economical way and of how to develop the process e.g. by new choice of catalyst. 1.1 Reactions: Thermodynamic control vs.

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1 Introduton Reaton knets s the part of physal hemstry whh seeks explanatons for the tme-dependent laws whh hemal reatons obey. Thermodynams gave us answers to the queston: What knd of reatons an happen n a reaton vessel f we know the amounts and physo-hemal propertes of the substanes n the vessel? Chemal knets s able to gve us the answer to the queston: How fast an the reatons, foretold by thermodynams, proeed at dfferent temperatures? In addton, hemal knets tres to gve good hnts to answer the queston: In whh way do the reatons really happen? Chemal knets s stll rather expermental sene. Wthout the results from well planned experments t s dffult to say anythng about the dynams of reatons. From ndustral pont of vew the results obtaned by hemal knet methods are extremely valuable. They are the bass of desgnng the heart of any hemal proess, the reator n whh hemal reatons are arred out n ndustral sale. We an defne three man goals for hemal engneerng reaton knets. Frstly, t s neessary to determne the tme dependeny of the hemal reaton of nterest. Ths means that we are seekng a mathematal expresson whh explans the degree of advanement of the reaton. Ths expresson s alled the reaton rate law n hemal knets. Seondly we seek for a smple relatonshp for the dependeny of the rate of reaton and temperature. For ths purpose we use Arrhenus equaton, whh adequately explans most of these needs. Thrdly, we are nterested n to have some dea of the mehansm of reaton. Although a detaled dvson of the total reaton to reaton steps s very demandng task we should stll try to model at least those reatons steps, whh essentally affet the rate law. Ths usually gves us both a good dea of how to run the overall reaton n the most

2 eonomal way and of how to develop the proess e.g. by new hoe of atalyst.. Reatons: Thermodynam ontrol vs. knet ontrol The man task of physal hemstry and espeally hemal thermodynams s to get an answer to the queston: To what extent does ertan hemal reaton or proess run before equlbrum s reahed and how temperature T, pressure p and the onentratons [A ] of the omponents n the reaton mxture nfluene the equlbrum state. The seond law of thermodynams deals wth the natural dreton of proesses and the queston whether a hemal reaton an our by tself. The rteron for spontaneous (natural) hange of the reaton gven by Eq. (.) aa + bb = dd + ee (.) an be mpled by Eq. (.2) d [ ] [ ] a [ A] [ B] ΔG = RT lnk ln D E e b 0 (.2) The general ondton for the equlbrum and the equlbrum onstant K of reaton (.) are gven by: o ΔG(T) = 0 and K (T) = exp( Δ G (T) / RT) (.3) At ertan values of T and p dfferent from the standard state temperature T 0 = 298 K and standard state pressure p 0 = 0 kpa we an obtan the equlbrum omposton for the reaton mxture from the value of equlbrum onstant K defned n Eq. (.4).

3 K d [ D] [ E] a [ A] [ B] e = b (equlbrum at T and p) (.4) Ths thermodynam expresson gves us a powerful tool for analyzng the reatons and proesses whh go to equlbrum but t has nothng to do wth tme. Knowledge of the exat value of the thermodynam equlbrum onstant, K gves us the possblty to alulate the maxmum possble yeld e.g. of NH 3 obtanable at any gven T and p from the reaton between N 2 and H 2. If, however, the rate of the reaton between N 2 and H 2 s too slow, the reaton wll not be eonomally feasble to be arred out beause equlbrum s not reahed wthn reasonable tme of reaton. In preparatve reatons of organ hemals several possble ompetng reatons an our and the relatve rates of these reatons nfluene the yeld of eah produt. Reaton rates are fundamental to funtonng of lvng organsms. Bologal atalysts (enzymes) ontrol the atvty of an organsm by seletvely speedng up ertan reatons and nhbtng others. Consequently, to understand and predt the behavor of hemal reatons and proesses one must onsder arefully both the thermodynam and knet lmtatons of the advanement of reatons. Reaton knets - hemal knets - s the study of the rates and mehansms of hemal reatons. The mehansm of the reaton s the sequene of elementary reatons (reaton steps) that add up and together gve the overall reaton. Mehansm s a hypothess about the elementary steps through whh the hemal reaton ours.

4 .2 The degree of the advanement of reaton Consder for example the homogeneous reaton presented n Eq. (.). It s assumed that the reaton ours n a losed system. In general we an present any hemal reaton by Eq. (.5) N 0= ν Y (.5) = 0 In ths equaton ν 's are the general stohometr numbers of omponents,.e. -a, -b, d, and e and Y are the hemal spees A, B, D, E nvolved n the reaton gven by Eq. (.). Note that the general stohometr numbers are postve for produts and negatve for reatants. The amount of reaton that has ourred wthn some perod of tme s expressed by the degree of the advanement (extent) of reaton, ξ # whh s defned by Eq. (.6) n = n +νξ # (.6) 0 Here n 0 s the amount of substane present ntally n the reaton mxture, and n s the amount of substane present at some later moment of tme. Sne n s expressed n moles and ν s a dmensonless quantty the advanement of the reaton ξ # s expressed n moles. The rate of reaton dξ # / for the reaton gven n Eq. (.) s defned as

5 dξ # dna dnb dnd dne = = = (.7) a b d e where n A and n B and respetvely n D and n E are the numbers of moles of reatants and produts. To nterpret the rate of reaton we need materal balane for the reaton mxture. From Eq. (.6) we obtan dn =νdξ # and dn d = ν ξ# (.8) Usually we use volume onentratons [mol/dm 3 ] n desrbng hemal knets and therefore we dvde the rate of reaton dξ # / by total the volume V of the reaton mxture. Now we an defne the rate of reaton r as follows: r # dξ dξ = = = V dn ν V (.9) In Eq. (.9) ξ = ξ # /V denotes the degree of the advanement of reaton per unt volume. In many (but not all) systems studed, the volume V s ether onstant or hanges by a neglgble amount durng the reaton. If V s onstant we have: d n V d = (.0) and thus e.g. for the reaton gven by Eq. (.) we obtan (when V = onstant):

6 r [ ] db [ ] dd [ ] de [ ] da = = = = (.) a b d e da In everyday language the quantty [ ] s often alled "the rate of reaton". Common unts used for r are: mol dm 3 s and kmol m 3 s. However, n ndustral proesses (espeally n proesses ontanng reatons between gases) the total volume of reaton mxture V hanges durng the reaton and Eq. (.2) must be used n areful analyss of reaton knets: r = d n V dn n = ν ν V ν V 2 dv (.2) For heterogeneous reatons we have to nlude n the rate law some parameter whh haraterzes the type of the heterogeneous reaton and makes the rate of reaton an ntensve property. Espeally for surfae reatons we ommonly use the defnton: r' = S dξ # (.3) where S s the surfae area. In heterogeneous atalyss ether the followng defnton r'' = W dξ # (.4)

7 where W s the mass of the atalyst or the defnton gven by Eq. (.5) where V s the total volume of atalyst s ommonly used r = V dξ # (.5)

Series Solutions of ODEs 2 the Frobenius method. The basic idea of the Frobenius method is to look for solutions of the form 3

Series Solutions of ODEs 2 the Frobenius method. The basic idea of the Frobenius method is to look for solutions of the form 3 Royal Holloway Unversty of London Department of Physs Seres Solutons of ODEs the Frobenus method Introduton to the Methodology The smple seres expanson method works for dfferental equatons whose solutons