Thermodynamics of Brownian Ratchets Reem Mokhtar Thermodynamics 1. First law of thermodynamics: You can’t win. 2. Second law of thermodynamics: You can’t even break even. 3. Third law of thermodynamics: You can’t stay out of the game. Source: Voet, Donald; Voet, Judith G. (2010-11-22). Biochemistry, 4th Edition (BIOCHEMISTRY (VOET)) (Page 52). John Wiley & Sons, Inc. Thermodynamics( 2nd(law,(by(Sadi(Carnot(in(1824.( ( • Zeroth:(If(two(systems(are(in(thermal(equilibrium(with( a(third(system,(they(must(be(in(thermal(equilibrium( with(each(other.(( • First:(Heat(and(work(are(forms(of(energy(transfer.( • Second:(The(entropy(of(any(isolated(system(not(in( thermal(equilibrium(almost(always(increases.( • Third:(The(entropy(of(a(system(approaches(a(constant( value(as(the(temperature(approaches(zero.( ( hLp://en.wikipedia.org/wiki/Laws_of_thermodynamics( Thermodynamics • First: the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the amount of work done by the system on its surroundings. à The law of conservation of energy can be stated: The energy of an isolated system is constant. • Second: The entropy of any isolated system not in thermal equilibrium almost always increases. • Third: The entropy of a system approaches a constant value as the temperature approaches zero, which implies that it is impossible for any procedure to bring a system to the absolute zero of temperature in a finite number of steps Source: Wikipedia, http://en.wikipedia.org/wiki/Laws_of_thermodynamics What(is(a(Ratchet?( A(device(that(allows(a(sha9(to(turn(only(one(way( hLp://www.hpcgears.com/products/ratchets_pawls.htm( Feynman,(R.(P.,(Leighton,(R.(B.,(&(Sands,(M.(L.((1965).(The$Feynman$Lectures$on$Physics:$ Mechanics,$radia7on,$and$heat((Vol.(1).(AddisonKWesley.( The(Ratchet(As(An(Engine( Ratchet,(pawl(and(spring.( Feyman’s Ratchet Feynman’s (Brownian) ratchet: what is it and what was the point of explaining it? 1. Rod 2. Paddle wheel: the fins on the left 3. A pawl: that yellow thing that forces the ratchet not to go back in the other direction 4. A ratchet: that saw toothed gear on the right 5. A spring/bearing (m) low friction, to allow rotation. (From the fun explanation @: http://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetual- motion-gambling-scheme/) Forward rotation Backward rotation energy to lift the pawl ε ε energy to lift the pawl ε Lθ work done on load θ ε + Lθ energy to rotate wheel θ Lθ work provided by load by one tooth f 1 L / − −(ε+ θ ) τ1 ε + Lθ energy given to vane L fB = Z e Boltzmann factor for work provided by vane b −1 −ε /τ2 f fB = Z e Boltzmann factor for ν fB ratching rate with ν T2 T2 attempt frequency tooth slip ν f b slip rate with f fL power delivered B ν B θ attempt frequency ν ε energy provided to ratchet T1 T1 Equilibrium and reversibility Ratchet Brownian motor b f ε+ Lθ ε ratching rate = slip rate f f τ1 − − B = B f b Leq θε= −1 τ1 τ2 Angular velocity of ratchet: Ω =θν (fB − fB )= θν e − e τ 2 Reversible process by increasing the ε ε load infinitesimally from equilibrium − − ΩL=0 →θν e τ1 − e τ 2 Leq. This forces a rotation leading to Without load: heating of reservoir 1 with τ2 ε Lθ dq1=ε+Leq ? and cooling of − − Ω(L)τ1=τ 2=τ→θν e τ e τ −1 reservoir 2 as dq2=−ε: Equal temperatures: L dq1 ε + eq θ τ1 = = τ1 − dq 2 ε τ 2 τ Ω 1 rectification dq dq 12dS dS 0 +=+=12 L ττ12 τ isentropic process 2 Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998) 2 Forward rotation Backward rotation ε energy to lift the pawl ε energy to lift the pawl ε Lθ work done on load θ ε + Lθ energy to rotate wheel θ Lθ work provided by load by one tooth f 1 L / − −(ε+ θ ) τ1 ε + Lθ energy given to vane L fB = Z e Boltzmann factor for work provided by vane b −1 −ε /τ2 f fB = Z e Boltzmann factor for ν f ratching rate with ν T2 T2 B tooth slip attempt frequency ν f b slip rate with f fL power delivered B ν B θ attempt frequency ν ε energy provided to ratchet T1 T1 Equilibrium and reversibility Ratchet Brownian motor b f ε+ Lθ ε ratching rate = slip rate f f τ1 − − B = B f b Leq θε= −1 τ1 τ2 Angular velocity of ratchet: Ω =θν (fB − fB )= θν e − e τ 2 Reversible process by increasing the ε ε load infinitesimally from equilibrium − − ΩL=0 →θν e τ1 − e τ 2 Leq. This forces a rotation leading to Without load: heating of reservoir 1 with τ2 ε Lθ dq1=ε+Leq ? and cooling of − − Ω(L)τ1=τ 2=τ→θν e τ e τ −1 reservoir 2 as dq2=−ε: Equal temperatures: L dq1 ε + eq θ τ1 = = τ1 − dq 2 ε τ 2 τ Ω 1 rectification dq dq 12dS dS 0 +=+=12 L ττ12 τ isentropic process 2 Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998) 2 Forward rotation Backward rotation energy to lift the pawl ε ε energy to lift the pawl ε Lθ work done on load θ ε + Lθ energy to rotate wheel θ Lθ work provided by load by one tooth f 1 L / − −(ε+ θ ) τ1 ε + Lθ energy given to vane L fB = Z e Boltzmann factor for work provided by vane b −1 −ε /τ2 f fB = Z e Boltzmann factor for ν f ratching rate with ν T2 T2 B tooth slip attempt frequency ν f b slip rate with f fL power delivered B ν B θ attempt frequency ν ε energy provided to ratchet T1 T1 Equilibrium and reversibility Ratchet Brownian motor b f ε+ Lθ ε ratching rate = slip rate f f τ1 − − B = B f b Leq θε= −1 τ1 τ2 Angular velocity of ratchet: Ω =θν (fB − fB )= θν e − e τ 2 Reversible process by increasing the ε ε load infinitesimally from equilibrium − − ΩL=0 →θν e τ1 − e τ 2 Leq. This forces a rotation leading to Without load: heating of reservoir 1 with τ2 ε Lθ dq1=ε+Leq ? and cooling of − − Ω(L)τ1=τ 2=τ→θν e τ e τ −1 reservoir 2 as dq2=−ε: Equal temperatures: L dq1 ε + eq θ τ1 = = τ1 − dq 2 ε τ 2 τ Ω 1 rectification dq dq 12dS dS 0 +=+=12 L ττ12 τ isentropic process 2 Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998) 2 Forward rotation Backward rotation energy to lift the pawl ε ε energy to lift the pawl ε Lθ work done on load θ ε + Lθ energy to rotate wheel θ Lθ work provided by load by one tooth f 1 L / − −(ε+ θ ) τ1 ε + Lθ energy given to vane L fB = Z e Boltzmann factor for work provided by vane b −1 −ε /τ2 f fB = Z e Boltzmann factor for ν f ratching rate with ν T2 T2 B tooth slip attempt frequency ν f b slip rate with f fL power delivered B ν B θ attempt frequency ν ε energy provided to ratchet T1 T1 Equilibrium and reversibility Ratchet Brownian motor b f ε+ Lθ ε ratching rate = slip rate f f τ1 − − B = B f b Leq θε= −1 τ1 τ2 Angular velocity of ratchet: Ω =θν (fB − fB )= θν e − e τ 2 Reversible process by increasing the ε ε load infinitesimally from equilibrium − − ΩL=0 →θν e τ1 − e τ 2 Leq. This forces a rotation leading to Without load: heating of reservoir 1 with τ2 ε Lθ dq1=ε+Leq ? and cooling of − − Ω(L)τ1=τ 2=τ→θν e τ e τ −1 reservoir 2 as dq2=−ε: Equal temperatures: L dq1 ε + eq θ τ1 = = τ1 − dq 2 ε τ 2 τ Ω 1 rectification dq dq 12dS dS 0 +=+=12 L ττ12 τ isentropic process 2 Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998) 2 Feyman’s Ratchet Feynman’s (Brownian) ratchet: what is it and what was the point of explaining it? In one sentence: The first moral is the Second Law itself: it is impossible to extract directed motion from a random process (a single heat reservoir). Anyone who claims they can do so is either mistaken or a charlatan. (From the fun explanation @: http://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetual- motion-gambling-scheme/) In other words…. Well, the Brownian ratchet was a ‘thought’ experiment, along with Maxwell’s demon. It was used historically to attempt to explain away the 1) potential existence of a molecular machine that can perform…. 2) perpetual motion, without violating the laws of thermodynamics, namely the conservation of energy and of molecular machines. In other words, a fraud J as Feynman found…. and if anyone tries to convince you otherwise … just say: But, by decree of thermodynamics, Feynman’s ratchet cannot work as a heat engine. It plainly violates the Second Law, which says that useful work can only be obtained by the flow of energy from high to low temperature. This device purports to get energy from a single temperature reservoir: that of the air around it.