Entropy and Free Energy: Predicting the direction of spontaneous change The approach to Chemical equilibrium

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1 Lectue 8-9 Entopy and Fee Enegy: Pedicting the diection of spontaneous change The appoach to Chemical equilibium Absolute entopy and the thid law of themodynamics To define the entopy of a compound in absolute tems it is necessay to define a efeence value. We can define a zeo of entopy at K. Accoding to the thid law of themodynamics the entopy of a pefect cystal at T = K is zeo. The standad entopy S 98 of a substance is defined as the mola entopy at T = 98 K and 1 ba pessue. Units ae J mol -1 K -1. S 98 values ae temed absolute o thid law entopies. 1

2 S 98 = S1 + S + S3 + S4 + S5 The entopy of a substance inceases on heating. Thee ae sudden inceases in entopy on melting at T m and vapoization at T b. When C P,m /T is plotted against T the entopy changes due to heating the solid, liquid and gas ae given by the shaded aeas. Assume that substance emains in same phase between T = 98 K and T. Standad entopy at temp. T. S T = S 98 + C P, m T / K ln 98K

3 Entopy change in chemical eactions Sum of standad enopies of eactants ( p) S ( x) S98 = ν js98 ν j 98 Sum of standad entopies of poducts Stoichiometic coefficient (fom balanced equation) Reaction entopy At temp. T S T = S 98 + C P, m T / ln 98 K K C = ν jcp, m ( p) C ( x) P, m ν j P, m See Chemisty 3 woked example 15.5, p.718. Chemisty 3 section 15.4, pp Gibbs Enegy Change Kotz, section 19.5, 19.6 pp Chemisty 3, section 15.5, pp We now discuss the way in which the spontaneity of a pocess may be detemined. Fom nd law of themodynamics S ( tot) = S( sys) + S( su) Assume pocess occus at const. T and P. ( su) q ( sys) qev S( su) = = T q ( ) ( ) Hence ev P sys = H sys, H ( ) ( sys) S su = ev T Suoundings System J.W. Gibbs, whee we define the Gibbs Enegy as T G = H T S Now nd law themodynamics implies that Fo a spontaneous pocess at const T,P S( tot) = S( sys) + S( su) > S( tot) > H S tot = S sys G( sys) < ( ) ( ) ( sys) T Multiplying acoss by -T T S tot = H sys T S sys = G sys ( ) ( ) ( ) ( ) 3

4 At low temp T the enthalpy change pat Of G has a geate magnitude than T S. The sign of G depends on the sign of H. T S T S H H At high temp T the enthalpy change pat of G has a smalle magnitude than T S. The sign of G depends on the sign of S. G < fo a spontaneous pocess. H < (exothemic change) makes favouable contibution to spontaneity. H > (endothemic change) makes unfavouable contibution to Spontaneity. S <, entopy deceases, makes unfavouable contibution to Spontaneity. S >, entopy inceases, makes favouable change to spontaneity. 4

5 G < : eaction o pocess is spontaneous G > : eaction o pocess is not spontaneous G = : eaction o pocess is at equilibium Standad Gibbs enegy change of fomation f G 98 is defined as the change in Gibbs enegy when 1 mol of a compound is fomed at P = 1 ba and at T = 98 K fom its constituent elements in thei standad states. Standad Gibbs enegy of eaction G 98 is computed fom f G fo the eactants and poducts using the following ule. G = ν G ( p) ν G ( ) 98 j f 98 j f 98 x See Chemisty 3 woked example 15.8, pp Once the standad Gibbs enegy of eaction G 98 is known at 98 K then it is possible to compute the coesponding Gibbs enegy of eactionat any othe temp T using the definition of the Gibbs enegy function. T 98 P, m G = H T S T T T H = H + C T ( ) = H + C T 1 S T 98 P, m 98 = S 98 + C P, m T / ln 98 K K See Chemisty 3 Woked example 15.9, pp C = ν jcp, m ( p) C ( x) P, m ν j P, m Moe of this type of calculation in SF Themodynamics. 5

6 Chemical equilibium. What is chemical equilibium? How much poduct will fom unde a given set of stating conditions? What is the composition of a eaction mixtue when a chemical eaction has attained equilibium? What is the effect of tempeatue on the composition of a eaction mixtue at equilibium? Chemical Equilibium. fowad eaction N (g) + 3 H (g) NH 3 (g) evese eaction Kinetics Equilibium Reactant concentations decease with time ; poduct concentations incease with time. Afte a long enough time eactant and poduct concentations attain steady, time invaiant values. A state of chemical equilibium has been attained. If a plot of eaction ate vesus time is examined we note that the ate of the fowad eaction deceases and the ate of the evese eaction inceases with inceasing time, until, at long times they become equal. At this stage the eaction ates no longe change with time and the eaction is said to be at equilibium. Habe Pocess: Ammonia synthesis Kinetic definition of Equilibium. 6

7 Kinetics vesus Equilibium. Kinetics applies to the speed of a eaction, the concentation of poduct that appeas (o of eactant that disappeas) pe unit time. Equilibium applies to the extent of a eaction, the concentation of poduct that has appeaed afte an unlimited time, o once no futhe change occus. A system at equilibium is dynamic on the molecula level; no futhe net change is obseved because changes in one diection ae balanced by changes in the othe. At equilibium: ate fowad step = ate evese step Reaching Equilibium on the Macoscopic and Molecula Level NO N O 4 N O 4 (g) colouless NO (g) bown 7

8 Popeties of an equilibium Reaction Equilibium systems ae dynamic (in constant chemical change) and evesible (chemical change can be appoached fom eithe diection). Pink Blue Co( H O) Cl 6 Co( H O) Cl 4 Co( H O) Cl + H O Co( H O) Cl 4 + H O 6 Blue Pink PLAY MOVIE ( ) ( )( ) Fe H O + SCN Fe SCN H O + H O Afte a peiod of time, the concentations of eactants and poducts ae constant. The fowad and evese eactions continue afte equilibium is attained. PLAY MOVIE PLAY MOVIE 8

9 Examples of Chemical equilibia : Phase Change H O s ( ) H O( l) PLAY MOVIE Chemical Equilibium : a kinetic definition. Countless expeiments with chemical systems have shown that in a state of equilibium, the concentations of eactants and poducts no longe change with time. This appaent cessation of activity occus because unde such conditions, all eactions ae micoscopically evesible. We look at the dinitogen tetaoxide/ nitogen oxide equilibium which occus in the gas phase. N O 4 (g) colouless Equilibium: Kinetic analysis. t R = k[ NO4 ] R = k NO [ ] Valid fo any time t NO (g) bown R = R Concentations vay with time concentation k[ N O ] k 4 = [ NO ] eq [ NO ] eq k = = K [ N O ] k 4 eq [ NO] t [ N O ] 4 Equilibium constant Kinetic egime t time Concentations time invaiant [ NO ] eq [ NO4 ] eq Equilibium state NO t N O 4 9

10 m eactants n poducts ate ate fowad evese = k = k fowad evese [eactants] [poducts] n m ate fowad = ate evese k fowad [eactants] m = k evese [poducts] n k fowad k evese [poducts]n = = K the equilibium constant [eactants] m This is also known as the LAW OF MASS ACTION. The values of m and n ae those of the coefficients in the balanced chemical equation. The ates of the fowad and evese eactions ae equal, NOT the concentations of eactants and poducts. Initial and Equilibium Concentations fo the N O 4 -NO System at 1 C Initial Equilibium Ratio [N O 4 ] [NO ] [N O 4 ] [NO ] [NO ] [N O 4 ] The value of the atio of initial concentations vaies widely but always gives the same value fo the atio of equilibium concentations. The individual equilibium concentations ae diffeent in each case but this atio of equilibium concentations is constant. Constant values Equilibium constant K 1

11 The size of the equilibium constant indicates whethe the eactants o the poducts ae favoued. Reactants favoued when K c is small Reactants and poducts ae in almost equal abundance when K c nea unity Poducts favoued when K c is lage The Reaction Quotient, Q In geneal, ALL eacting chemical systems ae chaacteized by thei REACTION QUOTIENT, Q. a A + b B p P + q Q Q = p [ P] [ Q] t t a [ A] [ B] t t q b If Q = K, then system is at equilibium. 11

12 THE EQUILIBRIUM CONSTANT Fo any type of chemical equilibium of the type a A + b B p P + q Q the following is a CONSTANT (at a given T) Equilibium constant K = p [ P] [ Q] eq a [ A] [ B] eq q eq b eq Poduct concentations Reactant concentations If K is known, then we can pedict concentations. of poducts o eactants in the eaction mixtue at equilibium and hence the yield of the eaction. Relationship between Gibbs Enegy and Equilibium Constant. We now deive an expession which elates the change in Gibbs enegy fo a eaction as a function of the composition of the eaction mixtue at any stage in the eaction. We can define Gibbs enegy in tems of the activity a k of the species k. aa + bb pp + qq Hence afte some algeba and simplification the change in Gibbs enegy fo eaction can be computed. ( pg + qg ) ( ag bg ) G = + G = p P Q ( GP + RT ln ap ) + q( GQ + RT ln aq ) a( G + RT ln a ) b( G + RT ln a ) G = G A A p q ap. aq + RT ln a b aa. ab A B B = G B + RT lnq Activity = genealised concentation G k = G k + RT ln a If the eaction is allowed to poceed To equilibium then we eplace Q by The equilibium constant K and set G = by definition. G G + RT ln K = = RT ln K k ( pg + qg ) ( ag bg ) G = P Q A + B Reaction quotient Q 1

13 = + G G RT ln Q G = RT ln K G = RT ln K + RT ln Q = RT (ln Q ln K) This is the most useful fom of the equation fo intepetation. Q G = RT ln K If Q/K <1 then Q < K and G is negative, the eaction tends to poceed in fowad diection and the eaction is said to be poduct favoued. If Q/K > 1 then Q > K and G is positive. Hee the eaction does not poceed spontaneously in the fowad diection and is said to be eactant favoued. If Q = K then since ln 1 = G = and the eaction is at equilibium. Gibbs enegy change fo eaction mixtue a A + b B Unde non-equilibium conditions Gibbs enegy change is : G = G + RT ln = G + RT ln Q c C + d D c [ C] [ D] a [ A] [ B] d b Expession shows how G vaies with composition of eaction mixtue. c d Q = eaction quotient [ C] [ D] Q = [ A] a [ B] b Q defines eactant and poduct concentation atio (i.e. eaction composition) at any stage in chemical tansfomation. Q < K Q > K When G = at constant T and P we have equilibium. Hence Q = K c. = G + RT ln K c [ C] [ D] [ A] eq a[ B] d G RT ln eq G = RT ln K = + b eq eq These ae vey impotant elations! G K = exp RT 13

14 Poduct- o Reactant Favoed Pocesses K comes fom themodynamics. G < : eaction is poduct favoed G > : eaction is eactant-favoed If K > 1, then G is negative. If K < 1, then G is positive. G = RT ln K Poduct-favoed K > 1 Reactant-favoed K < 1 Gibbs enegy and chemical equilibium. G Reaction not spontaneous In fowad diection G = Equilibium Q=K Q lage, Q>K [P]>>[R] G positive Q small, Q<K [P]<<[R] G negative Reaction spontaneous In fowad diection G = G Standad state Q=1 lnq= ln Q G = G + RT ln Q 14

15 G G = ξ p, T fowad eaction spontaneous evese eaction spontaneous ξ = ξ = 1 Extent of eaction ξ Reaction Gibbs enegy G G = ξ p, T extent of eaction = ξ K =1 K << 1 K >> 1 G = G = G = ξ = ξ = 1 G is the slope of the G vesus ξ gaph at any degee of advancement ξ of the chemical eaction. 15

16 16

17 Key stages in the Habe-Bosch synthesis of ammonia. Fitz Habe Nobel Pize, 1918 NH 3 synthesis is exothemic ( H = kjmol -1 ). Hence K deceases as T inceases. N (g) + 3 H (g) NH 3 (g) Cal Bosch Nobel Pize, 1931 Opeating conditions dictated by a balance between kinetics and themodynamics. Yield of NH 3 is high when pessue P is lage and tempeatue T is low. Rate of fomation of NH 3 is low when T is low. High pessue and continuous emoval of NH 3 used to incease yield. Tempeatue is aised and catalyst employed to enhance ate of NH 3 fomation. Effect of Tempeatue on K c fo Ammonia Synthesis T (K) K c x 1 15 K C as T How can we explain this? We need to be able to pedict the way that K vaies with tempeatue T. This is given by the van t Hoff equation x x x x x x 1-17

18 Tempeatue dependence of equilibium constant: van t Hoff equation. G G = RT ln K = H T S H T S We assume that H and S ae independent of tempeatue T ove the tempeatue ange of inteest. Assume that K = K 1 when T = T 1 and K = K when T = T. Note that T > T 1. = RT ln K ln K H ln K1 = RT ln K ln K H ln K = RT 1 H = RT 1 K = ln K 1 S + R S + R H = R S + R 1 1 T1 T This can be used to undestand the tempeatue dependence of the equilibium constant. Now T >T 1 so the tem 1/T 1-1/T is positive since 1/T < 1/T 1. Also the tem ln{k /K 1 } depends on the sign of H. Endothemic eaction: H is positive, the exponential tem is also positive, and so K /K 1 >1 and K > K 1. The equilibium constant fo an endothemic pocess inceases with tempeatue. Incease in T favous poducts. Exothemic eaction: H is negative, and K < K 1. The equilibium constant fo an exothemic pocess deceases with an incease in tempeatue. Incease in T favous eactants. 18

19 Pecent Yield of Ammonia vs. Tempeatue ( C) Optimizing Ammonia Synthesis Best NH 3 yield at low T and high P. 7 4 T/ C 6 14 P / atm 34 NH 3 yield / mol % Pessue / atm Tempeatue / C Low T : slow eaction kinetics High P : expensive plant N (g) + 3 H (g) NH 3 (g) 19

20 Themodynamic citeia of spontaneity. G = RT ln K If the eaction is exothemic ( H < ) and S > then G < and K > 1 at all tempeatues. If the eaction is exothemic ( H < ) and S < then G < and K > 1 povided that T < H / S. If the eaction is endothemic ( H > ) and S > then G < and K > 1 povided that T > H / S. G poducts dominant at equilibium = H T S K >> 1 if G eaction themodynamically feasible < If the eaction is endothemic ( H > ) and S < then G < and K > 1 at no tempeatue. eactants dominant at equilibium K << 1 if G > eaction not themodynamically feasible

21 K c Chemical Equilibium Poblems. Suppose that.15 mol PCl 5 is placed in a eaction vessel of volume 5 cm 3 and allowed to each equilibium with its decomposition poducts phosphoous tichloide and chloine at 5 C. If the equilibium constant K c is 1.8, detemine the composition of the eaction mixtue at equilibium. PCl 5 (g) Phosphoous(V) chloide PCl 3 (g) + Cl (g) K c = [ PCl3][ Cl ] [ PCl ] Initial concentation of PCl 5 =.15 mol /.5 L =.3 M. Let an amount x of PCl 5 be used up in eaction to fom poducts. Species PCl5 PCl3 Cl = [ PCl3][ Cl ] [ PCl ] x + 1.8x.54 = 1.8 ± x = 5 ( 1.8) x = = x 4.1.(.54) =.6 Phosphoous(III) chloide and We choose the positive oot of the quadatic -.6 Initial concentation Change in concentation Equilibium concentation equation and so x =.6. [ PCl ] 5 =.3 x = [ PCl3] = x =.6 [ Cl ] = x =.6 M M 5.3 -x +x +x.3 - x x x.3.6 =.38 M The fomation of NO fom N and O contibutes to ai pollution wheneve a fuel is bunt in ai at a high tempeatue as in a gasoline engine. At 15 K the equilibium constant K = 1 x 1-5. Suppose a sample of ai has [N ]=.8 mol/l and [O ]=. mol/l befoe any eaction occus. Calculate the equilibium concentations of eactants and poducts afte the mixtue has been heated to 15 K. N ( g) + O ( g) NO( g) ICE Table K 5 = = = x x [ ] ( ) ( )( ) NO x 4x [ N ].[ O ].8 x. x.8. N /M O /M NO /M Initial.8. Change -x -x +x Equilibium.8-x.-x x x = amount eacted We assume that x << 1, i.e. less than 1% of initial eactant concentation [R] then ([R] -x) [R] Appoximation valid when equilibium Constant K is small and << 1. 4 [ ] = N x M 4 [ ] O x M [ ] NO x M 1

22 K c Unde cetain conditions nitogen and oxygen eact to fom dinitogen oxide N O. Suppose that a mixtue of.48 mol N and.933 mol O is placed in a eaction vessel of volume 1 dm 3 and allowed to fom N O at a tempeatue fo which K c = x Detemine the composition of the eaction mixtue at equilibium. N (g) + O (g) N O(g) K c = 13 = 1 [ NO] [ N ] [ O ] = [ N O] vey small ( x) (.48 x) (.933 x) When eaanged this expession yields a cubic equation in x. Since K c is small we can assume that x is also vey small and assume : K C x ( x) (.48) (.933) K (.48) C = Hence ou appoximation is OK.48 x x.933 Hence at equilibium: K c = [ NO] [ N ] [ O ] Initial concentations [ N ] [ O ].48 mol = =.48 M 3 1 dm.933 mol = =.933 M 3 1 dm [ N O] = Species [N] [O] [NO] Initial concentation Change in concentation Equilibium concentation [ N] =.48 x [ O ] =.933 x 9 [ N O] = x M.48 M.933M x -x +x.4-x.933-x x Le Chatelie s Pinciple Tempeatue, catalysts, and changes in concentation affect equilibia. The outcome is govened by LE CHATELIER S PRINCIPLE...if a system at equilibium is distubed, the system tends to shift its equilibium position to counte the effect of the distubance. Heni Le Chatelie Note : only Q esponds to addition of R o P ; K emains the same. When a eactant is added to a eaction mixtue at equilibium, the eaction tends to fom poducts. When a eactant is emoved, moe eactant tends to fom. When a poduct is added, the eaction tends to fom eactants. When a poduct is emoved, moe poduct is fomed. Q < K Q > K Q = K = [ P] [ R] [ P] eq [ R] eq

23 Le Chatelie s Pinciple Change T change in K theefoe change in P o concentations at equilibium Use of catalyst: eaction comes moe quickly to equilibium. K not changed. Add o take away eactant o poduct: K does not change Reaction adjusts to new equilibium position Effect of Vaious Distubances on an Equilibium System Distubance Net Diection of Reaction Effect on Value of K Concentation Incease [eactant] Towad fomation of poduct Decease [eactant] Towad fomation of eactant Pessue (volume) Incease P Decease P Tempeatue Towad fomation of lowe amount (mol) of gas Towad fomation of highe amount (mol) of gas None None None None Incease T Towad absoption of heat Inceases if H xn> Deceases if H xn< Decease T Towad elease of heat Inceases if H xn< Deceases if H xn> Catalyst added None; ates of fowad and evese eactions incease equally. None 3

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