CHEMICAL REACTIONS AND DIFFUSION A.K.A. NETWORK THERMODYNAMICS BACKGROUND Classcal thermodynamcs descrbes equlbrum states. Non-equlbrum thermodynamcs descrbes steady states. Network thermodynamcs descrbes dynamcs. Mod. Sm. Dyn. Sys. Reactons & dffuson page 1
SIMPLE SUBSTANCE: usng molar unts for mass (unts of N: e.g., Kg-moles, pound-moles) U = U(S,V,N) Varable mass makes ths a 3-port capactor Generalzed dsplacement assocated wth mass varaton (.e., the mass flow port ): mass, N correspondng flow varable: mass flow rate, dn/dt = N Correspondng effort varable: evaluate total dfferental du = ( U/ S)dS + ( U/ V)dV + ( U/ N)dN du = TdS PdV + µdn µ U N S,V T Ṡ C µ N P V Remember that the mass flow and entropy flow ports are coupled. Mod. Sm. Dyn. Sys. Reactons & dffuson page 2
MULTI-COMPONENT SYSTEM: Apples to reactons, dffuson, phase changes, etc. U = U(S,V,m 1,...m n ) Rewrte n terms of moles usng m = M N where N are numbers of moles and M are molecular weghts. U = U(S,V,N 1,...N n ) Ths descrbes a mult-port capactor wth a port for each mass varable. T Ṡ C P V µ 1 µ n N 1 N n µ 2 N 2 For each dstnct component, defne a dstnct effort. du = TdS PdV + µ dn U µ N S,V Mod. Sm. Dyn. Sys. Reactons & dffuson page 3
Typcal boundary condtons: constant temperature and pressure. The correspondng co-energy functon s the Gbbs functon. G = U - TS + PV = G(P,T,N 1,...N n ) dg = du TdS SdT + PdV + V dp dg = µ dn SdT + VdP G µ N P,T thus the effort varable s chemcal potental Gbbs free energy per mole. Asde: G Note that = N P,T U N S,V Mod. Sm. Dyn. Sys. Reactons & dffuson page 4
CAPACITOR CONSTITUTIVE EQUATIONS In general, the consttutve equatons of ths multport capactor are coupled µ = µ (N 1,...N n, T,P) In some smple cases, the consttutve equatons are uncoupled In most of what follows, for clarty we wll gnore the temperature and pressure ports EXAMPLE: IDEAL GAS MIXTURE Consttutve equaton for each deal component of a mxture of deal gases (vald for dlute solutons, low pressure gas mxtures) µ (T,P ) = µ (T,P) + RT ln(p /P) where P s the partal pressure of consttuent, P and T are the pressure and temperature of the mxture (usually assumed constant) R s the unversal gas constant. Mod. Sm. Dyn. Sys. Reactons & dffuson page 5
Rewrte usng mole fractons y of the consttuents (whch determne the composton of the mxture) where P /P = N /N = y N = N s the total number of moles n the mxture. µ = µ + RT ln y where µ = µ (T,P) s the reference chemcal potental. In general the total number of moles need not be constant. If so, the consttutve equaton s that of a coupled mult-port capactor. µ = µ (N 1,...N n ). (The pressure and temperature ports have been neglected for smplcty.) To smplfy we wll consder only reactons for whch N s constant. Consttutve equatons are then those of n uncoupled one-port capactors µ = µ (N ) Mod. Sm. Dyn. Sys. Reactons & dffuson page 6
GENERAL REACTION: The relatve magntudes of the molar flows of the consttuents are constraned by the reacton. Analogous to a knematc constrant Stochometrc coeffcents descrbe the constrant. Reactants Products ν R1 S 1 + ν R2 S 2 + ν P1 S 1 + ν P2 S 2 + S denotes speces subscrpt R denotes reactants P denotes products. EXAMPLE: 2 H 2 + O 2 2 H 2 O Defne a stochometrc coeffcent for speces ν = ν P ν R General equatons descrbng progress or advancement of the reacton n terms of mole numbers are N (t) = N (0) + ν ξ(t) ξ(t) s an advancement coordnate where ξ(0) = 0 Mod. Sm. Dyn. Sys. Reactons & dffuson page 7
EXAMPLE: A + B 2 D 1 A + 1 B + 0 D 0 A + 0 B + 2 D Stochometrc coeffcents: ν RA = 1; ν RB = 1; ν RD = 0 ν PA = 0; ν PB = 0; ν PD = 2 ν A = -1; ν B = 1; ν D = 2 The tme varaton of numbers of moles of the varous speces are related as follows N A (t) = N A (0) ξ(t) N B (t) = N B (0) ξ(t) N D (t) = N D (0) + 2 ξ(t) where ξ(0) = 0. In general the total number of moles may vary. N(t) = N (t) = N(0) + ν ξ(t) where ν = ν = ν Ρ ν P In the followng we wll assume that ν = 0 and the total number of moles s constant. Mod. Sm. Dyn. Sys. Reactons & dffuson page 8
AFFINITIES: The advancement coordnate plays the role of a dsplacement. The correspondng effort s an affnty. dg = G/ N dn = µ dn = dξ µ ν P,T P,T dg P,T = dξ µ (ν P ν R ) = dξ µ ν P µ ν R Defntons: Forward affnty: A F = µ ν R Reverse affnty: A R = µ ν P Affnty: A = A F A R Thus dg = A dξ Note that affnty s an effort n the usual sense. Just lke force, A s the negatve gradent of a (co- )energy functon. At equlbrum, dg = 0 and A = 0 or A F = A R. Mod. Sm. Dyn. Sys. Reactons & dffuson page 9
Meanng of varables N s mass (n moles) Ṅ s a mass flow rate (moles/sec) ξ s also a mass flow rate µ N s a power flow A ξ s also a power flow EXAMPLE: A + B 2 D A F = µ A + µ B A R = 2 µ D Equlbrum A F = A R µ A + µ B = 2 µ D Ths reacton may be depcted as follows µ A A: C 0 TF :1/ ν N A A A F A R µ D 1 TF C:D ξ ξ ν D N µ D B B: C 0 TF :1/ ν B N B The stochometrc coeffcents are represented as the modul of transformers. Mod. Sm. Dyn. Sys. Reactons & dffuson page 10
CHANGE OF VARIABLES: If total number of moles s constant, we may descrbe reacton advancement n terms of mole fractons y S (t) = y S (0) + ν x(t) The dmensonless number x(t) = ξ(t)/n s termed the degree of reacton. The tme varaton of mole fractons of the varous speces are related as follows y A (t) = y A (0) x(t) y B (t) = y B (0) x(t) y D (t) = y D (0) + 2 x(t) where x(0) = 0 Mod. Sm. Dyn. Sys. Reactons & dffuson page 11
Ths change of varables may be depcted as follows. µ A A: C 0 TF :1/N ν A Note: Ṅ A ẏ A A F A R µ D 1.. TF C:D x x Nν D ṄD ẏ B µ B TF :1/N ν B B: C 0 Ṅ B It has been mplctly assumed that the affntes are scaled by the total number of moles. : Mod. Sm. Dyn. Sys. Reactons & dffuson page 12
REACTION KINETICS Asde: The followng results are usually derved from knetc theory. Ths alternatve dervaton s fully equvalent. So far we have descrbed equlbrum energy storage. There s, as yet, no representaton of the process. The process wll be represented by a dsspatve element. Consder an even smpler two-component reacton µ A µ B A: C C:B ẏ A ẏ B A F = µ A A R = µ B A = µ A µ B ẏ A = ẋ = ẏ B Mod. Sm. Dyn. Sys. Reactons & dffuson page 13
To formulate dynamc equatons for a state determned system we need rate to be a functon of state. Consder ẋ = g(a) µ A µ B 1 ẏ A ẏ A ẋ B R ẋ = g (µ A µ B ) + RT ln(y A /y B ) If so, the rate of advancement depends only on the rato of number of moles e.g., double both of the mole fractons (or concentratons) and the advancement rate s unchanged. In general, ths s only vald near equlbrum. Typcally, doublng both of the concentratons wll double the reacton rate. Mod. Sm. Dyn. Sys. Reactons & dffuson page 14
POINT: A one-port resstor cannot descrbe reacton rates far from equlbrum. That requres a coupled 2-port resstor wth consttutve equaton ẋ = g(a F,A R ). Ths general relaton descrbes many dfferent models of reacton knetcs, ncludng the law of mass acton. Mod. Sm. Dyn. Sys. Reactons & dffuson page 15
LAW OF MASS ACTION Net reacton rate s proportonal to the product of mole fractons (concentratons) of reactants and products. Alternatvely, net reacton rate s the dfference between a forward reacton rate proportonal to the product of reactant mole fractons and a reverse reacton rate proportonal to the product of product mole fractons In general ẋ = k F y ν R k R y ν P k F and k R are forward and reverse reacton rate constants. Mod. Sm. Dyn. Sys. Reactons & dffuson page 16
At equlbrum, the rate of reacton advancement s zero (forward and reverse reacton rates balance) The reacton rate constants are related as follows ẋ = 0 k F y *ν R = k R y *ν P y *ν P k F = = y *(ν P ν R ) = y *ν k R y *ν R The astersk denotes equlbrum. The equlbrum constant of the reacton s κ = k F /k R = y *ν It depends only on temperature. Mod. Sm. Dyn. Sys. Reactons & dffuson page 17
ASIDE: In general the equlbrum product of mole fractons depends on temperature and pressure. Strctly speakng, the equlbrum constant that s ndependent of temperature s defned n terms of partal pressures as (P */P)ν = κ(t) If the total number of moles may vary and ν 0, the equlbrum product of mole fractons s dfferent from the equlbrum product of partal pressures as follows. y *ν = (P /P)ν (P */P)ν = (P /P)ν κ(t) where P denotes a reference pressure. If the total number of moles s constant, ν = 0 and the two constants are dentcal. Mod. Sm. Dyn. Sys. Reactons & dffuson page 18
EXAMPLE: For the smple reacton A B ẋ = k F y A k R y B Usng the defnton of the rate of advancement for ths reacton we obtan two coupled frst-order dfferental equatons descrbng the tme varaton of the composton (mole fractons) of the mxture. ẋ = ẏ B = k F y A k R y B ẋ = ẏ A = k F y A k R y B The equlbrum constant s defned by ẋ = 0 κ = k F = y B* k R y A * Mod. Sm. Dyn. Sys. Reactons & dffuson page 19
Usng the consttutve equatons of the capactors we may relate the equlbrum constant to the equlbrum chemcal potentals and the net change of Gbbs free energy at equlbrum. At equlbrum µ A * = µ B * (astersk denotes equlbrum) µ A µ B = RT ln (y B */y A *) κ(t) = y B* µ A µ B RT = exp G * = exp RT y A where G = µ A µ B The consttutve equaton for the 2-port dsspator s ẋ = k F e µ A /RT ea F /RT ea R /RT = ẏ A = ẏ B Thus the smple two-component reacton may be depcted as follows. µ A µ B A: C R C:B ẏ A ẏ B Mod. Sm. Dyn. Sys. Reactons & dffuson page 20
EXAMPLE: Reconsder the reacton A + B 2 D Law of mass acton: ẋ = k F y A y B k R y D 2 at equlbrum ẋ = 0 k F y A *y B * = k R y D * 2 µ A * + µ B * = 2 µ D * µ A + µ B 2 µ D = RT ln (y D * 2 /y A *y B *) κ (T) = k F k R y D * = 2 ya *y B * = exp where G = µ A + µ B 2 µ D G RT Mod. Sm. Dyn. Sys. Reactons & dffuson page 21
Substtute for y A, y B and y D n the law of mass acton usng the capactor consttutve equatons µ A µ A y A = exp RT µ B µ B y B = exp RT µ D µ D y D = exp RT κ = exp((µ A + µ B )/RT) exp(2µ D /RT) k F k R = we may defne the temperature-dependent constant G = k F exp( (µ A + µ B )/RT) = k R exp( 2µ D /RT) Mod. Sm. Dyn. Sys. Reactons & dffuson page 22
The consttutve equatons for the 2-port R and the assocated juncton structure are ẋ = G ea F /RT ea R /RT = ẏ A = ẏ B = 2 ẏ D The reacton may be depcted as follows. µ A A: C 0 TF :1/N ν A Ṅ A ẏ A A F A R µ D 1 R TF C:D ẋ ẋ Nν D Ṅ ẏ D B µ B B: C 0 TF :1/N ν B Ṅ B : Mod. Sm. Dyn. Sys. Reactons & dffuson page 23
Are we justfed n treatng the reacton knetcs as a two-port resstor? If so, the net power nto the resstor must be postve defnte. P net,n = P n - P out = A F ẋ A R ẋ = Aẋ by defnton, G > 0 f A F > A R then ea F /RT > ea R /RT and ẋ > 0 f A F < A R then ea F /RT < ea R /RT and ẋ < 0 f A F = A R then ea F /RT = ea R /RT and ẋ = 0 Aẋ 0 The law of mass acton does, ndeed, descrbe a dsspator, a 2-port resstor. Mod. Sm. Dyn. Sys. Reactons & dffuson page 24
ONSAGER RECIPROCITY For convenence, temporarly change notaton so that power s postve nto both ports of the 2-port R. Now lnearze A F A R R ẋ ż = ẋ ẋ A F G ż = RT ea F /RT= A F ẋ G ż = A R RT ea R /RT = AR G G RT e A F /RT RT ea F /RT ẋ A F = ż G G RT e A R /RT RT e A R /RT A R Mod. Sm. Dyn. Sys. Reactons & dffuson page 25
Ths s a conductance matrx. ẋ = Y A F ż A R Y s non-negatve defnte As above, Aẋ 0 Y s sngular because ẋ = ż The constrant on the flows may look lke a zero juncton but the consttutve equatons determnng the flows cannot (n general) be expressed as functon of effort dfferences In general, ths s not a nodc 2-port. Now consder equlbrum A F * = A R * Y becomes symmetrc. Symmetry of the conductance matrx at equlbrum s known as Onsager recprocty. Mod. Sm. Dyn. Sys. Reactons & dffuson page 26
At (or near) equlbrum, the law of mass acton may be represented by the one-port resstor and the one juncton above. The 1-port R consttutve equaton s G ẋ RT exp(a F */RT) However, far from equlbrum, Y s not symmetrc. Thus the 2-port R contans an embedded gyrator Reacton knetcs are closely analogous to transstor behavor. In fact, transstor crcuts may be used to model complex reactons. Mod. Sm. Dyn. Sys. Reactons & dffuson page 27
DIFFUSION Dffuson (e.g. across a membrane) may be treated exactly lke a reacton A A In that reacton the total number of moles s constant. ν = 0 k F = k R = p (permeablty) The equlbrum constant s k F κ = kr = 1 Mod. Sm. Dyn. Sys. Reactons & dffuson page 28