CHE 309: Chemicl Rection Engineering Lecture-8 Module 2: Rte Lw & Stoichiomtery (Chpter 3, Fogler)
Topics to be covered in tody s lecture Thermodynmics nd Kinetics Rection rtes for reversible rections Equilibrium Constnt Stoichiometric Tble Why do we need them Stoichiomteric Tble for Btch Rectors Constnt volume Vrible volume Stoichiometric Tble for Flow Rectors
Rection Rte Constnt (k) rrhenius Lw mjority of rection rte constnts cn be expressed by n empiricl reltionship tht ws developed by Swedish scientist, Svnte rrhenius. k = exp(-e/rt) = frequency fctor or pre-exponentil fctor E = ctivtion Energy R = Universl gs Constnt Svnte rrhenius rrhenius studied rection rtes s function of temperture, nd in 889 he introduced the concept of ctivtion energy s the criticl energy tht chemicls need to rect. He lso pointed out the existence of "greenhouse effect" in which smll chnges in the concentrtion of crbon dioxide in the tmosphere could considerbly lter the verge temperture of plnet. The following equivlent formultion of rrhenius' greenhouse lw is still used tody F = α Ln(C/Co) Here is crbon dioxide (CO 2 ) concentrtion mesured in prts per million by volume (ppmv); denotes bseline or unperturbed concentrtion of CO 2, nd F is the rditive forcing, mesured in wtts per squre meter. The constnt lph (α) hs been ssigned vlue between five nd seven
Some Interprettions nd Implictions of rrhenius Lw
Interprettion of prmeters in the rrhenius eqution k = exp(-e/rt) For simple rection in which two molecules collide nd rect, the pre-exponentil term in the rrhenius eqution cn be thought of s the frequency of collision. lso, the exponentil term cn be thought of s the frction of the rectnts tht posses energy greter thn E.
Frequency Fctor ccording to Collision Theory The pre-exponentil fctor cn be estimted for gs-phse rection from the collision theory, which is essentilly bsed on the Kinetic Theory of Gses. From kinetic theory of gses, the bimoleculr collision of unlike molecules in mixture of & B cn be written s: σ + σ 2 B 2 Z B = ( ) 8π kt ( + ) 6 2 0 M M B C C B o. of collision/sec-cm 3 σ i = vogdro's number k= Boltzmnn Constnt (.30 0 M i = dimeter of molecule of species i, cm = Moleculr mss of species i -6 erg / K)
Why is there n ctivtion energy In the cse of free rdicls: o need ctivtion energy In generl cse : in order to rect.the molecules need energy to distort or stretch their bonds so tht they brek them nd thus form new bonds. 2.The steric nd electron repulsion forces must be overcome s the recting molecules come close together The ctivtion energy: brrier to energy trnsfer between recting molecules tht must be overcome.
ctivtion Energy - E + BC B C B + C -B-C +BC E H R B + C Rection Coordinte ctivtion Energy: simple description of E would be the energy brrier tht rectnts must overcome for rection to proceed.
Illustrtion of ctivtion Energy explined by Trnsition Stte Theory
Why does incresing temperture result in incresed rection rte? E : minimum energy tht must be possessed by recting molecules before the rection will occur From: Chemicl Kinetics nd Rection Dynmics by Pul L. Houston
Implictions of rrhenius Lw k = exp(-e/rt) ln( k) = ln - (E/R) x /T k is most sensitive to temperture t lower tempertures k with higher E is more sensitive to temperture thn those with low E Rxn: High E Rxn 2: Low E /T
Kinetics of Mny Processes in ture follow rrhenius Reltionship Some Exmples Cricket chirping nt wlking Tumour growth Diffusion in solids [D = D o exp (-E D /RT)] References: M. I. Msel, Chemicl Kinetics nd Ctlysis Octve Levenspiel, Chemicl Rection Engineering Rte of Cricket Chirping
The Cse of nt wlking: Cn we relly represent it with rrhenius Lw? M. I. Msel, Chemicl Kinetics nd Ctlysis
The Cse of nt wlking 8 Running Speed (cm/s) 7 6 5 4 3 2 Rw dt 0 0 5 0 5 20 25 30 35 40 45 Temperture (C) 2.5 Processed dt ln (wlking speed) 2.0.5.0 0.5 0.0-0.5 -.0 3.5 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 000/T (K - )
Exmple 3- Determintion of the ctivtion Energy C 6 H 5 2 C C 6 H 5 C + 2 k (s - ) 0.00043 0.0003 0.0080 0.00355 0.0077 T (K) 33.0 39.0 323.0 328.0 333.0 ln k = ln E R ( ) T -4-5 ln k = -462*(/T) + 38.925 R 2 = 0.9999 ln k -6-7 -8-9 0.003 0.003 0.003 0.003 0.0032 0.0032 0.0033 /T Slope = -E/R, y-intercept = ln
If two k vlues re known t two different tempertures, ) ( ln ) exp( ) exp( ) exp( 2 2 2 2 2 2 T T R E k k RT E RT E k k RT E k RT E k = + = = = or k(t) = k (T 0 ) e E/R (/T 0-/T)
Fctors ffecting concentrtion In rel rectors, the temperture nd pressure my vry with time nd/or position. These chnges my ffect rection rte constnt nd/or concentrtion nd, thereby, the net rtes of rection. We hve seen erlier, how the rte constnt (k) is ffected by temperture k = exp (-E/RT) We will now evlute wht fctors ffect concentrtion of rectnts or product species.
Effect of pressure nd temperture on gs phse concentrtion Concentrtion, by definition is moles of species per unit volume. C i = ni V The ide here is to find out if nd how concentrtion my vry with pressure nd temperture. Eqution of Stte for Rel Gs PV = n = V z n RT z RT P z= for idel gs Concentrtion of species in gs phse is function of both pressure nd temperture.
Effect of pressure nd temperture on liquid phse concentrtion Let us sy tht we hve n i moles of species in totl liquid volume of V. ccordingly, the concentrtion of species i is: C i = ni V Our interest is in determining how the volume of liquid chnges with temperture nd/or pressure. The liquid density does not chnge significntly with temperture nd pressure, therefore, the volume of fixed mss of liquid does not chnge significntly either. Conclusion: Concentrtion of species in liquid-phse remins constnt with chnges in pressure nd temperture.
Wht is Stoichiometric Tble & Why do we need them? To nswer why, let us review our current sttus for designing rector: t = X dx r V 0 V = F ( r ( X ) ) Exit V = F X dx r 0 Wht is stoichiometric tble? Species Initilly (mol) Chnge (mol) Remining (mol) B -- D I Why do we need stoichiometric tble? If the rte lw depends on more thn one species, we must relte the concentrtions of the different species to ech other. => to express the rte lw s function of X.
Stoichiometric Tbles
Stoichiometric Tble for Btch Rectors Rection: b c + B C + d D Species : bsis of clcultion (limiting rectnt) θi = i0 0 Species Initil Chnge Remining B C D I (Inert) Totl BO CO DO IO - ( X ) = ( ) = 0 0 X = 0( θ X ) = θ B b b - ( X ) B = ( bθ B X ) b B = B0 ( 0 X ) = 0( θ B = θ C c c ( X ) C = ( θ c c+ X ) c C = C + ( 0 X ) = 0( θc + = θ D d d d d ( X ) D = ( θ D+ D = D + ( 0 X ) = 0( θd+ θ I I = IO d c b TO T = TO + ( + ) X = - 0 X 0 X 0 X ) ) ) We now hve reltionship between conversion nd number of moles of ech species