0.0J /.77J / 5.60J hermodynamics of Biomolecular Systems 0.0/5.60 Fall 005 Lecture #6 page he Second Law First Law showed the euivalence of work and heat U = + w, du = 0 for cyclic process = w Suggests engine can run in a cycle and convert heat into useful work. Second Law Puts restrictions on useful conversion of to w Follows from observation of a directionality to natural or spontaneous processes Provides a set of principles for - determining the direction of spontaneous change - determining euilibrium state of system Need a definition: Heat reservoir Definition: A very large system of uniform, which does not change regardless of the amount of heat added or withdrawn. Also called a heat bath. Real systems can come close to this idealization. wo classical statements of the Second Law: Kelvin Clausius and a Mathematical statement
0.0J /.77J / 5.60J hermodynamics of Biomolecular Systems 0.0/5.60 Fall 005 Lecture #6 page I. Kelvin: It is impossible for any system to operate in a cycle that takes heat from a hot reservoir and converts it to work in the surroundings without at the same time transferring some heat to a colder reservoir. (hot) >0 w<0 -w= -w (hot) -w >0 w<0 <0 =-w- IMPOSSIBLE!! - OK!! (c old) II. Clausius: It is impossible for any system to operate in a cycle that takes heat from a cold reservoir and transfers it to a hot reservoir without at the same time converting some work into heat. >0 <0 - = (hot) - (hot) - w >0 w> 0 <0 - = w+ IMPOSSIBLE!! (c old) (c old) OK!! Alternative Clausius statement: All spontaneous processes are ersible. (e.g. heat flows from hot to cold spontaneously and ersibly)
0.0J /.77J / 5.60J hermodynamics of Biomolecular Systems 0.0/5.60 Fall 005 Lecture #6 page 3 Mathematical statement: đ = 0 and đ < 0 đ is a state function = ds ds = đ S ENROPY ds = 0 S = S S = > for cycle [] [] [] + S < 0 = S > < 0 Kelvin and Clausius statements are specialized to heat engines. Mathematical statement is very abstract. Let s Link them through analytical treatment of a heat engine. he Carnot Cycle - a typical heat engine All paths are ersible p adiabat isotherm ( ) 4 isotherm ( ) adiabat 3 (hot) (c old) w
0.0J /.77J / 5.60J hermodynamics of Biomolecular Systems 0.0/5.60 Fall 005 Lecture #6 page 4 isothermal expansion at (hot) U = + w 3 adiabatic expansion ( = 0) U = w 3 4 isothermal expansion at (cold) U = + w 4 adiabatic compression ( = 0) U = w Efficiency = work output to surroundings heat in at (hot) = ( w w w w ) + + + st Law du = 0 + = ( w + w + w + w ) + Efficiency = = + Kelvin: < 0 Efficiency ε < (< 00%) -w = ε = work output Note: if the cycle were run in erse, then < 0, > 0, w > 0. It s a refrigerator! Carnot cycle for an ideal gas V = w = pdv = R ln V = U 0; 3 = w = ( ) 3 4 0; CV γ V Rev. adiabat = V3 = = = 4 V4 U 0; w pdv = Rln 3 V3 4 w = ( ) = 0; CV Rev. adiabat V4 = V γ
0.0J /.77J / 5.60J hermodynamics of Biomolecular Systems 0.0/5.60 Fall 005 Lecture #6 page 5 ( V4 V ) ( ) ln 3 = ln V V γ γ V V V 4 V = = = = V4 V3 V3 V đ or + = 0 = 0 this illustrates the link between heat engines to the mathematical statement of the second law Efficiency ε = + = 00% as 0 K For a heat engine (Kelvin): > 0, w < 0, < otal work out = w = ε = ( w) < Note: In the limit 0 K, (-w), and ε 00% conversion of heat into work. 3 rd law will state that we can t reach this limit! For a refrigerator (Clausius): > 0, w > 0, < otal work in = w = But = w = Note: In the limit 0 K, w. his means it takes an infinite amount of work to extract heat from a reservoir at 0 K 0 K cannot be reached (3 rd law).
0.0J /.77J / 5.60J hermodynamics of Biomolecular Systems 0.0/5.60 Fall 005 Lecture #6 page 6 he efficiency of any ersible engine has to be the same as the Carnot cycle, this can be shown by running the ersible engine as a refrigerator, using the work output of a Carnot engine to drive it so that the total work out is zero, and showing that, if the efficiency of the ersible engine is higher, then the second law is broken. Additionally: We can approach arbitrarily closely to any cyclic process using a series of only adiabats and isotherms. So, for any ersible cycle đ 0 = his is consistent with the mathematical statement of the second law, which defines Entropy, a function of state, with đ đ ds = S = S S = Note: Entropy is a state function, but to calculate S from reuires a ersible path. An ersible Carnot (or any other) cycle is less efficient than a ersible one. p adiabat 4 ersible isotherm with pext = p isotherm (.) adiabat 3 ( w) ( w) w w ir U = + w = + w < > <
0.0J /.77J / 5.60J hermodynamics of Biomolecular Systems 0.0/5.60 Fall 005 Lecture #6 page 7 ** An ersible isothermal expansion reuires less heat ** than a ersible one. ε = + < + = ε ( < 0) also đ đ đ < < 0 his leads to the Clausius ineuality đ 0 contains đ đ = 0 < 0 Important corollary: he entropy of an isolated system never decreases (A) ersible (A): he system is isolated and ersibly (spontaneously) changes (B) ersible from [] to [] (B): he system is brought into contact with a heat reservoir and ersibly brought back from [] to [] Path (A): = 0 (isolated) Clausius = 0! đ 0 đ đ + 0
0.0J /.77J / 5.60J hermodynamics of Biomolecular Systems 0.0/5.60 Fall 005 Lecture #6 page 8 đ = = S S S 0 S = S S 0 his gives the direction of spontaneous change! For isolated systems S > 0 Spontaneous, ersible process S = 0 Reversible process S < 0 Impossible S = S S independent of path But! Ssurroundings depends on whether the process is ersible or ersible (a) Irersible: Consider the universe as an isolated system containing our initial system and its surroundings. S = S + S > 0 universe surr system S > S sys surroundings (b) Reversible: S = S + S = 0 univ surr sys S = S surr sys S universe 0 for any change in state (> 0 if ersible, = 0 if ersible)