THERMODYNAMICS OF REACTING SYSTEMS

Transcription

1 ChE 505 Chapter N THERMODYNAMICS OF REACTING SYSTEMS. Introduction Thi chapter explain the baic of thermodynamic calculation for reacting ytem. Reaction toichiometry, heat and entropy of reaction a well a free energy are conidered here. Only ingle phae ytem are conidered here. Phae equilibrium will be addreed in the next chapter. Example and MATLAB calculation of the chemical equilibria are provided at the end of the chapter. 5. Reaction Stoichiometry The principle of conervation of each atomic pecie applied to every well defined chemical reaction lead to reaction toichiometry. Imagine that we have placed an inviible envelope around a finite ma of reactant and the content of that envelope are our ytem in it initial tate. We can count the atom of each atomic pecie preent in each reactant pecie. A chemical reaction take place in the ytem. Upon reaction completion, we recount the number of each atomic pecie. The total number of atom of each of the element preent in remaining reaction and product formed mut remain contant. The principle of conervation of ma applied to each atomic pecie yield the ratio in which molecule of product are formed and molecule of reactant are reacted. The repreentation of chemical pecie by a chemical formula indicate how many atom of each pecie are there in a molecule of the pecie under conideration. Hence, in a molecule of carbon dioxide (CO), there are: one atom of carbon, C, and two atom of oxygen, O. In a molecule of methane, CH 4, here i one atom of carbon and 4 of hydrogen, etc. In engineering application, a mole of the pecie under conideration i ued rather than a chemical formula (e.g., CH 4, O, etc.) repreenting an individual molecule of a particular chemical pecie. One hould recall that a mole i a baic unit of the amount of ubtance. The SI definition of a mole i: The mole i the amount of ubtance of a ytem that contain a many elementary entitie a there are carbon atom in 0.0 kg of carbon. The elementary entity (unit) may be an atom, a molecule, an ion, an electron, a photon, etc. The Avogadro contant i L = 6.03 x 0 3 (mol -) ). To obtain the number of mole of pecie, n (mol) in our ytem, we mut divide the ma of in the ytem m (kg), with the molecular weight of, M (g/mol), and multiply the reult by 000. m ( kg) m g n ( mol) x000 () M M

2 ChE 505 Chapter N Thi i equivalent to dividing the ma of the pecie expreed in gram with the molecular weight, a indicated by the econd equality in equation (). So the SI mole i the ame amount of ubtance a the old CGS gram mole that appear in old chemitry and phyic text. In the US we frequently ue a pound-mole (lb mol) a the meaure of the amount of ubtance. n * (lb mol) m (lb) M (a) It i important to note that the molecular weight of a pecie alway ha the ame numerical value independent of the ytem of unit. For example, the molecular weight of carbon i M c = (g/mol) = (lb /lb mol) = (kg/kmol). Therefore, (kmol) i thouand time larger than a mole (e.g. (kmol) = 0 3 mol)) and lbmol i time larger than a mole, i.e. (lb / mol) = mole. Accordingly, the Avogadro contant for a lb mole i L =.7308 x 0 6 (lbmole ) and for a kmole i L = 6.03 x 0 6 (kmol - ). To illutrate how reaction toichiometry i developed, conider the complete combution of methane (CH4) to carbon dioxide, CO. Thi i a reaction between methane, CH 4, and oxygen, O, that create carbon dioxide, CO, and water H O by complete combution. So we have at tart at end CH 4 + O CO + H O To develop a toichiometric equation we aume that we tart with one mole of methane. Thi implie that one mole of carbon mut be found both on the left hand ide and on the right hand ide of the toichiometric equation. So, one mole of CH 4 reacted mut produce one mole of CO. Since hydrogen i only contained in methane on the reactant ide, and there are H (two mole of hydrogen) in a mole of methane on the reactant ide of the toichiometric equation, there mut be two mole of water formed on the product ide in order to balance the amount of hydrogen. Now we have one mole of oxygen (O ) in the mole of carbon dioxide (CO ) on the product ide and another mole of oxygen in two mole of water. Therefore, we mut ue two mole of oxygen on the reactant ide to balance the amount of oxygen. methane: Thi lead to the following toichiometric equation for complete combution of CH 4 O CO H O () Hence, the requirement to balance out the atomic pecie, i.e. the application of the principle of conervation of ma of atomic pecie lead to the etablihment of tiochiometric coefficient (multiplier that multiply the mole of variou reactant and product pecie). The above toichiometric

3 ChE 505 Chapter N equation remain unchanged if multiplied with a common multiplier ay /: CH 4 O CO H O (a) or ay by: where CH 4 4O CO 4H O (b) Reaction toichiometry, for a ingle reaction, uch a that of equation () can now be repreented S S A A 0 (3) total number of peciein the ytem (e.g.4in caeof reaction ) chemicalformula for the - th pecie(e.g.ch toichiome tric coefficient for the - th pecie defined a 0 for product, 0 for reactant 4, O, CO,etc.) The toichiometric equation atifie the overall ma balance for the ytem S M 0 where M = molecular weight of pecie. (4) For example, for reaction () of methane combution we have CH 4 ; 0 ; CO ; H O If a reaction ytem can be decribed by a ingle reaction, then it generalized toichiometry i given by eq (3). Alternatively, for a ingle reaction between two reactant, A and B, and two product, P and S, the toichiometry can alo be repreented by: aa + bb = pp + where A, B, P, S are chemical pecie and a,b,p, are their toichiometry coefficient, repectively. A CH 4, B O, P CO and S H O in our example Naturally, thi implie that molecular weight of pecie A, B, P, S are uch that equation (4) i atified with v, v, v, v. 3 A (3a) B p Thi impler form i convenient for ingle reaction and alo in repreenting kinetic a dicued later. Single reaction implie that moleof Areacted a moleof B reacted b moleof P reacted p moleof S reacted Hence, a ingle reaction implie that the ratio of product produced and reactant conumed, or, a (3b)

4 ChE 505 Chapter N ratio of one reactant conumed to the other reactant conumed, are contant e.g. (mole of P produced) (moleof A reacted) p a ; (moleof A reacted) (mole of B reacted) a b (3c) If that i not the cae, then multiple reaction mut be ued to decribe the ytem. In a generalized form thi can be done a: S i A 0; i,...r (5) where R i toichiometriccoefficientof pecie in reactioni total numberof independentreaction For example, if in combution of carbon, there i alo carbon monoxide preent, then two reaction are needed to decribe the ytem. They can be a given below: C O CO (5a) CO O CO (5b) Here, we have a total of S = 4 pecie ( =,,3,4 for C, O, CO, CO, repectively), which are involved in two (R = ) independent reaction (i =,). If the above combution reaction involve air intead of oxygen, then nitrogen i the fifth pecie and hence S = 5, but it toichiometric coefficient in each reaction i zero ( i 5 0) for i=,) ince nitrogen doe not participate in thee reaction. Note that the matrix of toichiometric coefficient (for a ytem without nitrogen) 0 0 ha rank two (recall that the rank of a matrix i defined a the ize of the larget nonzero determinant). Adding a third reaction C O CO (5c) would add a third row to the above matrix of toichiometric coefficient, namely (- - 0 ) but the rank of the matrix would remain unchanged at. Clearly, eq (5c) i a linear combination (to be precie the exact um) of (5a) and (5b). Hence, we do not have a third independent reaction, and the toichiometry of the ytem can be decribed by any choice of two reaction of the above three reaction given by eq. (5a), (5b) and (5c). Finding the rank of the matrix of toichiometric coefficient will alway tell how many independent reaction are needed to characterize the toichiometry of the ytem. At the end of the chapter we will how how to find the rank of a matrix uing MATLAB. 4

5 ChE 505 Chapter N In ummary, in any reaction ytem we hould trive to etablih the reaction toichiometry by uing the principle of conervation of element. Then, if more than one reaction i preent, the number of independent reaction can be etablihed by determining the rank of the matrix of toichiometric coefficient..3 Meaure of reaction progre Let u conider firt a ingle reaction S A 0 occurring either in a batch ytem (i.e no material flow croe the boundarie of the ytem during reaction) or in a continuou flow ytem at teady tate (e.g. no variation in time). If n denote the mole of pecie in the batch at ome time t, and no i the initial number of mole of at time to, then reaction toichiometry dictate that mole of all pecie can be related to their initial mole via (molar) extent of reaction X. n n o X (mole of preent) = (mole of originally preent) + (mole of produced by reaction) Mole of produced by reaction are given by the product of the toichiometric coefficient,, and molar extent of reaction, X, which repreent "mole equivalent that participated in reaction". For reactant, 0, and mole produced are a negative quantity, hence, they are mole reacted. For product v X i clearly a poitive quantity. For a flow ytem at teady tate,. F Fo X (6b) where F, F o (mol / ) are molar flow rate of at exit and entrance, repectively, and X (mol/ ) i the molar extent of reaction. Equation (6) indicate that in a ingle reaction if we can determine the change in mole of one component (ay = A), then the molar extent of reaction can be calculated X (n A n Ao ) / A. Mole of all other pecie n can now be found provided their initial mole no were given. Equation (6) alo indicate that reaction progre, i.e. it extent, i limited by the limiting reactant. The limiting reactant i the one preent in amount le than required by toichiometry and limit the reaction extent to Xmax where (3) (6a) 5

6 ChE 505 Chapter N X max n o malletvalue over all (7) Uually, the limiting reactant i denoted by A o that X max = n A o / A For multiple reaction S i A 0 ; i,...r (5) molar extent are defined for each independent reaction i o that the mole of are given by n n o R i v i X i (8) (moleof preent) (mole of initially preent) (mole of producedby reaction i) all reaction Now, the change in mole of R pecie mut be determined to evaluate the R extent. Thi involve the olution of R linear algebraic equation. Then the mole of other pecie can be calculated by eq (8). Maximum extent are now thoe that yield zero mole of one or more reactant. Dealing with reaction toichiometry and reaction progre reult in linear algebraic equation to which all rule of linear (matrix) algebra apply. We are intereted only in non-negative olution. of the ytem Sometime other meaure of reaction progre are ued, uch a (molar) extent per unit volume mol L, which ha unit of concentration, or molar extent per mole which i dimenionle, or extent per unit ma. We will define them a we go along and when we need them. reaction or Often converion of a limiting reactant i ued in ingle reaction ytem to meaure progre of x A n Ao n A n Ao (9a) F F Ao A x A (9b) FAo The relationhip between converion and extent i readily etablihed n Ao x A v A X or F x X (0) Ao A A Hence, in a ingle reaction mole of all pecie can be given in term of converion, x A. Note that converion i unitle while extent ha unit of mole, i.e. of amount of ubtance. 6

7 ChE 505 Chapter N.4 Heat of reaction Heat of reaction i calculated a the difference between the heat of formation of product and the reactant. H r,i = H f(product) - H f(reactant) () where H r,i i the heat of the reaction i. The product and reactant react proportionally to their toichiometric coefficient. Therefore, Eqn.() can be written a H r,i = i H f, (a) The heat of formation data i uually available of tandard condition of 98 K and atm. Correponding tandard heat of reaction denoted a H r o,i can be calculated a H r,i = i H f, () where H f, i the heat of formation of the pecie at tandard condition. Tabulated value for ome pecie are hown in Table. The extenive databae chemkin available in the public domain i another ource. We will explain later how to ue it. Table : Heat of formation, entropy of formation, and Gibb free energy of formation at tandard condition for elected pecie. State = ga, temperature, T=98K and preure atm. Ga H J mol S J mol K G H T S J mol K Oxygen Water 4, ,600 Methane -74, ,83 Nitric oxide NO 90, ,686 Nitrogen dioxide NO 33, ,96 Carbon monoxide -0, Carbon dioxide ,355 Sulfur dioxide 96, ,39 Sulfur trioxide -395, ,34 Ozone 4, ,789 7

8 ChE 505 Chapter N Group contribution method are alo available to predict thee for new compound. Uing the value of tandard condition, the value at any other condition can be calculated uing He Law. Thi tate that the enthalpy change i indifferent of the path taken a long a one land up in the ame pot. A reaction carried out at any temperature T i equivalent in term of energy change to the um of following three path (i) Cool reactant form T to T ref. (ii) Carry out reaction at T ref with an aociated enthalpy change of H r, i (iii) Heat the product back form T ref to T. The tep are chematically hown in the Figure. T (Reactant) Reaction at T T (Product) Reactant Cool reactant T ref Same a T Heat product Reaction at T ref Product of T ref Figure. Schematic of He Law: Enthalpy change i indifferent of the path taken a long a one land up in the ame pot. Hence, the enthalpy change due to reaction (i.e. the heat of reaction) at temperature different from the tandard temperature can be calculated by equation (3) which add a temperature correction term to the tandard heat of reaction. Thi temperature correction then i the um over all pecie of the algebraic product of the toichiometric coefficient with the integral of the molar pecific heat for each pecie from the reference (tandard!) temperature to the temperature of interet. H r,i = H r,i + [ T T ref i, C p d T ] (3) The variation of C p i uually expreed a a polynomial function of temperature C pi = max 0 A T (4a) where A are the coefficient for pecie. 8

9 ChE 505 Chapter N or C p = A + BT + CT + DT - (4b) Thi i another approximate form of C p a a function of temperature. Up to even contant are ued in chemkin databae ( max = 7) while 4 contant are ued a an approximation in many book a hown in (4b). Subtituting we find H i,r = H i,r + i max A [ T T ] (5) ref The above expreion provide a method for calculation of heat of reaction at any given temperature T in an exact manner. The information needed i the heat of formation of all pecie at T ref and the coefficient A in eq (5a) for temperature variation of molar pecific heat (e.g. heat change) C p for each pecie..5 Entropy change in reaction S r i the denoted a the entropy change of reaction at tandard condition (T ref ) and can be calculate from the entropy of formation S f, for each pecie participating in the reaction. S r = S f, (6) The entropy change at any other temperature T can be calculated a S r = S r (T ref ) + T T ref Cp dt (7) T Uing the polynomial expreion for C p a a function of temperature and integrating we obtain S r = S r + A 0 ln T T ref + max A [ T - T ref ] (8) Knowing the entropy and enthalpy change, at any given temperature, the free energy change of reaction i calculated a : G r = H r - T S r (9) Knowledge of how G r change with temperature i important in determining the direction that the reaction will favor. If G r < 0 reaction to the right or forward reaction i highly favorable, while if G r > 0 the revere reaction i more favorable. The temperature at which G r = 0 i where the tranition occur. For an exothermic reaction, G r typically increae with temperature. 9

10 ChE 505 Chapter N G r G r T T Exothermic Endothermic Figure : Variation of free energy a a function of temperature for exothermic and endothermic reaction: Example: i) Thermodynamic calculation for CO oxidation to CO. Thi i an exothermic reaction. CO + O CO T(K) H r (J/mole) S r (J/mole K) G r (J/mole) , , , , , , , ,659 Reaction become le favorable at elevated temperature a the reaction i exothermic. ii) Steam reforming of methane CH 4 + H O CO + 3H Thi i an endothermic reaction T(K) H r (J/mole) S r (J/mole K) G r (J/mole) 500 5, , , , , , , ,960 Reaction become favorable only at higher temperature. Thi i alway true for an endothermic proce. Note that G r decreae with increaing the temperature which i conitent with trend hown in Figure. 0

11 ChE 505 Chapter N.6 Chemical Thermodynamic: Brief Review of Chemical Equilibria For implicity conider an iothermal, ingle phae ytem ubect to a ingle reaction. A 0 () ~ The equilibrium tate i then defined by the minimum in Gibb free energy of the ytem (min n G ) which can be expreed by i) ~ G 0 (0) where ii) n n o X e for all () iii) appropriate equation of tate ~ G ( T, P, ) i the partial molal Gibb free energy of and i the function of temperature, preure and x compoition. (It i often called chemical potential and denoted by ) In general ~ G G ~ RT ln a () where 0 G i the Gibb free energy of pure pecie (function of temperature only). a i the activity of pecie ; n i mole of ; no i initial mole of ; i toichiometric coefficient of (poitive for product, negative for reactant); T i temperature of the ytem; Xe i the equilibrium reaction molar extent; Subtitution of the econd equality of equation (4) for into equation () yield G RT ~ ln a 0 (3) Recognizing that na i na and that the um of logarithm S n a i the logarithm of the product n S a we get the following equation

12 ChE 505 Chapter N S S n a G (3a) RT ~ We define the tandard Gibb free energy of reaction at temperature T by G r G 0 (4) with all G ~ being evaluated at the temperature T of interet. Then the thermodynamic equilibrium contant, K, which i a function of temperature only, i obtained by taking the anti-logarithm of equation (3a) and i given by: K a e G r RT where a i the activity of pecie need to: In order to calculate the equilibrium reaction extent, Xe, and the equilibrium compoition we a) calculate K at the temperature of interet, b) relate the activity of each pecie,, a, to a meaure of compoition (e.g. mole fraction) by an appropriate model for the mixture. c) relate the choen meaure of compoition to reaction extent uing toichiometric relation indicated by ii) above. Since the Gibb free energy of formation i tabulated for all chemical pecie at T ref = 98K (5 C) (or can be found from H and S data a hown in previou ection) it i convenient to calculate the Gibb free energy change due to reaction G rt0 o G r at thee tandard condition a: G f (6) (5) temperature: The equilibrium contant at tandard temperature i then obtained from equation (5). Van Hoff' equation etablihe the rate of change of the equilibrium contant K with d nk dt H r RT (7) T To ( 98 K); K K98 exp( GrT / RT0 ) (8) 0 where H r i the tandard heat of reaction at temperature T (calculated a hown in previou ection),

13 ChE 505 Chapter N K98 i the equilibrium contant at the tandard tate temperature of T0 (mot often 98K) and G rro i the tandard Gibb free energy of reaction at T0 which i obtained from tabulated Gibb free energie of formation G f For gae (tandard tate pure ga at atm) we ue y for mole fraction of, a i the activity of, p i partial preure of, P i total preure of the ytem, partial molal fugacity of. i the fugacity coefficient of while f i the The needed relation are included below: a y P f y P / atm p f y P /atm (30) a y P /atm p /atm (30a) K a P atm y (3) K P atm K y K P P 0 K y K where P0= or atm. (3a) K a p / atm (3) K a K P K / atm (3a) The generalized fugacity coefficient, f y P, would have to be evaluated from an appropriate equation of tate. If Lewi-Randall rule i ued For gae at low preure K f P. K P atm K y K P / atm (3b) For liquid (auming tandard tate of unit activity, i.e. the tandard tate of each component i the pure component tate) the following relation hold: a x (33) where i the activity coefficient 3

14 ChE 505 Chapter N K a x K x K (34) Since x = C / C K C K c K (35) For an ideal mixture K Above C i the molar concentration of pecie and C Example: Reaction Sytem: SO + O = SO 3 Converion of 99% i deired C i the total molar concentration. Condition: T = 600 o C = 873 o K P = atm Stoichiometric Feed of Pure Reactant No Nitrogen Calculate equilibrium converion of SO and ee if 99% can be reached. G f and H f for variou pecie are given in ( kcal/mol) below. Bai: mole of SO. Total number of pecie S = 3 3 ntot n o 3ntot n 3 0 X Specie Name No. Stoich. Coeff. G f H f n o n n o X SO X O 0 0 X SO X Aume ideal ga mixture and write the equilibrium contant in term of K y : K K y P P o K y P P o K y P o P where (A) 4

15 ChE 505 Chapter N 3 y K y y y y 3 y3 y y y S03 y S0 y0 Evaluate the mole fraction of each pecie in term of reaction extent X n y X n to t 3 X ; y X 3 X ; y 3 X 3 X Subtitute thee mole fraction in the expreion for K y K y 4 X 3 X 4 X X 3 X 3 X X 3 X X 3 where X = Xe = equilibrium extent From (A) K X 3 X X 3 P 0 P (B) Calculate K98 at 98 K Calculate G r K 98 e x 7.7 x 0 x kcal mol G r o RT o e H r 33, x98 e x0 4 x 70.9 x 0 x kcal mol Aume for implicity (in order to find a firt etimate for equilibrium condition) that H r cont H r Then d nk dt n K T K T K 98 H r RT ; T T o 98 K K 98 K 98 e H r R H r R 98 47, K 873 e K T 98 T 873 e

16 ChE 505 Chapter N Note the dramatic drop in K with temperature due to the exothermicity of the reaction H r 0. While equilibrium would have been all the way to the right at 98 K we cannot operate at uch condition becaue the rate i too low. Let u ee what are the equilibrium limitation at 873 K. Solve equation (B) for Ky expreed in term of extent (ee equation (C)) Solve by trial and error equation (C) for equilibrium extent Xe = X. X 3 X X 3 K 873 P P 0 (C) where K ; P P o. Note that elevated preure, a predicted by L Chatelier principle, would help move the equilibrium to the right..rearrange equation (C) to ue Newton-Raphon procedure X 0 X 3 X.78 X 3 D X 3X X X X n X n X n D X n (D) Uing a tarting gue of Xo = 0.5 ( mol) the Newton Raphon algorithm (D) yield: Iteration No. n Extent (mol) X = Xe = (mol) Converion of SO i: n SO, 0 n SO X x SO n SO, 0 x SO eq X X (mol)/ (mol) Equilibrium converion of 0.99 required by emiion control cannot be reached under thee condition. Equilibrium extent could be increaed by - lowering the temperature (but rate are lower) - increaing the preure (ome of it can be recovered by running a turbine at the end of reactor a done in the former USSR) 6

17 ChE 505 Chapter N.7 Solution Thermodynamic The equilibrium ditribution of inorganic compound in aqueou ytem i of great importance in environmental engineering. Thee calculation are more complex than the ga phae cae becaue of two additional conideration. (i) charge neutrality ha to be maintained. (ii) ytem i often non-ideal. The importance of thi in environmental application i illutrated by a number of example below. Example : Heavy metal have toxicological impact and often get incorporated in biota uch a fih. The free ion form M ++ for example, i often the toxic form while the complexe uch a MOH +, M(OH) or ligand with SO 4 or other negative ion are conidered to be le harmful. The ditribution of the metal in variou form i a function of the total concentration and the ph of the olution. Solution thermodynamic provide u with a mean of doing uch calculation and help in making udiciou policy deciion. Example : Conider a oil ytem that ha ome lead contamination or naturally occurring lead. The quetion i whether the lead i going to remain there or diolve in ground water and enter into drinking water. The lead may remain in the olid phae depending on the ph or the preence of other ion (common ion effect decribed later). Example 3: Conider SO generated in a pollutant plume in the ga phae. Thi i a highly oluble ga and enter the aqueou phae with rain water. It then react to form variou ionic pecie uch a HSO 3, SO -- 3, SO -- 4 etc. The equilibrium ditribution i important in many application. Example 4: Conider mercury, a highly volatile toxic metal, that i highly poionou. Depending on the ph of the aqueou environment, mercury can be found in variou form uch a HgH, Hg, Hg ++, Hg ++, HgO, HHgO and Hg(OH) form. When converted to the organic form, monomethyl mercury (CH 3 Hg) can be inerted by microorganim, it become lipophyllic and get accumulated in body fat of fih. Conumption of contaminated fih reult in damage to the central nervou ytem, liver and kidney and caue impaired child development. In 950, 00 people in Minamata Bay in Japan died from mercury poioned fih. 7

18 ChE 505 Chapter N.8 Rank and checking linear independence of reaction uing MATLAB To appreciate fully what the MATLAB program doe for you look firt at Appendix A. In that appendix we decribe the procedure involved in obtaining the rank of the matrix and identifying independent reaction. Conider the carbon dioxide formation conidered earlier. The reaction cheme i C O CO (5a) CO O CO (5b) C O CO (5c) Step : type in the toichiometric matrix a follow: Each row i reaction index and each column i the pecie index. Thu each row repreent one reaction in the ytem. We take carbon, C, to be pecie, oxygen to be, carbon monoxide i 3, carbon dioxide i 4. coefficient i: v = [- - 0 ; ; ] Step : find the rank by typing the following rank(v) Then the matrix of toichiometric For the above cae the anwer will be o that there are only two independent reaction. In other word the third reaction i a linear combination of the firt two reaction. To find thi combination go to tep a. Step a; do thi if the rank i le than the number of reaction. Type rref(v') Note that v' i the tranpoe of the toichiometric matrix and the above command reduce the tranpoed matrix to the Echelon form (in thi form the main diagonal of the ha one on it). The reult will be

19 ChE 505 Chapter N The lat column give the linear combination for the third reaction. Thu reaction (5c) i equal to 0.5 time the firt reaction plu 0.5 time the econd reaction. In thi imple example it i obviou by inpection a well. Step 3: To find the ytem invariant (thee will be relation between mole produce or conumed among variou pecie), type rref(v) Thi reduce the toichiometric matrix to the echelon form. The reult for our example i: The lat two column here provide the invariant. Thu F 3 = - * F + * F and F 4 = * F - * F Here F repreent the change in molar flow rate between the exit and entrance for pecie I. Note that the lat row i all zero indicating that the third reaction i redundant and doe not contribute to the invariant. (recall that the rank wa two). Let u ee what the implication of all thi i; the above equation tell u that only two value F and F are independent. For example, in an experiment uppoe we find that 5 mole/ec of carbon ha reacted. Then F 5. We alo find that 4 mole/ec of oxygen i conumed. F 4. Then uing the equation above we find Alo F 5 4 = mole/ec i the amount of CO formed. 3 F = 3 mole/ec of CO i formed. 9

20 ChE 505 Chapter N If the meaurement how otherwie, then either the meaurement are in error or ome other C bearing pecie i being formed (not likely here). Hence in thi cae the meaurement would be in error. Thu, the invariant of a reaction are ueful in proper bookkeeping of the variou pecie..9 MATLAB PROGRAM FOR CALCULATION OF EQUILIBRIUM COMPOSITION OF A REACTING GAS MIXTURE The following program calculate the equilibrium compoition of a ga mixture. The program i written for the SO example and can be modified eaily for other cae. The bet way to learn thi i for the tudent to TYPE out the program a it i and execute it on matlab. Thi way they get familiar with the programming a well. The program alo illutrate the ue of the olver FSOLVE to olve a et of non-linear algebraic equation. The program i interpaced with ome explanation and thee tatement need to be omitted in the actual program. PREAMBLE SECTION % filename gaeq; created on an 3-03by P.A.Ramachandran. % compute the equlibirum compoiton of a reacting mixture. % preamble global ng n nr tempin prin ctot xg nu... global keq98 keq delhr global rga global n ntot At thi part of the program the number of ga phae pecie ng and number of reaction nr are to be entered. %number of pecie and number of reaction ng = 3 ; nr = ; DIMENSIONING THE VECTORS Thi i required but no action by uer i needed. % intialize the vector. keq98 = zero(nr, ); keq = zero(nr, ); 0

21 ChE 505 Chapter N delhr = zero(nr, ); xg = zero(ng,); nu = zero ( nr, ng); zeta = zero(nr,) ; USER DATA SECTION A: REACTIONS Here the reaction pecific variable are entered Thee are the toichiometric matrice and the equilibrium contant for each reaction at 98K and the heat of reaction at 98K. The program aume that the heat of reaction i independent of temperature. Note that the ga contant rga mut have conitent unit with the heat of reaction. % provide value for toichimetry and eq contant. nu(,) = -. ; nu(,) = -. nu(,3) =. ; keq98() = 6.593E+4 ; delhr() = ; %aumed contant (not a function of T) rga =.97 ; % ga contant; ue conitent unit B: PROCESS CONDITIONS Here the temperature, preure and the mole of each pecie preent in the initial mixture are pecified. The uer mut provide a gue value for the extent of reaction (zeta vector). Try to provide realitic value, for example the converion of a key component can not be greater than one. Thi will fix the maximum value for zeta % provide the feed condition. tempin = 873.0; prin =.0e+05; % initial mole. xg() =.0; xg() =.0; xg(3) = 0.0;

22 ChE 505 Chapter N % provide initial value for extent of reaction zeta() = 0.4 CALCULATION SECTION At thi tage the matalb take care of the ret of the calulation % mtlab function folve olver i called to find the root % the required equation are programmed in a file fun_eq.m zeta = folve ( 'fun_eq', zeta) % pot proce the reult. % molar fow rate at exit = TYPE n % total mole = TYPE ntot Pot proceing can be done by typing n which give the mole of pecie at Equilibrium,. The converion and other required information can be calculated eaily on the matlab command window. FUNCTION SUBROUTINE The program require a function ubroutine which calculate the function to be olved. The ubroutine i in a file fun_eq.m and the liting i hown below. No change in thi by uer i needed. Student may want to tudy how thi i written by following thi with the text earlier function fvec = fun_eq( zeta) % filename fun_eq; created on an 3-03by P.A.Ramachandran. % Define the (nr) function to be olved. % preamble global ng n nr tempin prin ctot xg nu... global keq98 keq delhr global n ntot global rga fvec = zero ( nr,) ; ptotatm = prin /.0e+05 ;

23 ChE 505 Chapter N % find K at the deired condition for i = : nr keq(i)=keq98(i)*exp( delhr(i) /rga *(./98-./tempin)) ; end %number of mole of ecah pecie for the given extent % n0 = for thee calculation for = : ng n() = xg(); for i=:nr n() = n() + nu(i,) * zeta(i); end end % find ntot ntot = 0.0; for = : ng ntot = ntot + n(); end % find the mole fraction for each pecie pp = n /ntot * ptotatm; % et up dicrepancy function for each reaction for i = : nr prod =.0; for = : ng prod = prod * pp()^nu(i,); end fvec(i) = keq(i) - prod end _ The tudent hould run the program for other cae for example to tudy the effect of inert, like nitrogen, preure or other ytem involving multiple reaction. We will now illutrate ue of thi program for a combution application. 3

24 ChE 505 Chapter N Example: Chemical equilibrium et an upper limit on the compoition of combution gae. Given a mixture of CO, CO and O we need to find the CO concentration a a function of exhaut ga temperature. Once CO i formed, it i unlikely that CO will be formed a the ga cool ince the reaction rate are lowed down. Hence the maximum CO in the exhaut ga can be found from the equilibrium calculation. A an example, conider an exhaut ga mixture with 8%CO. 3.3%O and the ret N. Temperature of thi ga i raied to 600K. Find the equilibrium compoition of thi ga auming the reaction CO CO / O We run the program with 4 pecie the fourth being inert (N ). The ample reult are a follow: ***** COMPUTED RESULTS ********* Equilibrium contant of reaction i e-05 Heat of reaction i (J/mol) Free energy change of reaction i (J/mol) Program converged The extent of reaction i e-03 (mole) We ue 00 mole total a the input condition. Mole of CO in the exhaut ga i therefore.955e-03% of the total mole or 9.5 ppm (by mole or volume). Problem for dicuion Predict the effect of temperature (range of 000 to 000K) on the CO content of exhaut ga and plot the reult. If CO reduction wa the only goal, would you operate at high temperature or low temperature Example: Conider a ga to be 3.3%O and ret N. Thi preumably imulate and exhaut ga from an internal combution engine. The following reaction take place N O NO NO / O NO Find the concentration of NO and NO at800k and atm. 4

25 ChE 505 Chapter N Again we ue the fortran program ince all thee gae are in the chmkin databae. We modify the datafile and run the program to find the following reult. Equilibrium contant of reaction i.98983e-04 Heat of reaction i (J/mole) Free energy change of reaction i Equilibrium contant of reaction i E-03 Heat of reaction i (J/mole) Free energy change of reaction i (J/mole) program converged The extent of reaction i E-0 (J/mole) The extent of reaction i.77360e-04 (J/mole) NO formed = x x N O Formed = x The above are percentage ince the total tarting mole are aumed a00. Hence the ga ha 7ppm of NO and.77 ppm of NO.0 STUDY QUESTIONS What i the ignificance of the rank of the toichiometric matrix? State the meaure of progre of reaction for multiple reaction. How doe the equilibrium contant vary with temperature? How i Keq and K y related to each other? Show one example? 5

26 ChE 505 Chapter N. EXERCISES. The principle of conervation of element mut be applied in order to obtain the toichiometric coefficient in a ingle reaction. Aume that there are S chemical pecie that are either product or reactant in the reaction and there are N element preent a contituent. Let the number of atom of element i in pecie be i. Let the yet unknown toichiometric coefficient of pecie be S. Show i that the conervation principle above require 0 for all i =,,.N. Apply thi to the reaction of complete combution of methane and how how to determine the unknown '.. In a reaction ytem coniting of methane, oxygen, carbon monoxide, water, hydrogen and carbon monoxide, the following 6 reaction (R = 6) may occur: CH 4 + 3/ O = CO + H O () CO + / O = CO () CH 4 + H O = 3H + CO (3) H + / O = H O (4) CO + H O = CO + H (5) CH 4 + O = CO + H O (6) Find the number of independent reaction R..3 Carbon (0 mole initially) i burned with oxygen. At the end of reaction 4 mole of carbon are left and 4 mole of CO are produced. The produced amount of CO i unknown. (i) Write the toichiometric matrix with carbon and CO a component () and () (ii) (iii) Find the matrix in echelon form. Find the mole of O conumed and CO produced in the proce..4 The equilibrium contant for CO CO + O at 000 K i 6.046*0 -. Find the value at 600 K if the heat of reaction i 8 k/mole..5 Find the thermodynamic data for the reaction NO + O NO 6

27 ChE 505 Chapter N Uing a uitable databae (report which one you ued) over a temperature range of 500 to 500 K, determine at what temperature would you expect a larger NO concentration. Low or high? What i the equilibrium compoition..6 What i the effect of preure on CO formation in a combution ytem? At 600 K, find the CO content of the exhaut ga if the preure were atm. Initial compoition i the ame 8 % CO, 3.3 % O,78.7 % N. A in the Matlab program example..7 Propane i burned in air with.5 time the toichiometric air. Find the compoition of exhaut gae auming that only CO and H O are formed..8 For the compoition in Quetion (.7) find the equilibrium compoition of CO if the combution take place at 800 K and atm preure..9 Find the equilibrium compoition for the above cae uing the net program. 7