ERT 316: REACTION ENGINEERING CHAPTER 3 RATE LAWS & STOICHIOMETRY 1 Lecturer: Miss Anis Atikah Ahmad Email: anisatikah@unimap.edu.my Tel: +604 976 3245 OUTLINE PART 1: Rate Laws Relative Rates of Reaction Reaction Order & Rate Law Reaction Rate Constant, k PART 2: Stoichiometry Batch System Stoichiometric Table Flow System Stoichiometric Table Calculation for Concentration in terms of Conversion 1. RELATIVE RATES OF REACTION aA bB cC dD rC rD rA rB a b c d Reaction Stoichiometry EXAMPLE 2 NO O2 2 NO2 rNO rO2 rNO2 2 1 2 If NO2 formed at 4 mol/m3/s (r NO = 4 mol/m3/s), what is 2 the rate of formation of NO? 1. RELATIVE RATES OF REACTION 2 NO O2 2 NO2 If NO2 formed at 4 mol/m3/s (r NO = 4 mol/m3/s), 2 what is the rate of formation of NO? rNO rO2 rNO2 2 1 2 rNO rNO2 2 2 3 rNO 4mol / m / s 2 2 rNO 4mol / m3 / s 4mol / m3 / s 2 2 1. RELATIVE RATES OF REACTION EXERCISE The Reaction: 2 A 3B 5C is carried out in a reactor. If at a particular point, the rate of disappearance of A is 10 mol/dm3/s, what are the rates of B and C? 1. RELATIVE RATES OF REACTION 2 A 3B 5C The relative rates are rA rB rC 2 3 5 Given, the rate of disappearance of A, -rA, is 10mol/dm3/s 3/s r = -10 mol/dm A Thus, solving the rates of B & C; rA rC 2 5 rA rB 2 3 3 rB 10mol / dm3 / s 2 15mol / dm3 / s 5 rC 10mol / dm3 / s 2 25mol / dm 3 / s 2. REACTION ORDER & RATE LAW Rate law is a kinetic expression that gives the relationship between reaction rate, -rA, and concentration. The reaction rate (rate of disappearance) depends on temperature and composition. It can be written as the product of reaction rate constant, kA and a function of concentrations (activities) of the reactants involved in the reaction: rA k A T fnCA , CB ... 2. REACTION ORDER & RATE LAW Rate law is a kinetic expression that gives the relationship between reaction rate, -rA, and concentration. For reaction in which the stoichiometric coefficient is 1 for ALL species: 1NaOH 1HCl 1NaCl 1H 2O we shall delete the subscript on the specific reaction rate, (e.g.; A in kA) to let k k NaOH k HCl k NaCl k H 2O 2.1 POWER LAW MODELS & ELEMENTARY RATE LAWS Power Law Model: rA kCA CB The rxn is 𝛂 order wrt reactant A AND The rxn is 𝛃 order wrt reactant B The overall order of the reaction, n; n 2.1 POWER LAW MODELS & ELEMENTARY RATE LAWS The unit of the specific reaction, k, will vary with the order of reaction. A Products Concentration k 1 n Time rA k A k mol / dm3 s Second order (n=2) rA k ACA rA k AC A2 Third order (n=3) rA k AC k s 1 k dm3 / mol s 2 3 k dm / mol s 1 Zero order (n=0) First order (n=1) 3 A 2.1 POWER LAW MODELS & ELEMENTARY RATE LAWS Elementary reaction: a chemical reaction in which one or more of the chemical species react directly to form products in a single reaction step and with a single transition state. Unimolecular reaction A Products A B Products Bimolecular reaction Elementary rate law: The rxn is said to follow the elementary rate law if the stoichiometic coefficients are IDENTICAL to the reaction order of each species. 2 NO O2 2 NO2 rNO k NOC CO2 2 NO Nonelementary rxn But follows the elementary rate law! EXAMPLES OF REACTION RATE LAWS EXAMPLES OF REACTION RATE LAWS EXAMPLES OF REACTION RATE LAWS 2.2 NON-ELEMENTARY RATE LAWS Non-elementary rate laws: reactions that do not follow simple rate laws (power rate laws). Example 1: Homogeneous Rxn CO Cl2 COCl2 Gas phase synthesis of phosgene The kinetic rate law is: rCO kCCOC 32 Cl 2 Rxn order: first order wrt to CO, three-halves order wrt Cl2, five-halves order overall. 2.2 NON-ELEMENTARY RATE LAWS Example 2: Heterogeneous Rxn C6 H 5CH 3 H 2 C6 H 6 CH 4 Gas-solid catalyzed rxn: Hydrodemethylation of toluene (T) cat T H2 B M cat The rate of disappearance of toluene per mass of catalyst is: r ' T kPH 2 PT 1 K B PB KT PT In terms of partial pressure rather than concentrations where KB & KT is the adsorption constants. 2.3 REVERSIBLE REACTIONS aA bB ⇌ cC dD For reversible rxn, all rate laws must reduce to the thermodynamic relationship relating the reacting species concentrations at equilibrium. c Ce a Ae d De b Be C C KC C C Thermodynamic Equilibrium Relationship 2.3 REVERSIBLE REACTIONS EXAMPLE: combination rxn of 2 mol of benzene to form 1 mol H2 and 1 mol diphenyl. kB 2C6 H 6 ⇌ C12 H10 H 2 k-B symbolically; 2B kB ⇌k D H 2 -B The rate of disappearance of benzene; rB, forward kBC 2 B OR rB, forward kBC 2 B The reverse rxn btween diphenyl & hydrogen; k-B C12 H10 H 2 2 ⇌ C6 H 6 The rate of formation of benzene (in reverse direction); rB ,reverse k B C D C H 2 2.3 REVERSIBLE REACTIONS The net rate of formation of benzene is; rB rB ,net rB , forward rB ,reverse kBC k BCDCH 2 2 B Multiplying both sides by -1, we obtain the rate law of disappearance of benzene, -rB rB kBC k BCDCH 2 2 B 2 k B k B CB CDCH 2 kB 2.3 REVERSIBLE REACTIONS 2 k B rB k B CB CDCH 2 kB Replacing the ratio of the reverse & forward rate law constant by equilibrium constants; 2 CDCH 2 rB k B CB KC where kB Concentration K C equilibrium constant kB 3. THE REACTION RATE CONSTANT k A T Ae E / RT Arrhenius equation A= preexponential factor or frequency factor E= activation energy, J/mol or cal/mol R=gas constant = 8.314 J/mol-K = 1.987 cal/mol-K T= absolute temperature, K A -no of collision e E / RT -probability that the collision will result in a reaction 3. THE REACTION RATE CONSTANT k A T Ae E / RT Activation energy is a measure of the minimum energy that the reacting molecules must have in order for the reaction to occur (energy required to reach transition state). Transition state Energy barier e Reactants Products k - no of collision that result in a rxn A -total no of collision E / RT probability that - the collision will result in a rxn 3. THE REACTION RATE CONSTANT k A T Ae E / RT Taking a natural logarithm; E1 ln k A ln A R T E ⬆, k ⬆, -r = ⬆ The larger the activation energy, the more temperature sensitive k and thus the reaction rate. 4. BATCH SYSTEMS STOICHIOMETRIC TABLE Purpose of developing stoichiometric table: To determine the no of moles of each species remaining at a conversion of X. 4. BATCH SYSTEMS STOICHIOMETRIC TABLE Components of stoichiometric table: Species A B C D I Totals Initially (mol) Change (mol) refers to moles of species reacted or formed Remaining (mol) 4. BATCH SYSTEMS STOICHIOMETRIC TABLE aA + bB cC + dD Recall from Chapter 2: N A0 N A X N A0 moles of A reacted N A N A0 N A0 X Factorizing; N A N A0 1 X moles of A remaining in the reactor at a conversion of X 4. BATCH SYSTEMS STOICHIOMETRIC TABLE Moles B reacted, NB Moles C formed, NC Moles D formed, ND Moles B reacted Moles A reacted b N A0 X a c N A0 X a d N A0 X a Moles A reacted 4. BATCH SYSTEMS STOICHIOMETRIC TABLE b moles B remaining N N A0 X B 0 in the system, NB a moles of B initially in the system NC ND c N C 0 N A0 X a d N D 0 N A0 X a moles of B reacted moles of C formed moles of D formed 4. BATCH SYSTEMS STOICHIOMETRIC TABLE Species Initially (mol) A N A0 B C D I Totals N B0 NC 0 N D0 NI0 NT 0 Change (mol) Remaining (mol) N A0 X N A N A0 N A0 X b N A0 X a c N A0 X a d N A0 X a b N A0 X a c N C N C 0 N A0 X a d N D N D 0 N A0 X a NI NI 0 - N B N B0 d c b NT NT 0 1 N A0 X a a a 4. BATCH SYSTEMS STOICHIOMETRIC TABLE Total no of moles per mole of A reacted can be calculated as: d c b NT NT 0 1 N A0 X a a a N T 0 N A0 X where Change in the total number of moles per mole of A reacted d c b 1 a a a 4. BATCH SYSTEMS STOICHIOMETRIC TABLE rA kCA rA kCA 2 rA k AC 3 A Can we express concentration of each species?? Species A B C D I Totals Initially Change Remaining Concentration 4. BATCH SYSTEMS STOICHIOMETRIC TABLE Concentration of each species in terms of conversion can be expressed as: Recall from stoichiometric table N A N A0 1 X CA V V N B N B 0 b / a N A0 X CB V V N C N C 0 c / a N A0 X CC V V N D N D 0 d / a N A0 X CD V V Remaining (mol) A B C D N A N A0 N A0 X b N A0 X a c N C N C 0 N A0 X a d N D N D 0 N A0 X a N B N B0 4. BATCH SYSTEMS STOICHIOMETRIC TABLE N B 0 b / a N A0 X CB V N A0 N B 0 / N A0 b / a X V N A0 B b / a X V N C 0 c / a N A0 X CC V N A0 N C 0 / N A0 c / a X V N A0 C c / a X V 4. BATCH SYSTEMS STOICHIOMETRIC TABLE N D 0 d / a N A0 X CD V N A0 N D 0 / N A0 d / a X V N A0 D d / a X V N i 0 Ci 0 yi 0 i N A0 C A0 y A0 4. BATCH SYSTEMS STOICHIOMETRIC TABLE Species Initially A B N A0 N B0 C NC 0 D N D0 I NI0 Change N A0 X Remaining N A N A0 N A0 X b N A0 X N B N B 0 b N A0 X a a c N A0 X N C N C 0 c N A0 X a a d N A0 X N D N D 0 d N A0 X a a - NI NI 0 Concentration N A0 1 X CA V N A0 B b / a X V N c / a X CC A0 C V CB N A0 D d / a X CD V C IO N B0 b b N B N B 0 N A0 X N A0 X a N A0 a b N A0 B X a i N i 0 Ci 0 y i0 N A0 C A0 y A0 4. BATCH SYSTEMS STOICHIOMETRIC TABLE Species Initially A B N A0 N B0 C NC 0 D N D0 I NI0 Change N A0 X Remaining N A N A0 N A0 X Concentration N A0 1 X CA V b N A0 X N B N A0 B b X CB N A0 B b / a X a a V c N A0 X NC N A0 C c X CC N A0 C c / a X a V a d N A0 X N D N A0 D d X CD N A0 D d / a X a a V C IO NI NI 0 - 4. BATCH SYSTEMS STOICHIOMETRIC TABLE EXAMPLE Given the saponification for the formation of soap from aqueous caustic soda & glyceryl stearate is: 3NaOH aq C17 H 35COO 3 H 5 3C17 H 35COONa C3 H 5 OH 3 Letting X the conversion of sodium hydroxide, set up a stoichiometric table expressing the concentration of each species in terms of its initial concentration and the conversion. 4. BATCH SYSTEMS STOICHIOMETRIC TABLE EXAMPLE We know that this is a liquid-phase reaction. Therefore, V=V0 3NaOH aq C17 H 35COO 3 H 5 3C17 H 35COONa C3 H 5 OH 3 3A B 3C D a 3 b 1 c 3 d 1 N A0 1 X N A0 1 X CA C A0 1 X V V0 N A0 B b / a X 1 CB C A0 B X V0 3 4. BATCH SYSTEMS STOICHIOMETRIC TABLE EXAMPLE Species Initially A B N A0 N B0 C NC 0 D N D0 I NI0 Total NT 0 Change Remaining N A0 X N A N A0 1 X 1 N A0 X N B N A0 B 1 X 3 3 N A0 X NC N A0 C X 1 1 N N N A0 X D A0 D 3 X 3 NI NI 0 - 0 NT N T 0 Concentration C A C A0 1 X 1 C B C A0 B X 3 CC C A0 C X 1 C D C A0 D X 3 C IO 5. FLOW SYSTEMS STOICHIOMETRIC TABLE Purpose of developing stoichiometric table: To determine the effluent flow rate of each species at a conversion of X. 5. FLOW SYSTEMS STOICHIOMETRIC TABLE Components of stoichiometric table: Species A B C D I Totals Feed rate to reactor (mol/time) Change within Effluent rate the reactor from reactor (mol/time) (mol/time) 5. FLOW SYSTEMS STOICHIOMETRIC TABLE Species A B C D I Totals Feed rate to reactor (mol/time) FA0 Change within the reactor (mol/time) FA0 X Effluent rate from reactor (mol/time) FA FA0 FA0 X Concentration (mol/L) CA FA0 1 X b FA0 B b / a X FB 0 B FA0 b F X FB FA0 B X A0 a CB a FC 0 C FA0 FD 0 D FA0 FI 0 i FA0 FT 0 c FA0 X a c FC FA0 C X C FA0 C c / a X C a d d F F FA0 X D A0 D a X CD FA0 D d / a X a F FI FA0 I C I A0 I FT FT 0 FA0 X QUIZ 5 Given a liquid phase reaction: A+ 2B C + D The initial concentration of A and B are 1.8 kmol/m3 and 6.6 kmol/m3 respectively. Construct a stoichiometric table for a flow system considering A as the basis of calculation. Fi 0 Ci 0 yi 0 i FA0 C A0 y A0 ANSWER FOR QUIZ 5 A+ 2B C + D Given: C A0 1.8kmol / m3 CC 0 0kmol / m3 CBO 6.6kmol / m3 CDO 0kmol / m3 Since C & D are products. From stoichiometry, we know that, a 1 b 2 c 1 d 1 Ci 0 i C A0 6.6 B 3.67 1.8 d c b 1 1 a a a 0 C 0 1 .8 0 D 0 1.8 ANSWER FOR QUIZ 5 Species A B Feed rate to reactor (mol/time) Change within the reactor (mol/time) FA0 FA0 X FB 0 FA0 B Effluent rate from reactor (mol/time) FA FA0 1 X 2FA0 X FB FA0 B 2 X C FC 0 FA0C FA0 X FC FA0 X D FD 0 FA0 D FA0 X FD FA0 X Totals FT 0 FT FT 0 FA0 X ANSWER FOR QUIZ 5 Substituting the numerical values; Species Feed rate to reactor (mol/time) Change within the reactor (mol/time) FA0 FA0 X FA FA0 1 X FB 0 3.67 FA0 2 FA0 X FB FA0 3.67 2 X C FC 0 0 FA0 X FC FA0 X D FD 0 0 FA0 X FD FA0 X A B Totals FT 0 Effluent rate from reactor (mol/time) FT FT 0 FA0 X 6. CONCENTRATION IN TERMS OF CONVERSION 1. For liquid phase: Batch System: V V0 CA N A N A0 1 X V V CB N B N A0 B b / a X N A0 B b / a X C A0 B b / a X V0 V V NC N A0 C c / a X C A0 C c / a X N A0 C c / a X CC V0 V V ND N A0 D d / a X N A0 D d / a X C d / a X CD A0 D V V 0 V 6. CONCENTRATION IN TERMS OF CONVERSION 1. For liquid phase: Flow System - CA FA CB FB CC FC CD FD 0 FA0 1 X FA0 B b / a X FA0 B b / a X C A0 B b / a X 0 FA0 C c / a X 0 C A0 C c / a X FA0 D d / a X FA0 D d / a X 0 C A0 D d / a X FA0 C c / a X 6. CONCENTRATION IN TERMS OF CONVERSION 2. For gas phase: Batch System CA NA V Need to substitute V from gas law equation From equation of state; At any time t, PV ZNT RT (1) At initial condition (t=0) P0V0 Z 0 NT 0 RT0 (2) T= temperature, K P= total pressure, atm (1 atm= 101.3 kPa) Z= compressibility factor R= gas constant = 0.08206 dm3-atm/mol-K 6. CONCENTRATION IN TERMS OF CONVERSION 2. For gas phase: Batch System PV ZNT RT (1) P0V0 Z 0 NT 0 RT0 (2) Recall from stoichiometric table Dividing (1) by (2); P0 T Z NT V V0 P T0 Z 0 NT 0 Dividing (4) by NT0 ; N A0 NT 1 X NT 0 NT 0 1 y A0 X NT NT 0 N A0 X (3) (4) 6. CONCENTRATION IN TERMS OF CONVERSION 2. For gas phase: Applies for both batch and flow systems Batch System NT 1 y A0 X NT 0 NT 1 X NT 0 Rearranging; d c b NT 1 a a a NT 0 y A0 Will be substitute in (3) NT NT 0 NT 0 X At complete conversion (for irreversible rxn): X=1, NT=NTf N Tf N T 0 NT 0 6. CONCENTRATION IN TERMS OF CONVERSION 2. For gas phase: Batch System P0 T Z NT V V0 P T0 Z 0 NT 0 (3) Substituting the expression for NT/NT0 in (3), P0 T Z V V0 1 X P T0 Z 0 T P V V0 0 1 X T0 P (5) If the compressibility factor are not change significantly during rxn, Z0⩳Z 6. CONCENTRATION IN TERMS OF CONVERSION 2. For gas phase: Flow System Cj Need to substitute υ from gas law equation Fj From gas law, at any point in the reactor, CT FT P ZRT (1) At the entrance of reactor; CT 0 FT 0 P0 0 Z 0 RT0 (2) Dividing (1) by (2) FT P0 T 0 FT 0 P T0 (3) 6. CONCENTRATION IN TERMS OF CONVERSION 2. For gas phase: Flow System Recall from stoichiometric table FT P0 T 0 FT 0 P T0 FT FT 0 FA0X Substituting for FT; FT 0 FA0X 0 FT 0 P0 T P T0 FA0 P0 T 0 1 X FT 0 P T0 P0 T P0 T 0 1 y A0X 0 1 X P T0 P T0 (4) 6. CONCENTRATION IN TERMS OF CONVERSION 2. For gas phase: Flow System Cj Need to substitute υ from gas law equation Fj P0 T 0 1 X P T0 Substituting υ & Fj; Cj FA0 j v j X 0 1 x C A0 P0 T P T0 v j X P T0 1 x P0 T j (4) F j F j 0 j v j X (5) Stoichiometric coefficient (d/a, c/a, -b/a, -a) 6. CONCENTRATION IN TERMS OF CONVERSION 2. For gas phase: aA + bB cC + dD Flow System Concentration for each species: FA0 1 X 1 X T0 P C A0 1 x T P0 0 1 X CA FA CB FB FA0 B b / a X C B b / a X T0 A0 1 x T CC FC FA0 C c / a X C C c / a X T0 A0 1 x T CD FD FA0 D d / a X C D d / a X T0 P A0 1 x T P0 FI FI 0 I C A0 I T0 P CI 1 x T P0 P P0 P P0 P0 P T T0 SUMMARY Relative rate of reaction: aA bB cC dD rA rB rC rD a b c d Power Law Model: rA kCA CB SUMMARY Elementary rate law: The rxn that in which its stoichiometic coefficients are IDENTICAL to the reaction order of each species. Non-elementary rate laws: The reactions that do not follow simple rate laws (power rate laws) in which its stoichiometic coefficients are NOT IDENTICAL to the reaction order of each species. Reversible reaction: All rate laws must reduce to the thermodynamic relationship relating the reacting species concentrations at equilibrium. SUMMARY Reaction Rate Constant, k k A T Ae E / RT E ⬆, k ⬆, -r ⬆ The larger the activation energy, the more sensitive k is, (towards the change in temperature) SUMMARY Stoichiometric Table for Batch Systems Species Initially A B N A0 N B0 C NC 0 D N D0 I NI0 Change Remaining N A0 X N A N A0 N A0 X b N A0 X N B N B 0 b N A0 X a a c N A0 X N C N C 0 c N A0 X a a d N A0 X N D N D 0 d N A0 X a a - NI NI 0 SUMMARY Stoichiometric Table for Flow Systems Species A B C D I Totals Feed rate to reactor (mol/time) FA0 FB 0 B FA0 FC 0 C FA0 FD 0 D FA0 FI 0 i FA0 FT 0 Change within the reactor (mol/time) FA0 X Effluent rate from reactor (mol/time) FA FA0 FA0 X b FA0 X a c FA0 X a b FB FA0 B X a d FA0 X a - d FD FA0 D X a c FC FA0 C X a FI FA0 I FT FT 0 FA0 X SUMMARY Expression of V and υ in calculating the concentration of each species: Batch systems Liquid phase: V V0 Gas phase: T P0 V V0 1 X T0 P Flow systems Liquid phase: 0 Gas phase: T P0 0 1 X T0 P QUIZ 6 Derive a concentration for each species for the isothermal gas phase reaction below, neglecting the pressure drop: A+BC