coil transition:
16. Лев Давидович Ландау (1908 - 1968)Нобелевская Премия 1962
17. TRANSITIONS:KINETICS
18. n# = n × exp(-ΔF#/kBT)n# n→TRANSITION TIME:t0→1
20. phase separationCoil- Coil- ≈Native≈
21. TRANSITION RATE = SUM OF RATES
22. t0→… →finish ≈ t0→#1→ finish + t0→#2→
23. __TRANSITION TIME IS ESSENTIALLY EQUAL FOR “TRAPS”
24. DIFFUSION:KINETICS
25. Mean kinetic energy of a particle:
26. Friction stops a molecule within picoseconds: m(dv/dt)
27. Friction stops a molecule within picoseconds:
28. The End
29. For “small part”: Pj(εj) = exp(-εj/kBT)/Z(T);
30. Thermostat: Tth = dEth/dSth“Small part”: Pj(εj,Tth)
31. Along tangent: S-S(E1) =
32. Separation of potential energyin classic (non-quantum) mechanics:P(ε)
33. P(εKIN+εCOORD) ~ exp(-εCOORD/kBT)•exp(-εKIN/kBT)P(εCOORD) = exp(-εCOORD/kBT) / ZCOORD(T)ZCOORD(T)
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THERMODYNAMISC&STATISTICAL PHYSICS

PROTEIN PHYSICSLECTURES 7-8Basics of thermodynamics & kinetics

WHAT IS “TEMPERATURE”?EXPERIMENTAL DEFINITION := t,oC + 273.15oEXPERIMENTAL DEFINITION

Benoît Paul Émile Clapeyron (1799 – 1864)William Thomson, 1st Baron Kelvin (1824

THEORYClosedsystem:energy E = constCONSIDER: 1 state of “small part” with ε

COMPARE:Probability1(ε1) = Mt(E-ε1) / M(E) =

Josiah Willard Gibbs (1839 –1903) Яков Григорьевич Синай, 1935

(dSth/dE) = 1/ TP1(ε1) ~ exp(-ε1/kBT)Pj(εj) = exp(-εj/kBT)/Z(T); Σj Pj(εj)

Separation of potential and kinetic energiesin classic (non-quantum) mechanics:P(ε) ~ exp(-ε/kBT)

IN THERMAL EQUILIBRIUM:TCOORD = TKIN = TouterWe may consider furtheronly potential energy:E

gradual transition“all-or-none” (or 1st order) phase transitioncoexistence& jump-liketransitioncoexistence(ΔE/kT*)(ΔT/T*) ~ 1Transition: |ΔF1|= |-ΔS×ΔT| ~ kT*ΔE-T*ΔS=0

Second order phase transitionchangeRecently observed in proteins;rare case

LANDAU: Helix-coil transition: Melting:NOT

Лев Давидович Ландау (1908 - 1968)Нобелевская Премия 1962

n# = n × exp(-ΔF#/kBT)n# n→TRANSITION TIME:t0→1 = t0→#1→ ≈ ≈ τ#→

t0→… →finish ≈ t0→#1→ finish + t0→#2→ finish + …##start_CONSECUTIVE REACTIONS:TRANSITION TIME

__TRANSITION TIME IS ESSENTIALLY EQUAL FOR “TRAPS” AT AND OUT OF PATHWAYS

Mean kinetic energy of a particle: ~ kBT

Friction stops a molecule within picoseconds: m(dv/dt) = -(3πDη)v [Stokes law],

Friction stops a molecule within picoseconds:

For “small part”: Pj(εj) = exp(-εj/kBT)/Z(T);

Thermostat: Tth = dEth/dSth“Small part”: Pj(εj,Tth) ~ exp(-εj/kBTth);

Separation of potential energyin classic (non-quantum) mechanics:P(ε) ~ exp(-ε/kBT)

P(εKIN+εCOORD) ~ exp(-εCOORD/kBT)•exp(-εKIN/kBT)P(εCOORD) = exp(-εCOORD/kBT) / ZCOORD(T)ZCOORD(T) = ΣCexp(-εC/kBT): depends ONLY

“all-or-none” (or first order) phase transitionF(T1)________________

Слайды презентации

Слайд 2 THERMODYNAMISC
&
STATISTICAL PHYSICS

Слайд 3 WHAT IS “TEMPERATURE”?

EXPERIMENTAL DEFINITION :
= t,oC + 273.15o
EXPERIMENTAL

DEFINITION

Слайд 4 Benoît Paul Émile Clapeyron (1799 – 1864)
William Thomson,

Benoît Paul Émile Clapeyron (1799 – 1864)William Thomson, 1st Baron Kelvin

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

1906)

Слайд 5 THEORY

Closed
system:
energy
E = const
CONSIDER: 1 state of

“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]

Слайд 6 COMPARE:
Probability1(ε1) = Mt(E-ε1) / M(E) =

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

Слайд 7 Josiah Willard Gibbs
(1839 –1903)

Яков Григорьевич Синай, 1935

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

Joseph Liouville
(1809 - 1882)

Слайд 8 (dSth/dE) = 1/ T
P1(ε1) ~ exp(-ε1/kBT)
Pj(εj) = exp(-εj/kBT)/Z(T);

Σj Pj(εj) ≡ 1
Z(T) = Σi exp(-εi/kBT)

partition function
СТАТИСТИЧЕСКАЯ СУММА

Слайд 9 Unstable (explodes, v → inf.)

Unstable (falls)
stable

?
unstable
?

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

Слайд 10 Separation of potential and kinetic energies
in classic (non-quantum)

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

Слайд 11 IN THERMAL EQUILIBRIUM:

TCOORD = TKIN = Touter

We may

IN THERMAL EQUILIBRIUM:TCOORD = TKIN = TouterWe may consider furtheronly potential

consider further
only potential energy:

E ⇒ ECOORD
M ⇒ MCOORD
S(E) ⇒

SCOORD(ECOORD )
F(E) ⇒ FCOORD , etc.

Слайд 12 TRANSITIONS:
THERMODYNAMICS

Слайд 13 gradual transition
“all-or-none” (or 1st order) phase transition
coexistence
& jump-like
transition
coexistence
(ΔE/kT)(ΔT/T)

~ 1
Transition: |ΔF1|= |-ΔS×ΔT| ~ kT*

ΔE-T*ΔS=0

Слайд 14 Second order phase transition
change
Recently observed in proteins;
rare case

Слайд 15 LANDAU: Helix-coil transition:

LANDAU: Helix-coil transition: Melting:NOT 1-s order phase

Melting:
NOT 1-s order phase transition

1-s order phase transition

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

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

Слайд 16 Лев Давидович Ландау
(1908 - 1968)
Нобелевская Премия 1962

Слайд 17 TRANSITIONS:
KINETICS

Слайд 18 n# = n × exp(-ΔF#/kBT)
n#
n
→
TRANSITION TIME:
t0→1 =

n# = n × exp(-ΔF#/kBT)n# n→TRANSITION TIME:t0→1 = t0→#1→ ≈ ≈

t0→#1→ ≈
≈ τ#→ (n/n#) =

τ#→ × exp(+ΔF#/kBT)

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

falls

τ#→

Слайд 19

Слайд 20
phase separation

Coil
- Coil

- ≈Native
≈

Слайд 21 TRANSITION RATE = SUM OF RATES

(or: ≈the highest rate)

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

PARALLEL REACTIONS:

RATE = 1/ TIME

Слайд 22 t0→… →finish ≈ t0→#1→ finish + t0→#2→ finish

t0→… →finish ≈ t0→#1→ finish + t0→#2→ finish + …##start_CONSECUTIVE REACTIONS:TRANSITION

+ …
#
#
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

Слайд 23 _
_
TRANSITION TIME IS ESSENTIALLY
EQUAL FOR “TRAPS” AT

__TRANSITION TIME IS ESSENTIALLY EQUAL FOR “TRAPS” AT AND OUT OF

AND OUT OF PATHWAYS OF CONSECUTIVE REACTIONS:

TRANSITION TIME ≅

SUM OF TIMES
(or: ≈the longest time)

# main

finish

start

“trap”: on

“trap”: out

main #

Слайд 24 DIFFUSION:
KINETICS

Слайд 25 Mean kinetic energy of a particle:

~ kBT = Σj Pj(εj) ∙ εj

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

in 3D

James Clerk

(1831 –1879)

Слайд 26 Friction stops a molecule within picoseconds:

m(dv/dt) =

-(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

Слайд 27 Friction stops a molecule within picoseconds:

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

→

Слайд 28 The End

Слайд 29 For “small part”: Pj(εj) = exp(-εj/kBT)/Z(T);

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

Слайд 30 Thermostat: Tth = dEth/dSth

“Small part”: Pj(εj,Tth) ~

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

Слайд 31 Along tangent: S-S(E1) = (E-E1)/

T1

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

Слайд 32 Separation of potential energy
in classic (non-quantum) mechanics:

P(ε) ~

Separation of potential energyin classic (non-quantum) mechanics:P(ε) ~ exp(-ε/kBT) Classic:

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