Basics of thermodynamics & kinetics PROTEIN PHYSICS LECTURES 7-8 Basics of thermodynamics & kinetics

THERMODYNAMISC & STATISTICAL PHYSICS

EXPERIMENTAL DEFINITION : WHAT IS “TEMPERATURE”? EXPERIMENTAL DEFINITION : EXPERIMENTAL DEFINITION = t,oC + 273.16o

WHAT IS “TEMPERATURE”? THEORY Closed system: energy E = const S ~ ln[M] CONSIDER: 1 state of “small part” with  & all states of thermostat with E-. M(E-) = 1 • Mt(E-) St(E-)  k • ln[Mt(E-)] St(E-)  St(E) - •(dSt/dE)|E Mt(E-)  exp[St(E)/k] • exp[-•(dSt/dE)|E/k]

______________________________________________________________ All-system’s states with E have equal probabilities For “small part’s” state: depends on e: COMPARE: Probability1(1) = Mt(E-1) / M(E) = exp[- 1• (dSt/dE)|E/k] and Probability1(1) = exp(-1/kBT) (BOLTZMANN) One has: (dSt/dE)|E = 1/ T k = kB ______________________________________________________________   -kBT, M  M  exp(1)  M  2.72

Pj(j) = exp(-j/kBT)/Z(T); j Pj(j)  1 (dSth/dE) = 1/ T P1(1) ~ exp(-1/kBT) Pj(j) = exp(-j/kBT)/Z(T); j Pj(j)  1 Z(T) = i exp(-i/kBT) partition function СТАТИСТИЧЕСКАЯ СУММА

stable Along tangent: S-S(E1) = (E-E1)/ T1 i.e., F = E - T1S = const (= F1 = E1 - T1S1) stable Unstable (explodes, v → inf.) Unstable (falls)  unstable 

KIN = mv2/2 : does not depend on coordinates Separation of potential energy in classic (non-quantum) mechanics: P() ~ exp(-/kBT) Classic:  = COORD + KIN KIN = mv2/2 : does not depend on coordinates Potential energy COORD: depends only on coordinates P() ~ exp(-COORD/kBT) • exp(-KIN/kBT) Z(T) = ZCOORD(T)•ZKIN(T)  F(T) = FCOORD(T)+FKIN(T) ======================================================================================================================== Elementary volume: (mv)x = ħ  (x)3 =(ħ/|mv|)3

IN THERMAL EQUILIBRIUM: TCOORD = TKIN = T We may consider further only potential energy: E  ECOORD M  MCOORD S(E)  SCOORD(ECOORD ) F(E)  FCOORD , etc.

TRANSITIONS: THERMODYNAMICS

“all-or-none” (or 1st order) phase transition gradual transition “all-or-none” (or 1st order) phase transition coexistence coexistence & jump-like transition E-T*S=0 Transition: |F1|= |-ST| ~ kT* (E/kT*)(T/T*) ~ 1

Second order phase transition change Recently observed in proteins; they are rare

Fhelix_n = Const + nf FICE_n = Cn2/3 + nf LANDAU: Helix-coil transition: Melting: NOT 1-s order phase transition 1-s order phase transition N N n n Helix & coil: 1D objects Ice & water: 3D objects Fhelix_n = Const + nf FICE_n = Cn2/3 + nf 1D interface 3D interface Mid-transition: f = 0 Shelix_n ~ ln(N) positions SICE_n ~ ln(N) N : very large; n ~ N, <<1 (~0.001) Const << ln(N) 2/3N2/3 >> ln(N) phases mix phases do not mix rule of contraries

TRANSITIONS: KINETICS

t01 = t0#1 = n# = n  exp(-F#/kBT) n# n  TRANSITION TIME: Not “slowly goes”, but climbs, falls and climbs again… n# # falls n  TRANSITION TIME: t01 = t0#1 = = # (n/n#) = #  exp(+F#/kBT)

TRANSITION RATE = SUM OF RATES (or: the highest rate) PARALLEL REACTIONS: TRANSITION RATE = SUM OF RATES (or: the highest rate) RATE = 1/ TIME 1/TIME = (1/#)  exp(-F1#/kBT) + (1/#)  exp(-F2#/kBT)

_ # # t0… finish = t0#1 finish + t0#2 finish + … CONSECUTIVE REACTIONS: TRANSITION TIME  SUM OF TIMES (or: the highest time) start finish t0… finish = t0#1 finish + t0#2 finish + … TIME = #  exp(+F1#/kBT) + #  exp(+F2#/kBT) + …

_ _ # main TRANSITION TIME IS ESSENTIALLY “trap”: on “trap”: out start start finish finish TRANSITION TIME IS ESSENTIALLY EQUAL FOR “TRAPS” AT AND OUT OF PATHWAYS OF CONSECUTIVE REACTIONS: TRANSITION TIME  SUM OF TIMES (or: the longest time)

DIFFUSION: KINETICS

Mean kinetic energy of a particle: mv2/2 ~ kBT <> = j Pj(j)  j v2 = (vX2)+(vY2)+(vZ2) Maxwell: in 3D

Friction stops a molecule within picoseconds: m(dv/dt) = -(3D)v [Stokes law] D – diameter; m ~ D3  1g/cm3 – mass;  – viscosity tkinet  10-13 sec  (D/nm)2 in water During tkinet the molecule moves somewhere 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

Friction stops a molecule within picoseconds: tkinet  10-13 sec  (D/nm)2 in water DIFFUSION: During tkinet the molecule moves somewhere 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  4•10-10 sec  (D/nm)3 in water

STATISTICAL MECHANICS For “small part”: Pj(j) = exp(-j/kBT)/Z(T); Z(T) = j exp(-j/kBT) j Pj(j) = 1 E(T) = <> = j j Pj(j) if all j =  : #STATES = 1/P, i.e.: S(T) = kBln(1/P) S(T) = kB<ln(#STATES)> = kBj ln[1/Pj(j)]Pj(j) F(T) = E(T) - TS(T) = -kBT  ln[ Z(T)] STATISTICAL MECHANICS

Thermostat: Tth = dEth/dSth “Small part”: Pj(j,Tth) ~ exp(-j/kBTth); E(Tth) = j j Pj(j,Tth) S(Tth) = kBj ln[1/Pj(j,Tth)]Pj(j,Tth) Tsmall_part = dE(Tth)/dS(Tth) = Tth STATISTICAL MECHANICS

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

KIN = mv2/2 : does not depend on coordinates Separation of potential energy in classic (non-quantum) mechanics: P() ~ exp(-/kBT) Classic:  = COORD + KIN KIN = mv2/2 : does not depend on coordinates Potential energy COORD: depends only on coordinates P() ~ exp(-COORD/kBT) • exp(-KIN/kBT) ZKIN(T) = K exp(-K/kBT): don’t depend on coord. ZCOORD(T) = Cexp(-C/kBT): depends on coord. Z(T) = ZCOORD(T)•ZKIN(T)  F(T) = FCOORD(T)+FKIN(T) ======================================================================================================================== Elementary volume: (mv)x = ħ  (x)3 =(ħ/|mv|)3

<KIN>   at T<0 P(KIN+COORD) ~ exp(-COORD/kBT)•exp(-KIN/kBT) P(COORD) = exp(-COORD/kBT) / ZCOORD(T) ZCOORD(T) = Cexp(-C/kBT): depends ONLY on coordinates P(KIN) = exp(-KIN/kBT) / ZKIN(T) ZKIN(T) = K exp(-K/kBT): don’t depend on coord. T<0: unstable (explodes) <KIN>   at T<0 due to P(KIN) ~ exp(-KIN/kBT)

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