Mathematical Expressions for Chemical Reaction Rates: Rate Laws, Extent, and Conversion, Summaries of Chemical Thermodynamics

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Chemical Reaction

Engineering

Thomas Rodgers

8.3 Problems.................................. 91

iii

List of Figures

vi

viii

Nomenclature

Roman A Arrhenius pre-exponential factor varies C Concentration mol m−^3 Eact Activation energy J mol−^1 G Generation mol s−^1 K Equilibrium constant varies k Rate constant varies M Ratio of CB 0 /CA 0 − N Moles mol n Molar flow rate mol m−^3 s−^1 R 8.314 J mol−^1 K−^1 r Reaction rate mol s−^1 S Selectivity − T Temperature K t Time s t 1 / 2 Half-life s V Volume m^3 v Volumetric flow rate m^3 s−^1 X Conversion − Y Yield − Greek α Reaction order − β Reaction order − δ Change in moles per mole of A −  Change in mole fraction per mole of A − ν Stoichiometric coefficient − 0 Related to the start e Refers to the equilibrium f Related to the end

ix

Reactions and Reactors

• 1 Reactions and Reactors Nomenclature ix
  • 1.1 Introduction
  • 1.2 Reaction Rate
    • 1.2.1 Reaction Equation or Rate Law
    • 1.2.2 Relative Rates of Reaction
  • 1.3 Extent of Reaction
  • 1.4 Conversion
  • 1.5 General Mole Balance for Reactors
  • 1.6 Use of Extent of Reaction and Conversion
    • 1.6.1 Extent of Reaction
    • 1.6.2 Conversion
  • 1.7 References
  • 1.8 Problems
• 2 Batch Reactors
  • 2.1 Introduction
  • 2.2 Mole Balance Applied to Batch Reactors
  • 2.3 Extent of Reaction in Batch Reactors
  • 2.4 Conversion in Batch Reactors
  • 2.5 Summary
  • 2.6 References
  • 2.7 Problems
• 3 Plug Flow Reactors (PFR)
  • 3.1 Introduction
  • 3.2 Mole Balance Applied to PFR
  • 3.3 Extent of Reaction in PFR
  • 3.4 Conversion in PFR
  • 3.5 Summary
  • 3.6 References
  • 3.7 Problems
• 4 Continously Stirred Tank Reactors (CSTR)
  • 4.1 Introduction
  • 4.2 Mole Balance Applied to CSTR
  • 4.3 Extent of Reaction in CSTR
  • 4.4 Conversion in CSTR
  • 4.5 Levenspiel Plots
    • 4.5.1 CSTR
    • 4.5.2 PFR
    • 4.5.3 Reactors in Series
  • 4.6 Summary
  • 4.7 References
  • 4.8 Problems
• 5 Determination of Rate Law Parameters
  • 5.1 Introduction
  • 5.2 Experimental Determination of Rates
  • 5.3 Determination of Reaction Orders by Inspection
  • 5.4 Differential Methods of Analysis
    • 5.4.1 For Rates Dependent on the Concentration of Only One Species
    • 5.4.2 For More Complex Reactions (Dependence on CA and CB )
    • 5.4.3 Method of Initial Rates
  • 5.5 Integral Methods
    • 5.5.1 Zeroth Order Reaction by Integral Method
    • 5.5.2 First Order Irreversible Reaction by Integral Method
    • 5.5.3 Second Order Reaction by Integral Method
  • 5.6 Method of Half-Lives
  • 5.7 References
  • 5.8 Problems
• 6 Temperature and Pressure Dependance of Reactions
  • 6.1 Introduction
  • 6.2 Arrhenius Equation
    • 6.2.1 Example Calculation of the Arrhenius Parameters
  • 6.3 Reactors with a Change in Pressure at Constant Volume
  • 6.4 Reactors with a Change in Volumetric Flowrate
  • 6.5 References
  • 6.6 Problems
• 7 Multiple Reactions
  • 7.1 Introduction
  • 7.2 Yield and Selectivity
  • 7.3 Reversible Reactions
  • 7.4 Parallel Reactions
  • 7.5 Series Reactions
  • 7.6 References
  • 7.7 Problems
• 8 Design Structure for Reactors
  • 8.1 Design Structure for Reactors
    • 8.1.1 Example
  • 8.2 References
• 1.1 Schematic representation of the design of a reactor volume.
• 2.1 Schematic representation of a batch reactor.
• 2.2 Mole-time trajectories for component A and B.
• 3.1 Schematic representation of a PFR.
• 3.2 Mole balance of component i in a differential segment of δV
• 3.3 Molar flowrate of component A and B as a function of volume in a PFR.
• 4.1 A typical CSTR showing the flow of the reacting material A.
• 4.2 Levenspiel plot of example reaction data.
• 4.3 Representation of (a) CSTR and (b) PFR volume calculations
• 4.4 Representation of multiple (a) CSTR and (b) PFR volumes
• 5.1 Concentration vs time curve.
• 5.2 Differential rate for a constant volume batch reactor
• 5.3 Initial concentration vs rate curve.
• 5.4 Zeroth order irreversible reaction in a constant volume batch reactor
• 5.5 First order reaction in a constant volume batch reactor
• 5.6 First order reaction in a constant volume batch reactor
• 5.7 Second order reaction in a constant volume batch reactor
• 5.8 Plot of data taken from a constant volume batch reactor
• 6.1 Temperature dependence of the rate constant k
• 6.2 Symbolic reaction showing the activation energy barrier
• 6.3 Plot of ln k vs 1 /T
• 7.1 Plot of ln (1 − X/Xe) vs t to determine −k 1 (M + 1) / (M + Xe)
• 7.2 Plot of − ln (CA/CA 0 ) vs t to determine (k 1 + k 2 )
• 7.3 Concentration plot based on Equation 7.4.6 to determine k 1 /k
• 7.4 Concentration plot for parallel reactions
• 7.5 Concentration plot for serial reactions
• 7.6 Concentration profiles for series reactions
• 7.7 Concentration profiles for parallel reactions
• Chapter
  • 1.1 Introduction Contents
  • 1.2 Reaction Rate
    • 1.2.1 Reaction Equation or Rate Law
    • 1.2.2 Relative Rates of Reaction
  • 1.3 Extent of Reaction
  • 1.4 Conversion
  • 1.5 General Mole Balance for Reactors
  • 1.6 Use of Extent of Reaction and Conversion
    • 1.6.1 Extent of Reaction
    • 1.6.2 Conversion
  • 1.7 References
  • 1.8 Problems

2

CHAPTER 1. REACTIONS AND REACTORS

The rate equation is independent of the type of reactor (e.g. batch or continuous flow) in which the reaction is carried out.

Rate Law

A −−→ products

The reaction rate may be a linear function of concentration, i.e. −rA = kCA or may be some other algebraic function of concentration, such as −rA = kC A^2 or −rA = (k 1 CA) / (1 + k 2 CA).

−rA depends on concentration and temperature. This dependence can be written in gen- eral terms as − rA = k (T ) × f (CA, CB , · · · ) (1.2.2)

where k (T ) (the rate constant) is a function of temperature. The rate law must be deter- mined experimentally. The common form of the rate law is:

− rA = kCAαCBβ (1.2.3)

where α is defined as the reaction order with respect to A and β the reaction order with respect to B. The overall order of the reaction is given by the sum, n = α + β.

The units of k depend on the form of the rate law. We know that the rate must have units of moles/(volume-time) so that k has units of (concentration)^1 −n/time. Thus when n is equal to 1, k has units of s−^1 , for n equal to 2, k has units of mol−^1 dm^3 s−^1 , and when n is equal to 3, k has units of mol−^2 dm^6 s−^1.

Rate Constant Units

Typical examples of reaction rates are a first order reaction given by,

− rA = kCA (1.2.4)

or a second order reaction, − rA = kC^2 A (1.2.5)

where k is called the rate constant,

Usually the order of the reaction provides some insight into the molecular mechanism for the reaction. A first order reaction corresponds to a uni-molecular process, whereas a second order reaction corresponds to reaction controlled by collisions between molecules. These rules are only strictly true when the reaction is an elementary step. Most reactions are combinations of elementary steps, which can lead to more complicated rate laws. In addition, the rate law can depend on the relative concentrations of the components. For instance, rate laws are independent of the concentrations of components, which occur at a large excess relative to another component (i.e. water).

For example, for the reaction of

2 NO + O 2 −−→ 2 NO 2

4 ©cT.L. Rodgers 2013

1.2 Reaction Rate

1.2 Reaction Rate

the rate law is given by − rNO = kNOC NO^2 CO 2 (1.2.6)

The overall order for the reaction is equal to 3, whereas the reaction is 2nd order with respect to NO and 1st order with respect to O 2. Note that the exponents α, β are not always equal to the stoichiometric coefficients except for the case where the reaction is an elementary step as discussed below.

An elementary reaction is one that involves only a single step. For instance:

H + Br 2 −−→ HBr + Br

occurs when one hydronium ion collides and reacts with one bromine molecule. In this case the rate of reaction is given by k [H] [Br 2 ]. Because the step involves the collision of the two molecules, the rate is proportional to each of the concentrations. A general rule is thus that for an elementary reaction, the reaction order follows from the stoichiometric coefficients. However, many reactions are not elementary but instead have a complex, multistep reaction mechanism. In this case the rate of reaction does not often follow from the stoichiometric coefficients. For example, for the reaction of

H 2 + Br 2 −−→ 2 HBr

the rate law is given by

− rHBr =

k 1 CH 2 C Br^3 / 22 CHBr + k 2 CBr 2

(1.2.7)

This reaction occurs due to a free radical mechanism, which actually corresponds to a series of elementary reaction steps as illustrated below.

Br 2 −−→ 2 Br• Br•^ + H 2 −−→ HBr + H• H•^ + Br 2 −−→ HBr + Br• H•^ + HBr −−→ H 2 + Br• 2 Br•^ −−→ Br 2

In this case, the rate law and reaction order must be determined experimentally. If a reac- tion has several steps then the slowest elementary reaction step often is the rate-limiting (or rate determining) step.

1.2.2 Relative Rates of Reaction

Consider the reaction:

(−νA)A + (−νB )B −−→ νC C + νDD

where νA, νB , νC , and νD are called stoichiometric coefficients. Note that stoichiometric coefficients are negative for reactants and positive for products.

or sometimes written as,

A +

(−νB ) (−νA)

B −−→

νC (−νA)

C +

νD (−νA)

D

© cT.L. Rodgers 2013 5

1.4 Conversion

Consider the simple reaction A −−→ B, if this is first order, then the rate can be given by equation 1.2.4, − ri = kCi

where Ci is the concentration of i and can be related to the number of moles, Ni, by,

Ni = CiV (1.3.4)

for a constant volume reactor, V , or, to the molar flow rate, n˙i, by

n ˙i = Civ (1.3.5)

for a constant volumetric flow rate, v, reactor. This means that for a constant volumetric flow rate reactor with a first order reaction, we can write,

− ri =

k v

n˙i (1.3.6)

From equation 1.3.3, we know that the number of moles in terms of the extent of reaction is, n ˙i = ˙ni, 0 + νiξ

Substituting this into equation 1.3.6 gives,

− ri =

k v

( ˙ni, 0 + νiξ) (1.3.7)

This can also be carried out for other reaction rate functions.

1.4 Conversion

In many cases reactions will not proceed to completion, due to equilibrium limitatons or reaction kinetics. The conversion is often used as an indication of how far a reaction has proceeded. Conversion, Xi, is the fraction of a specified reactant which is consumed by the reaction,

Xi =

−d Ni Ni, 0

=

Ni, 0 − Ni Ni, 0

= −

ξνi Ni, 0

(1.4.1)

Conversion

Thus the conversion of A in our reaction is defined as,

XA =

moles of A reacted moles of A fed

Usually we drop the subscript A so X ≡ XA.

© cT.L. Rodgers 2013 7

CHAPTER 1. REACTIONS AND REACTORS

• For irreversible reactions, at the start X = 0 and at the end X = 1, in cases where all of A is consumed (note that A will only be totally consumed if it is the limiting reactant in an irreversible reaction)
• For reversible reactions, the maximum conversion possible is controlled by the equilibrium constant; the maximum value is termed the equilibrium con- version Xe

Conversion

The conversion can be used in material balance calculations as was the extent of reaction, if the number of moles of a species i present before the reaction is Ni, 0 , then the number of moles after the reaction can be found using,

Ni = Ni, 0 (1 − Xi) (1.4.2)

This can also be written in terms of a molar flow as,

n ˙i = ˙ni, 0 (1 − Xi) (1.4.3)

Again let us consider the simple reaction A −−→ B, with, for a constant volumetric flow rate, a first order reaction given by equation 1.3.6,

− ri =

k v

n˙i

From equation 1.4.3, we know the number of moles in terms of the conversion, thus giving us,

− ri =

k v

n˙i, 0 (1 − Xi) (1.4.4)

1.5 General Mole Balance for Reactors

Ideal reactors normally refer to simplified models of reactors, in which various approxi- mations are made. These approximations allow us to describe the behaviour of the reactor using simple mathematical expressions. We will study three types of ideal reactors: the batch reactor, the continuously stirred tank reactor (CSTR), and the plug flow reactor (PFR). To do this a general mole balance is needed for a general reactor as in Figure 1.1. The general mole balance can be written as,

In − Out + Generation = Accumulation 



Rate of flow of i in (moles/time)



 (^) −





Rate of flow of i out (moles/time)



 (^) +

      Rate of generation of i by chemical reaction (moles/time)

     

=

   

Rate of accumulation of i (moles/time)

   

ni|V − ni|V +δV + Gi = d d^ N ti

where ni is the molar flowrate of species i (with units of moles sec−^1 ) and Ni is number of moles of component i in the reactor. The above mole balance is performed about

8 ©cT.L. Rodgers 2013