Chemical Equilibrium. Chapter 6 Spontaneity of Reactive Mixtures (gases) Taking into account there are many types of work that a sysem can perform
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1 Ths chapter deals wth chemcal reactons (system) wth lttle or no consderaton on the surroundngs. Chemcal Equlbrum Chapter 6 Spontanety of eactve Mxtures (gases) eactants generatng products would proceed up untl the macroscopc concentratons of all the speces nvolved cease to change wth tme. At that pont the reacton (system) has reached the equlbrum poston. eactons that would evolve to reach the equlbrum poston wth tme. The crtera that a chemcal reacton (system) should satsfy to n order to the reach equlbrum poston under condtons of constant temperature, volume and constant pressure, temperature s establshed n ths chapter. Constant temperature, pressure and volume are those at ntal and fnal state. For a process to be spontaneous: S S S That s, S S S unv sys surr sys surr Total For the system, n general ; S for reversble processes; dq TdS du dw for rreversble processes; TdS du dw. e. du dw TdS General condton for Spontanety: Takng nto account there are many types of work that a sysem can perform du dw TdS du dv dw TdS ext nonv Condton for spontanety s a functon of state functons and path functons. For an solated system; because w =, du = thus; TdS for spontanety; seen before! For an sothermal process - condton for spontanety: du dw dw TdS V nonv du TdS dw dw V nonv
2 Spontanety - reactons run at constant T and V: Also consderng the term TS: d() TS TdS SdT TdS du d() TS dw dw V d() U TS dw dw V () d U TS dw dw V nonv nonv nonv :sothermal process; dt= The combnaton of state functons (U TS) defnes a new state functon, Helmholtz energy, A. Then the general condton of spontanety for sothermal processes becomes; da dw dw V nonv For reactons run at constant T For reactons run at constant T and V (contnued): Maxmum work at constant T wll then be; da dw dw da dw V nonv Total For reactons are carred out at constant T and V, where dv = wth non -V work not possble, the condton for spontanety; da dw.e. da. Total Spontanety - for reactons run at constant T and : Chemcal reactons are run under constant pressure than at constant volume condtons. At constant and T, consder the nequalty for the general case, du dw TdS du dv dw TdS now the term TS: d() TS TdS SdT TdS and the term V: () d V dv Vd dv ext nonv Therefore, ()() du d V d TS dw d() U V TS dw usng H U V () d H TS dw nonv nonv nonv For reactons run at constant T and (contnued): Maxmum work at constant T wll then be; () d H TS dwnon V dg dw du dw nonv The combnaton of state functons (H TS) defnes a new state functon, Gbbs energy, A. For reactons are carred out at constant T and, where d = wth non -V work not possble, the condton for spontanety; dg. Total
3 A dfferent perspectve of Gbbs Free energy G G = H TS Dvdng by (-T) we get G = H T S G H S T T Each term can be vewed n terms entropy. If a change takes place then -ΔH/T s the change n entropy n the surroundngs, ΔS s the change of entropy n the system. Therefore, the total change n entropy s the sum of them. G H S T T S S S unv surr sys For reactons run at constant T and V: da dw da. For reactons run at constant T and : nonv A U TS =() H V TS A U TS = H V TS const T A H V n T TS G H TS dg dw dg nonv const T const V Spontanety and Temperature non-spontaneous G H TS non-spontaneous Dfferental forms of U, H, A, and G Defntons: G H ; S G H ; S T H ; S spontaneous T H ; S spontaneous Dfferental forms: du TdS dv dh TdS Vd da SdT dv dh SdT Vd.e. U f ( S,) V H f ( S,) A f ( T,) V H f ( T,)
4 dh TdS Vd means H f ( S,) Expanson gves: H H dh TdS Vd ds d S Comparng terms: S New defntons of T and V. Maxwell elatons: du TdS dv dh TdS Vd da SdT dv dh SdT Vd For example: Second partal dervatves Maxwell elatons: H H dh ds d S H ( S,)(,) H S S S S S S Maxwell elatons: Maxwell relatons have been derved usng the fact that U, H, A. and G are state functons. They are useful n transformng non-obvous partal dervatves n other partal dervatves that can be drectly measured.
5 Gbbs and Helmholtz Energy Dependence on, V, and T Gbbs Dependence on, V and T Usng: S and always postve values. Therefore Helmholz energy of a pure substance decreases as ether the temperature or the volume ncreases. We get; Gbbs energy decreases wth ncreasng temperature, t ncreases wth ncreasng pressure. for solds and lquds = for gases For gases the Gbbs free energy ncreases wth t s partal pressure. Gbbs Dependence on, V and T Because the thermodynamc equlbrum constant K s related to G/T the expresson for the temperature dependence of G/T s useful. G(298.15,) = emember Also: (Eq. 6.34)
6 2 2 d[ G / T ] d[ G / T ] H H d[1/ T ] d[ 1/ T ] d G G 1 H d H d d[1/ T ] T T T Gbbs Energy of a eacton Mxture: eacton mxtures are made of many components, say labelled 1,2,3,.,..,n The total Gbbs energy of the mxture would be a functon of many quanttes. G = f(t,, n 1, n 2,,n n ) T G 1 d H d T T T1 T1 T G2 G H : assumng H s a constant (not qute) T2 T1 T2 T1 Gbbs Helmholtz Equaton Defnng chemcal potental (G m, ) of speces as; The crteron for spontanety of chemcal reactons: dg For a process carred out at constant T and ; For a mxture of chemcals that of a composton (n ) the Total Gbbs energy at constant T, s If the mxture changes ts composton (chemcal reacton) the change n Gbbs energy; dg dn The crteron for spontanety of chemcal reactons: dg dn Thus spontaneous change s from hgh chemcal potental to one of low chemcal potental. The flow of materal wll contnue untl the chemcal potental has the same value n all regons of the mxture. The crteron for equlbrum n a multcomponent mxture s that the chemcal potental of each speces wll be the same throughout a mxture. dg
7 Gbbs Energy of a Gas n a Mxture Spontanety for a reacton however does not mply that the reacton goes to completon, that s the lmtng reacton beng consumed nearly completely at the end. At equlbrum the reacton mxture would contan more products than reactants. A Gbbs free energy of a reactve mxture (of deal) gases when mxed would be some from each component n the mxture. Consder the case where all reactants are ntally separated by a barrer and are allowed to react. The reacton can be envsaged to occur n two stages mxng of reactants and converson of reactants to products. In both stages the concentraton of ndvdual speces change and therefore ther chemcal potentals change. Mxng of reactants Would H 2 (A) move thro the membrane spontaneously? If so, at equlbrum stage the partal pressure of H 2 s the same on both sdes and from pont of vew H 2 t s chemcal potental s the same on both sdes. ; pure, end mxed H 2, left H 2, rght H 2 H 2 G G nt ln T ln ( T,) T ln start, pure start, pure H 2 H 2 H 2 H 2 :start start H 2 Intal pressure n both compartments = Sempermeable membrane Mxng of reactants ; pure, end mx H 2, left H 2, rght H 2 H 2 T ln T ln pure, end mxed H 2 H 2 H 2 H 2 H 2 Now gas A n a mx: A xa artal pressure :end Total pressure of mx Sempermeable membrane start H 2 end H 2 mx H 2
8 @ equlbrum; T ln mxed H 2 H 2 H 2 H 2 ln T T ln T ln x mxed H 2 H 2 H 2 H 2 A mxed A ( T,)(,) A Tln T ln T xa pure ( T,)(,) ln T T x mxed A A A mxed A ( T,) = chemcal potental of a component A of mole fracton x A, n an deal gas mxture wth a total pressure of (the general equaton). Note: x A < 1, so (T lnx A ) < mxed A ( T,)(,) pure A T Now x A <, therefore chemcal potental of a gas n a mxture s less than that of the pure gas f the total pressure s the same for the pure sample and the mxture. Because (T, ), dffuson of H 2 from the left sde to the rght sde of the system wll proceed untl the partal pressures of H 2 on both sdes of the membrane are equal. Applyng the same argument for Ar t can be concluded mxng of the two subsystems would be spontaneous f they were not separated by the membrane because both components would have a lesser free energy after mxng. mxed pure ( T,)(,) ln T T x Ar Ar Ar Gbbs Energy of Mxng G mx (Ideal Gases): Mxture of He, Ne, Ar, and Xe (), total number of moles n, at the same temperature and pressure. The gases are allowed to mx. All gases unformly dstrbutes n the contaner. The transformaton s the mxng (fnal state, f) of the (n, mol) four pure substances (ntal state, ). Gbbs Energy of Mxng G mx (Ideal Gases): Mxture of He, Ne, Ar, and Xe (), total number of moles n, at the same temperature and pressure. The gases are allowed to mx. All gases unformly dstrbutes n the contaner. The transformaton s the mxng (fnal state, f) of the (n, mol) four pure substances (ntal state, ). G mx G G n G n G n f m,, f m,, m,, f m,, m m n n T ln x n m,, m,, n n T n ln x T n ln x nt ln x n n n n T x n ln,,,, nt x ln x Spontaneous process
9 G mx Example: Bnary mxture G o for a Chemcal eacton: Consder a general reacton (balanced); X S mx x A =.5 x A x A All spontaneous processes are assocated wth loss of G. n mxng t s brougt about by large gan n S. G mx s at a mnmm when S mx at ts maxmum. Assume all pure separated gaseous components n the reacton mxture, each component wth a certan partal pressure (~concentraton). For the mxture there exsts only one set of partal pressures that would correspond to the equlbrum state partal pressures. In general f the system s not at equlbrum dg. and the system, as always tend to reach equlbrum, for the ncremental change n composton dn the correspondng change n G would be dg, wth dn s related to the stochometry ; dg dn By defnton G o s the free energy change when reactants at ther ST are converted to products at ST; G G O O f, The release of free energy, G o < ndcates a spontaneous reacton and vce versa. Also note: dg dn Equlbrum Constant of eactons (Ideal Gases): For a generc reacton (gaseous): Mx all speces are present n the reacton mxture. The reacton Gbbs energy for the arbtrary set of partal pressures A, B, C and D s, gven by, per stochometry dn = G G f, = Free energy of formaton of. where ΔG = ν ΔG O O f,
10 Q = eacton quotent The reacton quotent s such that G, the concentratons of the speces are such that reacton would not move ether drecton. That s the reacton s at equlbrum. The specal case of Q at equlbrum s K. G T ln K G T ln K The reacton quotent s such that G, the concentratons of the speces are such that reacton would move forward so that Q K. G G T lnq G T ln K T lnq T lnq T ln K Q K The reacton quotent s such that G, the concentratons of the speces are such that reacton would move backward so that Q K. G G T lnq G T ln K T lnq T lnq T ln K Q K G G T ln K G T ln K G ln K G T G o s a functon of T only therefore K s also a functon of T only. The thermodynamc equlbrum constant K does not depend on the total pressure or the partal pressure of reactants or products. relatonshp between the partal C D pressures of all reactants and products K A B Changng (s) of a equlbrum would change equlbrum concentratons. Q all values change to reestablsh the reacton quotent to K, a new eqlm. poston Note: also that K s a dmensonless number because each partal pressure s dvded by o. K
11 artal ressure n a Mxture of Ideal Gases: For a mxture of gases at a total pressure of tot wth each gaseous component wth n moles n the mxture, the partal pressure og component s, n x tot tot n Varaton of K wth Temperature: G T ln K ln K ln K G H TS H S T T T H S :General relatonshp T NH 3 synthess Temperature dependence of K s prmarly determned by H. If H, K ncreases wth ncreasng T and f H,K decreases wth T ncreasng. S has essentally no effect on the dependence of K vs T Coal Gasfcaton weakly dependent on T G H S T T ln K If S, (ln K ) s ncreased by the same amount relatve to all H / T at all T. If S, (ln K ) s decreased by the same amount relatve to allh / T at all T. G ln K See Eq T
12 Equlbra Involvng Ideal Gases and Sold or Lqud hases consdered constant. If the heat capactes of reactants and products are nearly the same, H nearly ndependent of temperature. Ch.3 sl 12 Some chemcal reactons nvolves a gas phase n equlbrum wth a sold or lqud phase. A general example would be; a A(s) + b B(g) = c C(l) +d D(g) Chemcal potental for solds and pure lquds are constant for low to moderate pressures. d D K and = for solds and pure lquds b B nt also G n T ln and ct V aa()()()() s bb g cc l dd g G c d a b C D A B D B cc d D T ln a A b B T ln D B GC GD GA GB dt ln bt ln D d b D B G T ln T ln G T ln B G G T ln K d b Other Expressons of K (conc. n alternate unts) Alternate expressons of K, n terms of mole fractons (x) or molartes (M), arbtrary K s consdered for the llustraton whch can be generalzed. = total pressure, V = reacton contaner volume. V d d d n T D xd d d d b ngas x n T D xd K c T K b b b b b x V x B xb B xb c T d d d d D D D d d D cdt c Tc c c d b c c c c T K K b b b b b b C B cbt cbtc c B c T c B c c c c T ngas
13 Extent of reacton ; s a quantty that measures the extent n whch the reacton proceeds. at the begnnng of a reacton =. n moles and n r remans after a materal change due to reacton, then the change n the extent of reacton d wll be ( s the stochometrc coeffcent of speces of the reacton, dn change n # moles of due to reacton.) nr n d and nr n dn n dn n dn d d dn 1 2 dn d dn The extent of reacton s a useful quantty n computatons equlbrum reactons, consder the reacton; the extent of reacton from ts defnton 3A 2B + C (2/3)(.5).5/3.( 5 /)( 2.)( 3.) / d Large eq at equlbrum mples progress of the forward reacton and therefore larger K eq values. dn d dg dn d d dg G d G T, G At a gven composton of the reacton mxture because each can be calculated, so would be G., Both G and G / are negatve, forward reacton spontaneous. T Both G and G / are postve, backward reacton - spontaneous. T, Both G and G /, are zero, nether drectons spontaneous, T system at equlbrum.
14 Dependence of K and K x on T and : K x K n gas H o K eq (ncrease - forward favored) < Decreases wth T decrease > Increases wth T ncrease n gas K x eq (ncrease - forward favored) > Decreases wth decrease < Increases wth ncrease = No change wth No effect So far ndvdual and mxtures of deal gases were consdered. Ideal gas condtons form an mportant reference pont for the study and the analyss of ndvdual and mxtures of real gases.
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