Ch 3. Rate Laws and Stoichiometry

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1 Ch 3. Rate Laws and Stoichiometry How do we obtain r A = f(x)? We do this in two steps 1. Rate Law Find the rate as a function of concentration, r A = k fn (C A, C B ). Stoichiometry Find the concentration as a function of conversion C A = g(x) Part 1: Rate Laws Basic Definitions: A homogenous rxn is the one that involves only one phase. A heterogeneous rxn involves more than one phase, rxn occurs at the interface between the phases, i.e., solid gas phase. An irreversible rxn proceeds in only one direction A B A reversible rxn can proceed in both directions A B (equilibrium) Molecularity of a rxn is the number of atoms, ions, or molecules involved in a rxn step. The terms unimolecular, bimolecular, and termomolecular refer to rxns involving, respectively, one, two or three atoms interacting in any one rxn step. Unimolecular: radioactive decay Bimolecular: rxns with free radicals Br. + X XBr + Y. Termolecular: rxn pathway following a series of bimolecular rxns. 1

2 A rate law describes the behavior of a reaction. The rate of a reaction is a function of temperature (through the rate constant) and concentration. 1. Relative Rates of Reaction aa + bb cc + dd Rate of formation of C = c/a (Rate of disappearance of A) The reaction : A + 3B 5C is carried out in a reactor. If at a particular point, the rate of disappearance of A is 10 mol/dm 3 s, what are the rates of B and C? The rate of disappearance of A, -r A, is or the rate of formation of species A is The relative rates are

3 Species B The rate of formation of species B (A + 3B 5C) The rate of disappearance of B, -r b, is Species C The rate of formation of species C, r c, is Rxn Order & Rate Law: Algebraic equation that relates r A to the concentrations of the reactants is called the kinetic expression or rate law. Usually, rate can be written as the product of rxn rate constant and concentrations. In these expressions, usually the limiting reactant is chosen as basis for calculations. -r A = k A (T) fn (C A, C B,..) In reality activities should be used r A = -k A a Aα a B β (a i = γ i C i ) r A = (-k A γ Aα γ Bβ )C Aα C B β k A 3

4 Power Law and Elementary Rate Laws In general -r A = k A C Aα C β B α : order in A β : order in B n = α + β = overall rxn order The unit of r A is always = concentration / time For a rxn with n order: {k} = (concentration) 1-n / time Therefore for a zero-, first-, second-, and third-order rxn zero order 1 3 st nd rd order order order ( n = 1) ( n = 0) ( n = ) ( n = 3) r A r r r = k = k C A A A A = k C ; = k C A A A A ; A A ; C mol { k} = 3 dm s 1 { k} = s B ; 3 dm { k} = mol s 3 dm { k} = s mol 1 Elementary Reactions A reaction follows an elementary rate law if and only if the stoichiometric coefficients are the same as the individual reaction order of each species. For the reaction in the previous example (A + B C + D), the rate law would be: -r A = k C A C B These rate laws can be derived from Collision Theory. if NO + O NO then r NO = k NO (C NO ) C O if elementary!!! Question What is the reaction rate law for the reaction A + ½ B C if the reaction is elementary? What is r B? What is r C? Calculate the rates of A, B, and C in a CSTR where the concentrations are C A = 1.5 mol/dm 3, C B = 9 mol/dm 3 and k A = (dm 3 /mol) (½) (1/s) 4

5 Solution: A + ½ B C Let s calculate the rate if, Then, For A + B C + D r A = k C A C B CC C K C D Elementary 5

6 Non Elementary Rate Laws: A large number of homogeneous or heterogeneous rxns do not follow simple rate laws. If the rate law for the non-elementary reaction A + B C + D is found to be r A = k C A C B, then the rxn is said to be nd order in A and 1 st order in B, and 3 rd order overall. For the homogeneous rxn CO + Cl COCl N O N + O r r CO N O = k C CO k C N = 1+ k' C C O O Cl 3/ k and k strongly temperature dependent In this case we can not state overall rxn order. Here, we can speak of reaction orders under certain limiting conditions as at very low conc n of O (1 >> k C O ) r N O = kn O C N O 1 st order (apparent rxn order) If 1 << k C O r N O r k C NO NO N O = k' CO 0 th order (apparently) These types are very common for liquid & gas phase rxns on solid catalysts 6

7 For the heterogeneous rxn C catalyst 6H5CH 3 + H C 6H 6 + CH 4 (toluene: T) (benzene: B) (methane: M) In these types of rxns, partial pressures are used instead of conc ns: r T Per mass of catalyst k PH PT = 1+ K P + K ' B B Specific rxn rate {k} = mol T / kg cat s atm T P T Adsorption constant {K T } = 1 / atm if ideal gas law is applied P i = C i R T Example: Gas Phase Catalytic Reactions When studying gas phase catalytic reactions the rate law is developed in terms of partial pressure, To rewrite the rate law, just use ideal gas law to relate to concentrations C A and C B and then write concentration in terms of conversion 7

8 The net rate of formation of any species is equal to its rate of formation in the forward reaction plus its rate of formation in the reverse reaction: rate net = rate forward + rate reverse At equilibrium, rate net 0 and the rate law must reduce to an equation that is thermodynamically consistent with the equilibrium constant for the reaction. Example: Consider the exothermic, heterogeneous reaction; A + B C At low temperature, the rate law for the disappearance of A is Recall P A = C A R T At high temperature, the exothermic reaction is significantly reversible: What is the corresponding rate law? If the rate of formation of A for the forward reaction (A + B C) is then we need to assume a form of the rate law for the reverse reaction that satisfies the equilibrium condition. If we assume the rate law for the reverse reaction (C A + B) is Then: and: 8

9 Deriving r A : The forward rate is: And the reverse rate law is: The net rate for species A is the sum of the forward and reverse rate laws: Substituting for r for and r rev : Solving for Kp: Does this rate law satisfy our requirement at equilibrium. For a rxn at equilibrium At equilibrium, r net 0, so; Solving for K p : The conditions are satisfied. 9

10 Rate Law for Reversible Reactions Example: Write the rate law for the elementary reaction Here k fa and k ra are the forward and reverse specific reaction rates both defined with respect to A. (1) () At equilibrium The equilibrium constant decreases as T increases (exothermic rxns) K e increases as T increases (endothermic rxns) 10

11 Reaction Rate Constant, k: k is the specific rxn rate constant given by Arrhenius Equation: k = A e E/RT where, E: activation energy (cal/mol) R: gas constant (cal/mol K) T: temperature (K) A: frequency factor (unit depends on rxn order) k T k A T 0 k 0 A T E A (Activation Energy) Molecules need some energy to distort/stretch their bonds so that they break them to form new bonds. The steric & repulsion forces must be overcome as the reacting molecules come close together. A A + B C A B A C a b+c a+b a+c 11

12 A rxn coordinate is usually used: A + BC A B C AB + C High energy complex E A-B-C Energy Barrier E A + E BC E AB + E C Rxn Coordinate The rxn coordinate denotes the energy of the system as a function of progress along the reaction path. More on Activation Energy The activation energy can be thought of as a barrier to the reaction. One way to view the barrier to a reaction is through the reaction coordinates. These coordinates denote the energy of the system as a function of progress along the reaction path. For the reaction the reaction coordinate is for the reaction to occur, the reactants must overcome an energy barrier or activation energy E A. 1

13 Why is there an Activation Energy? (1) the molecules need energy to distort or stretch their bonds in order to break them and to thus form new bonds () as the reacting molecules come close together they must overcome both steric and electron repulsion forces in order to react Energy Distribution of reacting molecules In our development of collision theory we assumed all molecules had the same average energy. However, all the molecules don t have the same energy, rather there is distribution of energies where some molecules have more energy than others. The distraction function f(e,t) describes this distribution of the energies of the molecules. The distribution function is read in conjunction with de f(e, T) de = fraction of molecules with energies between E and E + de One such distribution of energies in the following figure 13

14 Increase Temperature By increasing the temperature we increase the kinetic energy of the reactant molecules which can in turn be transferred to internal energy to increase the stretching and bending of the bonds causing them to be in an activated state, vulnerable to bond breaking and reaction. As the temperature is increased we have greater number of molecules have energies E A and hence the reaction rate will be greater. 14

15 Activation Energy The activation energy is a measure of how temperature sensitive the reaction is. Reactions with large activation energies are very temperature sensitive. If you know two points E A is known! ln k slope = 1 ( ) T slope = E R One can also write the specific reaction rate as: 15

16 Specific Reaction Rate Derivation Taking the ratio: This says if we know the specific rxn rate k o (T o ) at a temperature T o, and we know E, then we can find reaction rate k(t) at any other temperature T for that rxn. STOICHIOMETRY We shall set up Stoichiometric Tables using A as our basis of calculation in the following reaction. We will use the stoichiometric tables to express the concentration as a function of conversion. We will combine C i = f(x) with the appropriate rate law to obtain -r A = f(x). 16

17 1. Batch System Stoichiometric Reactor where; and Concentration -- Batch System: Constant Volume Batch: Note: if the reaction occurs in the liquid phase or if a gas phase reaction occurs in a rigid (e.g., steel) batch reactor Then if then and we have -r a =f(x) 17

18 Example: Write the rate law for the elementary liquid phase reaction 3A + B 4C solely in terms of conversion. The feed to the batch reactor is equal molar A and B with C A0 = mol/dm 3 and k A =.01 (dm 3 /mol) 4 1/s. 1) Rate Law: -r A =kc A3 C B ) Stoichiometry: Species A Liquid phase, v = v o (no volume change) Species B What is Θ B? Species A is the limiting reactant because the feed is equal molar in A and B, and two moles of B consumes 3 moles of A. We now have -r A =f(x) and can size reactors or determine batch reaction times. 18

19 . Flow System Stoichiometric Table Where: and Concentration -- Flow System: Liquid Phase Flow System: etc. If the rate of reaction were -r A = kc A C B then we would have This gives us -r A = f(x). Consequently, we can use the methods discussed in Chapter to size a large number of reactors, either alone or in series. 19

20 For Gas Phase Flow Systems: (Variable Volumetric Flow Rate) Combining the compressibility factor equation of state with Z = Z 0 we obtain with F T =C T V F To =C To V o The total molar flow rate Substituting for F T gives: y A0 δ = ε holds for both flow & batch reactors Example: Calculate ε For the gas phase reaction A + B C the feed is equal molar in A and B. Calculate ε. Solution A is the limiting reactant 0

21 Gas Phase Flow Systems etc. Again, these equations give us information about -r A = f(x), which we can use to size reactors. For example if the gas phase reaction has the rate law then with The molar flow rate of A: F A = F A0 + ν A (F A0 X) F A = F A0 (Θ + ν A X) Stoichiometric coefficient 1

22 Calculating the equilibrium conversion for gas phase reaction Consider the following elementary reaction with K C and = 0 dm 3 /mol and C A0 = 0. mol/dm 3. Pure A is fed. Calculate the equilibrium conversion, X e, for both a batch reactor and a flow reactor. Assume constant volume. Solution At equilibrium Stoichiometry Constant Volume V = V 0

23 Solving 3

24 Stoichiometric Table - Conversion Let s consider the production of ethyl benzene The gas feed consists of 5% toluene and 75% ethylene. Set up a stoichiometric table to determine the concentrations of each of the reacting species and then to write the rate of reaction solely as a function of conversion. Assume the reaction is elementary with k T =50(dm 6 /mol s). The entering pressure is 8. atm and the entering temperature is 7 C and the reaction takes place isothermally with no pressure drop. 4

25 Basis of Calculation The stoichiometric ratio is one toluene to two ethylene (1/). However, the feed is one toluene to three ethylene (1/3) and there is not sufficient toluene to consume all the ethylene. Therefore toluene is the limiting reactant and thus the basis of calculation. Entering Concentrations of ethylene and toluene Let A = toluene, B = ethylene, C = ethyl benzene and D = propylene Ethylene Since toluene, i.e. A, is the limiting reactant and has a stoichiometric coefficient of 1 Leaving F B 5

26 Complete the stoichiometric table including coolant flow rates Write the volumetric flow rate in terms of conversion P = P 0 and T = T 0 In terms of conversion For a flow system at constant T and P 6

27 In terms of conversion We now have r A solely as a function of X 7

28 Summary We learned to write the rate expression for a given reaction A B or A B Meaning of rate constant, activation energy k= A e {-E/RT} Summary 1. Batch System Stoichiometric Reactor (similarly for CSTR and PFR!!) where; and 8

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