CHEE 321: Chemical Reaction Engineering

Module 2: Rate Laws and Stoichiometry(Chapter 3, Fogler)

Design Equations for Isothermal Reactors

'0 ( )

A

AA

dXF rdW

= −

IDEAL DIFFERENTIAL ALGEBRAIC INTEGRAL REACTOR FORM FORM FORM

0 ( )AA A

dXN r Vdt

= − 00

AX

AA

dXt Nr V

′=

−∫

0 ( )AA A

dXF rdV

= −0

0

AX

AA

dXV Fr

′=

−∫

CSTR0 ( )

( )A A

A

F XVr

=−

BATCH

PFR

PBR 0 '0

A

A

X

AdXW F

r′

=−∫

Module 2: Rate Laws and Stoichiometery

Topics to be covered in this module

• Revisit reaction rate

• Rate law– Temperature dependency of Rate Law: Reaction Rate Coefficient

• Arrhenius factor and Activation energy– Concentration dependence of reaction rate or rate law

• Reaction order

• Elementary and non-elementary reaction rate laws– Reversible Reactions

• Stoichiometric tables for batch and flow reactors

aA + bB → cC + dD

How is (-rA) related to (-rB), (rC) and (rD) ?

Rate Law - Functional form

Rate Law: is an algebraic equation that relates reaction rate to species concentration

(-rA) = k · [f(CA, CB, ..)]

k is the reaction rate coefficient

The terms within the brackets [f(CA, CB, ..)] 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

Factors influencing rate of reaction (rj)

• Factors affecting rate coefficient (k)– temperature– catalyst type (if present in system)– pressure

• Factors affecting concentration– pressure (especially for gas-phase system)– temperature

Generally negligible

Temperature dependence of rate: the Arrhenius Law

A = frequency factor or pre-exponential factorEa = Activation EnergyR = Universal gas ConstantT = absolute temperature (K)

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".

A majority of reaction rate coefficients can be expressed as a function of temperature by an empirical relationship that was developed by Swedish scientist, Svante Arrhenius.

k = A exp(-Ea/RT)

Implications of Arrhenius Law

1/T (K-1)

Rxn1: High Ea

Rxn 2: Low Ea

ln(k) = ln(A) - (Ea/R) ·1/T

k = A exp(-Ea/RT)

• Rate increases exponentially with increasing T

• k with high value for activation energy Ea is more sensitive to temperature than those with lower values

T

Arrhenius Plot

Interpretation of Arrhenius parameters

• 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 (entropic factor)

• The activation energy can be thought of as the energy barrier that must be overcome as reactants go through the transition to products: transition state theory

k = A exp(-Ea/RT)

Illustration of Transition State

Activation Energy

Enzyme = catalyst

Hprod

Hreact

The role of catalyst is to lower the energy barrier to reaction, so that a higher fraction of molecules react when they collide.

Difference in energy content between products and reactants is the heat of reaction;Difference in energy content between transition state and reactants is the activation energy

Why does increasing temperature result in increased reaction rate ?

From: Chemical Kinetics and Reaction Dynamics by Paul L. Houston

Energy Distribution of Reactant Molecules

Kinetics of Many Processes in Nature follow Arrhenius Relationship

Some Examples• Cricket chirping• Ant walking• Tumour growth• Diffusion [D = Do exp (-ED/RT)]

Rate of Cricket Chirping

References:M. I. Masel, Chemical Kinetics and CatalysisOctave Levenspiel, Chemical Reaction Engineering

The case of ant walking: Can we really represent it with Arrhenius Relation?

•M. I. Masel, Chemical Kinetics and Catalysis

The Case of Ant walking

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60

1000/T (K-1)

ln (w

laki

ng s

peed

)

Raw data

Processed data

Units of rate coefficient

(rA) = k × [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) = kCA

2 k in m3/mol ·s or L/mol·s

Dependence of Reaction Rate on Concentration -Reaction Order

Consider the following reaction:

aA + bB → cC + dD

Note: The units of k depend on the reaction order.

The rate law for which may be written as follows:

(-rA) = k CAα CB

β (Power Law Model)

α = reaction order with respect to species “A”β = reaction order with respect to species “B”

n = α + β = overall order of reaction

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 CAα CB

β

Non-elementaryCO + Cl2 → COCl2 (-rCO) = k CCO CCl2

3/2

Now, if α = a and β = b , the reaction is considered to follow an elementary rate law.

Elementary ReactionH2 +I2 → 2HI (rHI) = k CH2 CI2

Reaction Rates for Reversible ReactionsReaction:

Reaction rates for reversible reactions. What’s so special about them?

Strictly speaking - All reactions are reversible !!

• The reaction occurs in both the forward and reverse directions

• The rate law must have expressions for both forward and reverse reactions

• The rate must be thermodynamically consistent. That is, rate laws (if expressed in terms of concentration) must reduce to thermodynamic relationship relating reacting species concentrations at chemical equilibrium.

, ,

, ,

c dC eq D eq

C a bA eq B eq

C CK

C C=

dDcCbBaA +⇔+

Thermodynamics: Equilibrium Constant

, ,

, ,

c dC eq D eq

C a bA eq B eq

C CK

C C=

dDcCbBaA +⇔+

Units are (mol/L)d+c-a-b

• For ideal gas system, concentration can be expressed in terms of partial pressureRTpC i

i =

, , ( )

, ,

( )

( )

( )

c dC eq D eq c d b a

C a bA eq B eq

aP

p pK RT

p p

K RT δ

− − + +

− ⋅

=

=

• See Fogler Appendix C for a discussion of chemical equilibrium. Various forms of reaction equilibrium constant are used in literature:─ K or Reaction Equilibrium Constant (in terms of activities)─ Kp or Pressure Equilibrium Constant─ KC or Concentration Equilibrium Constant

Reversible Reactions and Reaction Rate

Benzene Diphenyl

Let B = C6H6, D=C12H10

Rate of disappearance of B: Rate of formation of B:

Net rate of B formation:

At equilibrium:

Therefore,

2,B for B Br k C− =

6 6 12 10 22C H C H HB

B

k

k−

⎯⎯→ +←⎯⎯

2,B rev B D Hr k C C−=

2

2B B B B D Hr k C k C C−= − +

2

2, , ,0B B B eq B D eq H eqr k C k C C−= = − +

2, ,2

,

D eq H eqBC

B B eq

C Ck Kk C−

= =

22 D HB B B

C

C Cr k C

K⎛ ⎞

− = −⎜ ⎟⎝ ⎠

Define reaction rates wrt Benzene

Note: rD= −0.5 rB

Fogler 3.2.3

Now for the concentration terms…

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’ (i.e.; non-ideal) 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 coefficient (k) is affected by temperature → k = Aexp(-Ea/RT).

We will now evaluate how changes in temperature and pressure affect concentration of reactants or product species.

Effect of pressure and temperature on gas phase concentration

z=1 for ideal gas

The idea here is to find out if and how concentration may vary with pressure and temperature.

Concentration, by definition is moles of species per unit volume.

VnC i

i =

TRzP

Vn

= Concentration of a species in gas phase is a function of both pressure and temperature.

TRnzPV =

Equation of State for a Real Gas

Effect of pressure and temperature on liquid phase concentration

Our interest is in determining how the volume of liquid changes with temperature and/or pressure.

VnC i

i =

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:

Conclusion: Concentration of a species in liquid-phase can be considered to vary negligibly with changes in pressure and temperature.

We usually assume liquid is an incompressible fluid: density does not change significantly with temperature and pressure, therefore, the volume of a fixed mass of liquid does not change significantly either.

Stoichiometry

A B C D

A B C Db c d

a a a

a b c d

+

+ → +

+ →

For every mole of A consumed:b/a moles of B are consumed,c/a moles of C are produced,d/a moles of D are produced.

CA B Drr r ra b c d

= =− = −

; ;B A C A C Db c dr r r r r ra a a

= = − = −

Relative reaction rates Number of moles in a batch reactor0

0 0

0 0

0 0

(1 )A A A

B B A A

C C A A

D D A A

N N XbN N N XacN N N XadN N N Xa

= −

= −

= +

= +

Fogler 3.5

Stoichiometric Table for Batch Reactors

Species Initial Amount (mol)

Change (mol)

Remaining (mol)

A 0AN 0( )A AN X− 0 0( )A A A AN N N X= −

B 0BN 0( )A A

b N Xa

− 0 0( )B B A AbN N N Xa

= −

C 0CN 0( )A A

c N Xa

0 0( )C C A AcN N N Xa

= +

D 0DN 0( )A A

d N Xa

0 0( )D D A AdN N N Xa

= +

I (inert) 0IN 0 0I IN N=

Total 0TN 0 0T T A AN N N Xδ= +

1−−+=ab

ac

adδδ = increase in the total number of

moles per mole of A reacted

A B C Db c d

a a a++ →

Calculating Concentration for Batch ReactorsThe concentration of individual species (reactants and products) is required to calculate reaction rates: (-rA) = k · [f(CA, CB, ..)]

Since , we need to know the volume occupied by the reacting mixture.

The total number of moles in the reactor can change due to stoichiometry What about volume of the reaction mixture?

The answer depends upon whether it is a gas or liquid system, and also upon the reactor type (constant or variable volume).

ii

NCV

=

Calculating Concentration for Batch Reactors (Continued)

A. Constant volume reactors

Case I: Liquid Phase-Reactione.g. polymerization reaction

Case II: Gas Phase-Reaction with δ=0 (no change in total number of moles)e.g. water gas shift reaction: CO + H2O ↔ CO2 + H2

methane oxidation : CH4 + 2O2 ↔ CO2 + 2H2O

Case III: Gas Phase-Reaction with δ≠0 (V=const)e.g. ammonia synthesis: N2 + 3H2 ↔ 2 NH3 [δ <0]

propane oxidation (combustion): C3H8 + 5O2 ↔ 3CO2 + 4H2O [δ >0]

B. Variable volume reactors

Case IV: Gas Phase-Reaction in Variable Volume Reactore.g. combustion engine

Case I: Liquid Phase-Reaction: incompressible fluid means V=V0[volume of liquid in the reactor could be < reactor volume]

Case II: Constant Volume Reactor Gas Phase-Reaction with δ=0[gas occupies the total volume of the reactor]

No change in total moles and no change in reactor volume (V = V0) If isothermal, also no change in P (ideal gas)

Calculating Concentration for Batch Reactors Constant Volume Batch Reactors – Cases I & II (Continued)

VNC i

i =

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 and material balances.

N PV RT

=

Case III: Constant Volume Reactor Gas Phase-Reaction with δ ≠ 0V = V0 [gas occupies the total volume of the reactor]

However, total number of moles in the system changes. Therefore, the pressure or temperature in such systems vary with conversion. (Variation of pressure can be used to monitor the progress of a reaction).

Calculating Concentration for Batch Reactors Constant Volume Batch Reactors – Case III (Continued)

Pressure at any conversion X can be expressed according to the following relationship derived from the ideal gas law:

0 0

0 0 0 0 0

( )T A AT

T T

N N XN TP TP N T N T

δ+= =

Case IV: Variable Volume Reactor with Gas Phase-Reaction (δ≠0)

V ≠ V0

Calculating Concentration for Batch Reactors Variable Volume Batch Reactor – Case IV (Continued)

The pressure in such systems may be constant or vary with conversion. At any given conversion, X, the reactor volume is related to initial reactor volume (V0) and other operating parameters (P0, T0, P, and T). Additional information must be supplied!

00

0 0 0

( ) ( )T

T

PN T ZV VN P T Z

= For ideal gas 00

0 0

( )T

T

PN TV VN P T

=

0 0T T A AN N N Xδ= +

Stoichiometric Table for Flow Reactors

Reaction: Da

dC

a

c

a

b BA +→+

Species Feed Flow Rate (mol/s)

Change within Reactor (mol/s)

Effluent Rate from Reactor (mol/s)

A 0AF 0( )A AF X− 0 (1 )A A AF F X= −

B 0 0B B AF Fθ= 0( )A A

b F Xa

− 0 ( )B A B AbF F Xa

θ= −

C 0 0C C AF Fθ= 0( )A A

c F Xa

0 ( )C A C AcF F Xa

θ= +

D 0 0D D AF Fθ= 0( )A A

d F Xa

0 ( )D A D AdF F Xa

θ= +

I (inert) 0 0I I AF Fθ= - 0I IF F=

Total 0TF 0A AF Xδ 0 0T T A AF F F Xδ= +

Note the similarity between flow and batch reactor stoichiometric tables

Define 0 0/i i AF Fθ =(Limiting reagent A)

Calculating Concentration for Flow Reactors

From stoichiometry and mole balances, we have Fi = f(XA)

• What value of v (flowrate) should we use?

Liquid phase reactions (incompressible) v = v0

Gas phase reactions (ideal gas) 00

0 0

( )( )T

T

PF Tv vF P T

=

For isothermal and isobaric reactorswith no change in number of moles (i.e. δ=0)

v = v0

Concentration in Flow Reactors : ii

FCv

=

0 0T T A AF F F Xδ= +