CHEE 321: Chemical Reaction Engineering Module 2: Rate Laws and Stoichiometry (Chapter 3, Fogler)

1. CHEE 321: Chemical Reaction EngineeringModule 2: Rate Laws and Stoichiometry(Chapter 3, Fogler)

2. Module 2: Rate Laws and Stoichiometery Topics to be covered in this module • Re-visit reaction rate • Rate law • Temperature dependency of Rate Law: Reaction Rate Constant • Arrhenius factor and Activation energy • Concentration dependence of reaction rate or rate law • Reaction order • Elementary and non-elementary reaction rate laws • Stoichiometric tables for batch and flow reactors

3. Why do we need to know more about reaction rates? • Chemical reactors are designed to consume specific reactants while generating certain products. Quite simply, then, mole balance for chemical reactors include terms describing rate of consumption and/or generation of chemical species. • Let us take a look at the mole balance or design equations for isothermal reactors

4. BATCH CSTR Design Equations for Isothermal Reactors REACTOR DIFFERENETIAL ALGEBRAIC INTEGRAL FORM FORM FORM PFR PBR

5. Rate Law - Functional form Rate Law: or kinetic expression is an algebraic equation that relates reaction rate to species concentration (-rA) = k · [f´(CA, CB, ..)] k is the reaction rate constant The terms within the brackets [f´(CA, CB, ..)] simply denote dependency of reaction rate on the concentrations of the reactants (and for reversible reactions on the concentration of products as well) Note: k is constant at a given temperature RATE LAW IS INDEPENDENT OF REACTOR

6. Generally negligible Factors influencing rate of reaction (rj) • Factors affecting rate constant (k) • temperature • catalyst type • pressure • Factors affecting concentration • pressure • temperature

7. Relative Rates of Reaction For a reaction given below: aA + bB  cC + dD How is (-rA) related to (-rB), (rC) and (rD) ?

8. Temperature Dependency of Rate Law: Rate Constant

9. Temperature dependence of rate constant –Arrhenius Law A majority of reaction rate constants can be expressed as a function of temperature by an empirical relationship that was developed by Swedish scientist, Svante Arrhenius. A = frequency factor or pre-exponential factor Ea = Activation Energy R = Universal gas Constant k = A exp(-Ea/RT) Vignette on Svante Arrhenius Arrhenius studied reaction rates as a function of temperature, and in 1889 he introduced the concept of activation energy as the critical energy that chemicals need to react. He also pointed out the existence of a "greenhouse effect" in which small changes in the concentration of carbon dioxide in the atmosphere could considerably alter the average temperature of a planet. For his PhD thesis in 1884 he presented his "ionic theory", but it turned out to be a bit too revolutionary for his examiners' taste. He barely passed with a fourth class rank, "not without merit".

10. Interpretation of parameters in the Arrhenius equation k = A exp(-Ea/RT) • For simple reaction in which two molecules collide and react, the pre-exponential term in the Arrhenius equation can be thought of as the frequency of collision of the molecules. • Also, the exponential term can be thought of as the fraction of the reactants that posses energy greater than Ea.

11. Frequency Factor according to Collision Theory • The pre-exponential factor can be estimated for gas-phase reaction from the collision theory, which is essentially based on the Kinetic Theory of Gases. • From kinetic theory of gases, the bimolecular collision of unlike molecules in a mixture of A & B can be written as: No. of collision/sec-cm3

12. More on frequency factor, A For many bimolecular reactions, the pre-exponential factors are in the order of 1013 in appropriate units For selected reactions, the pre-exponential factors can be predicted within a factor of 2.

13. Ea A+B Energy HR C Reaction Coordinate Activation Energy - Ea Activation Energy: A simple description of Ea would be the energy barrier that reactants must overcome for a reaction to proceed.

14. Illustration of Transition State

15. Why does increasing temperature result in increased reaction rate ? From: Chemical Kinetics and Reaction Dynamics by Paul L. Houston

16. ln(k) 1/T Implications of Arrhenius Law k = A exp(-Ea/RT) T Rxn1: High Ea ln( k) = ln A - (Ea/R) x 1/T Rxn 2: Low Ea • k with higher Ea is more sensitive to temperature than those with low Ea

17. Kinetics of Many Processes in Nature follow Arrhenius Relationship • References: • M. I. Masel, Chemical Kinetics and Catalysis • Octave Levenspiel, Chemical Reaction Engineering Some Examples • Cricket chirping • Ant walking • Tumour growth • Diffusion in solids [D = Do exp (-ED/RT)] Rate of Cricket Chirping

18. The case of ant walking: Can we really represent it with Arrhenius Relation? • M. I. Masel, Chemical Kinetics and Catalysis

19. The Case of Ant walking Raw data Processed data

20. Units of rate constant (rA) = k x [conc. terms] units of k = units of (rA )/units of [conc. terms] Zero-order reaction (rA) = k k in mol/m3 ·s or mol/L·s First order reaction (rA) = kCA k in s-1 Second-order reaction (rA) = kCA2 k in m3/mol ·s or L/mol·s

21. Concentration Dependency of Rate Law

22. Dependence of Reaction Rate on Concentration - Reaction Order Consider the following reaction: aA + bB  cC + dD The rate law for which may be written as follows: (-rA) = k CAa CBb (Power Law Model) a = reaction order with respect to species “A” b = reaction order with respect to species “B” n = a + b = overall order of reaction Note: The units of k depends on the reaction order. You can find out the overall order of reaction from units of k

23. Rate Laws for Elementary and Non-Elementary Reactions In the previous example, i.e., aA + bB  cC + dD The rate law (in terms of the rate of consumption of A) was written as: (-rA) = k CAa CBb Now, if a = a and b = b , the reaction is considered to follow an elementary rate law. Elementary Reaction H2 +I2 2HI (rHI) = k CH2 CI2 Non-elementary CO + Cl2 COCl2 (-rCO) = k CCO CCl23/2

24. What factors affect concentration We know that the concentration of reactants and products in batch and plug flow reactors vary with time and position, respectively  as the reaction proceeds. In addition, for ‘real’ reactors, the temperature and pressure may vary with time and/or position. These changes may affect reaction rate constant and/or concentration and, thereby, the net rates of reaction. For instance, we have seen earlier, how the rate constant (k) is affected by temperature  k = A exp (-Ea/RT). We will now evaluate how changes in temperature and pressure affect concentration of reactants or product species.

25. Concentration, by definition is moles of species per unit volume. The idea here is to find out if and how concentration may vary with pressure and temperature. Equation of State for a Real Gas z=1 for ideal gas Concentration of a species in gas phase is a function of both pressure and temperature. Effect of pressure and temperature on gas phase concentration

26. Effect of pressure and temperature on liquid phase concentration Let us say that we have ni moles of a chemical species in a total liquid volume of V. Accordingly, the concentration of species i is: Our interest is in determining how the volume of liquid changes with temperature and/or pressure. The liquid density does not change ‘significantly’ with temperature and pressure, therefore, the volume of a fixed mass of liquid does not change significantly either. Conclusion: Concentration of a species in liquid-phase can be considered to vary negligibly with changes in pressure and temperature.

27. Stoichiometric Table

28. What is a Stoichiometric Table & Why do we need them? To answer why, let us review our current status for designing reactors:

29. Stoichiometric Table forBatch Reactors

30. Reaction: Stoichiometric Table for Batch Reactors

31. Reaction: Some Useful Definitions 1. Net mole change for the reaction For above reaction, let us define a term “d”, which represents net change in total number of moles per mole of A reacted 2. Relationship between ‘d’ and initial mole fraction of A

32. Concentration in Batch Reactors : Calculating Concentration for Batch Reactors From stoichiometric table, we have Ni = f(X). The concentration of individual species (reactants and products) can be calculated if we know the volume occupied by the reacting mixture. The calculation method for volume will depend on the reaction condition as well as on the reactor type. Reaction Condition (i) Gas-phase reaction (ii) Liquid-phase reaction Reactor types A. Constant volume reactors B. Variable volume reactors

33. Calculating Concentration for Batch Reactors (Continued) A. Constant volume reactors Case I: Liquid Phase-Reaction e.g. polymerization reaction Case II: Gas Phase-Reaction with ‘d=0’ e.g. water gas shift reaction: CO + H2O  CO2 + H2 methane oxidation : CH4 + 2O2 CO2 + 2H2O Case III: Gas Phase-Reaction with ‘d≠0’ e.g. ammonia synthesis: N2 + 3H22 NH3[d <0] propane oxidation: C3H8 + 5O2 3CO2 + 4H2O [d >0] B. Variable volume reactors Case IV: Gas Phase-Reaction in Variable Volume Reactor e.g. combustion engine

34. Calculating Concentration for Batch Reactors Constant Volume Batch Reactors – Cases I & II(Continued) Recall, our goal is to calculate volume for the different batch reactor operations. We can get the number of moles of any species (Ni) as a function of conversion from stoichiometry table. Case I: Constant Volume Reactor + Liquid Phase-Reaction For liquid phase reaction, the volume of interest for calculation of concentration is the volume that liquid occupies, which is usually the initial volume of liquid (V0). V = V0[volume of liquid in the reactor could be < reactor volume] Case II: Constant Volume Reactor Gas Phase-Reaction with ‘d=0’ V = V0 = VR[gas occupies the total volume of the reactor]

35. Calculating Concentration for Batch Reactors Constant Volume Batch Reactors – Case III (Continued) Case III: Constant Volume Reactor Gas Phase-Reaction with ‘d ≠ 0’ V = V0 = VR[gas occupies the total volume of the reactor] The pressure in such systems vary with conversion. Variation of pressure has been used to monitor the progress of a reaction. Pressure (P) at any conversion X can be expressed according to the following relationship:

36. Calculating Concentration for Batch Reactors Variable Volume Batch Reactor – Case IV (Continued) Case IV: Variable Volume Reactor with Gas Phase-Reaction (d≠0) V ≠ V0 The pressure in such systems may be constant or vary with conversion. At any given conversion, X, the reactor volume may be related to initial reactor volume (V0) and other operating parameters (P0, T0, P, and T) For ideal gas

37. Flow Reactors

38. Reaction: Stoichiometric Table for Flow Reactors I hope you see the similarity between flow and batch reactor stoichiometric tables

39. Concentration in Flow Reactors : Calculating Concentration for Flow Reactors • From stoichiometric table, we have • Fi = f(X) • What value of v should we use to calculate concentration? • Liquid phase reactions • v = vO • Gas phase reactions For isothermal and isobaric reactors with no change in number of moles (i.e. d=0) v = vo