CH 204: Chemical Reaction Engineering

CH 204: Chemical Reaction Engineering - lecture

notes

January-April 2010

Department of Chemical Engineering

Indian Institute of Science

Bangalore 560 012

Contents

1 Introduction page 1

2 Review of background material 8

References 49

iii

Introduction

Levenspiel (2004, p. iii) has given a concise and apt description of chemical

reaction engineering (CRE):

Chemical reaction engineering is that engineering activity concerned with the ex-

ploitation of chemical reactions on a commercial scale. Its goal is the successful

design and operation of chemical reactors, and probably more than any other ac-

tivity, it sets chemical engineering apart as a distinct branch of the engineering

profession.

The ingredients of CRE are (i) thermodynamics, (ii) kinetics, (iii)

tranport processes, (iv) types of reactors, (v) mode of operation and con-

tacting, (vi) modelling and optimization, and (vii) control. These topics are

brieﬂy discussed below.

1.1 Thermodynamics

1.1.1 Feasibility of the reaction

The standard free energy of formation ∆G

of gaseous NO at a temperature

T = 298 K and a reference pressure p

= 1 atm is 86.6 kJ/mol. Consider a

closed system that initially contains a mixture of N

and O

, and is main-

tained at a constant temperature T and a pressure p = p

. Consider the

reaction

(g) +

(g)  NO(g) (1.1)

Here the notation “(g)” implies that the species is present in the gas phase.

Similarly, “(l)” and “(s)” will be used to denote species present in the liquid

and solid phases. respectively. Some NO will be formed by (1.1), but the

equilibrium mole fraction of NO (y

NO,e

) is  1 as ∆G

/(R T )  1. Here R

is the gas constant, and y

NO,e

is the mole fraction attained at long times.

Hence we say that the reaction is not feasible under these conditions. The

conversion increases as T increases, but is less than 1 % even at T = 1780

K and p = 1 atm. Hence alternative reactions, such as the oxidation of NH

must be used to produce NO (Chatterjee and Joshi, 2008).

1.1.2 The heat of reaction

The sign of the heat of reaction ∆H determines whether the reactor should

be heated or cooled. The former applies for endothermic reactions (∆H >

0), and the latter for exothermic reactions (∆H < 0). The magnitude of

∆H determines the amount of heating or cooling required.

1.1.3 Allowance for thermodynamic non-idealities

For gaseous reactions at high pressure or low temperature, the equilibrium

constant K

, which is based on partial pressures, must be replaced by the

equilibrium constant K

, which is based on fugacities (Denbigh, 1971, p. 152,

see also section 2.8). For example, consider the ammonia synthesis reaction

(g) + 3 H

(g)  2 NH

(g) (1.2)

At T = 450

◦

C, the value of K

is 6.64 ×10

−3

at p = 10 atm and 8.84

×10

−3

at p = 300 atm (Denbigh, 1971, p. 152). Thus K

varies with the

pressure, whereas the value of K

is approximately constant in this pressure

range - it is 6.5 ×10

−3

at p = 10 atm and 6.6 ×10

−3

at p = 300 atm. The

slight variation of K

is caused by the use of approximate expressions for the

fugacities, based on the Lewis and Randall rule. Given the value of ∆G

, we

can calculate K

, and using the thermodynamic relations between fugacities

and partial pressures, the equilibrium composition can be calculated.

Similarly, for a liquid phase reaction involving the synthesis of methyl

tert-amyl ether (an additive for high octane gasoline) from methanol and

2-methyl-2-butene, the calculated activity coeﬃcient for methanol is in the

range 6.4-7.7 at T = 298 K (Heintz et al., 2007).

1.2 Reactions and kinetics

1.2.1 Classiﬁcation of reactions

Reactions may be classiﬁed by (a) the number of phases involved, (b) the

presence or absence of a catalyst, and (c) the nature of the overall reaction.

Introduction 3

If all the reactants and products, and catalysts, if any, are in a single

phase, the reaction is said to be homogeneous. An example is provided by

the thermal cracking of ethane to ethylene (Froment and Bischoﬀ, 1990,

p. 29)

(g)  C

(g) + H

(g) (1.3)

On the other hand, if more than one phase is involved, the reaction is said to

be heterogeneous. An example is provided by the chemical vapour deposition

(CVD) of Si on a substrate (Fogler, 1999, p. 675)

SiH

(g) → Si(s) + 2 H

(g) (1.4)

(silane)

Equation (1.3) represents a non-catalytic reaction, whereas ammonia

synthesis involves a solid catalyst. In some cases, a homogeneous catalyst

may be involved. For example, an enzyme called glucose isomerase catalyzes

the isomerization of glucose to fructose in the liquid phase (Fig. 1.1).

Fig. 1.1. Isomerization of glucose to fructose. Adapted from Schmidt (2005, p. 24).

Schmidt (2005, p. 24) notes that this is the largest bioprocess in the chemical

industry. As fructose is ﬁve times sweeter than glucose, the process is used

to make high-fructose corn syrup for the soft drink industries.

The overall reaction, as written, may represent either an elementary

reaction or a non-elementary reaction. An example of the former is given

by the gas-phase reaction (Laidler, 2007, p. 138)

(g) + CO(g)  NO(g) + CO

(g) (1.5)

Here NO is formed by the collision between molecules of NO

and CO, and

the rate expression conforms to the stoichiometry shown. On the other

hand, (1.4) represents a non-elementary reaction, as it actually proceeds by

the sequence of reactions shown below (Fogler, 1999, p. 666).

SiH

(g)  SiH

(g) + H

(g)

SiH

(g) + ∗ → SiH

∗

SiH

∗ → Si(s) + H

(g) (1.6)

where * represents an active site on the substrate.

1.2.2 The rate expression

The rate expression provides information about the rate at which a reactant

is consumed. The rate is usually expressed per unit volume of the ﬂuid for

ﬂuid-phase reactions, and per unit area (or unit mass) of the catalyst for

reactions involving solid catalysts. For example, the rate of formation of Si

by the mechanism (1.6) is given by

˙r

k p

SiH

+ K p

SiH

(1.7)

where p

SiH

and p

are the partial pressures of SiH

and H

, respectively.

Equation (1.7) can be derived from (1.6) by assuming that the reac-

tions follow mass action kinetics and invoking some other assumptions.

1.2.3 Alternative catalysts or alternative routes

The conventional process for the manufacture of 5-cyanovaleramide (an in-

termediate for a herbicide) by the hydrolysis of adiponitrile (Fig. 1.2) used

MgO as a catalyst (Pereira, 1999). The catalyst was diﬃcult to recover and

reactivate, and the conversion had to be limited to 20 % to avoid a low

selectivity. An alternative process based on a supported enzyme catalyst

gave a high conversion and a high selectivity.

Introduction 5

Fig. 1.2. Conversion of adiponitrile to 5-cyanovaleramide.

1.3 Transport processes

1.3.1 Balance equations

For ﬂuid-phase reactions, continuum equations are usually used. If the re-

actions involve two phases that are stratiﬁed, as in the case of a gas-liquid

reaction in a falling-ﬁlm reactor, separate equations can be written for each

of the phases. If one phase is dispersed in the other, as in the case of stirred

liquid-liquid dispersions or ﬂuidized beds, we can either write separate equa-

tions for each phase, or use some form of explicit or implicit averaging to

write continuum equations for each phase (see. for example, Jackson, 2000,

Yu et al., 2007).

1.3.2 Constitutive equations

For a ﬂuid phase consisting of simple ﬂuids such as air or water, the Navier-

Stokes equations are commonly used to describe momentum transfer, with

Fourier’s law for heat conduction. Diﬀusion is described either by Fick’s law

for binary mixtures, or by the Maxwell-Stefan equations for multicomponent

mixtures. For a stationary solid phase such as a bed of catalyst pellets, the

momentum balance is not required. However, Fourier’s law and Fick’s law

have to be modiﬁed by replacing the thermal conductivity and diﬀusivity by

the “eﬀective” thermal conductivity and “eﬀective” diﬀusivity, respectively.

For dispersed multiphase systems, constitutive equations are more

complicated, and not as ﬁrmly established as for single-phase systems.

1.4 Types of reactors

1.4.1 Ideal reactors

The adjective ideal refers to the state of mixing in the reactor. It is assumed

to be perfect in the case of ideal batch, semi-batch, and continuous stirred

tank reactors. The plug ﬂow reactor corresponds to the assumption of perfect

mixing in the radial direction, no mixing in the axial direction, and a ﬂat

axial velocity proﬁle. As discussed in Levenspiel (2004, pp. 283-287, 321-

334) and Fogler (1999, pp. 873-876, 893-904) a sequence of ideal reactors

can sometimes be used to model nonideal reactors.

1.4.2 Actual reactors

In addition to the conventional stirred vessels, “empty” tubular reactors,

and packed beds, there are many other types of reactors such as ﬂuidized

beds (Lee and Li, 2009), trickle beds (Wu et al., 2009), ﬂuidized catalytic

crackers (Yang et al., 2009), bubble columns (Tokumura et al., 2009), mem-

brane reactors (Rahimpour and Ghader, 2004), microchannel reactors (Wang

et al., 2009), and multifunctional reactors (Fan et al., 2009; Agar, 1999). The

references in brackets represent recent articles discussing such reactors.

1.5 Mode of operation and contacting

Reactors can be operated either in batch, semi-batch or continuous modes.

The ﬁrst two modes cause the concentrations of the species to vary with

time, whereas the latter can be operated in either a steady or unsteady

manner. Usually, startup, shutdown, and disturbances in feed ﬂow rate, etc.

lead to unsteady operation. For some systems, it may be advantageous to

deliberately operate in an unsteady manner to achieve higher selectivity or

conversion. For example, Sotowa et al. (2008) examined the eﬀect of forced

temperature cycling of a catalyst layer on propylene (C

) oxidation. They

found the forced operation led to a higher time-averaged conversion than

steady state operation, for the same rate of consumption of energy. The

Introduction 7

bombardier beetle provides an example of a natural system that relies on

forced periodic operation (Aneshansley et al., 1969).

For multiphase reactors, several modes of contacting, such as cocur-

rent, countercurrent, and cross-ﬂow are possible. Gillou et al. (2008) ex-

amined the eﬀect of introducing H

at various points along the length of

a microchannel reactor on the conversion of CO to hydrocarbons by the

Fischer-Tropsch process. Compared to the introduction of H

along with

CO at the inlet of the reactor, an increase in selectivity was obtained for

some hydrocarbons.

1.5.1 Modelling, control, and optimization

A mathematical model of the reactor permits prediction of the conversion,

selectivity (for systems with multiple reactions), ﬂow patterns and hot spots

or regions of high temperature. The eﬀect of changes in operating conditions

can also be examined. The availability of a model permits the development

of suitable control schemes to ensure product quality, and also provides a

valuable aid for the optimization of parameters or operating policies to sat-

isfy speciﬁc objective functions. For example, Altinten et al. (2008) modelled

a batch reactor used for the production of polystyrene. Using a suitable con-

trol scheme, the reactor temperature was varied with time so as to follow an

“optimum” proﬁle. This ensured that a polymer of the desired molecular

weight was obtained in the minimum possible time.

Review of background material

2.1 Representation of reactions

The following notation will be used to represent irreversible and reversible

reactions:

A + B → C irreversible

A + B  C reversible (2.1)

Let A

, i = 1, N represent N species participating in a single reaction

+ a

+ ...  a

+ ...a

where a

represents the number of moles of species A

. The reaction can be

written compactly as

i=1

= 0

where ν

is the stoichiometric coeﬃcient for A

. The usual convention is

< 0, for reactants; v

> 0, for products

For a reversible reaction, a species may be either a product or a reactant,

depending on the direction in which the reaction proceeds. In such a case,

the signs for the ν

are chosen in the usual manner, i.e. assuming that the

reaction proceeds from left to right. In case the reaction proceeds in the

opposite direction, the expression for the reaction rate will change sign and

the signs of the ν

can be left unchanged.

The above notation can be readily extended to multiple reactions.

If N species participate in M reactions, the reactions can be represented by

j=1

= 0, i = 1, M

Review of background material 9

where ν

is the stoichiometric coeﬃcient for the j

species participating in

the i

reaction.

2.2 The condition for reaction equilibrium

Consider an isolated system in which the reaction A

+ A

 A

occurs.

(An isolated system is one that does not interact with its surroundings. In

particular, there is no transfer of heat, mass, or work between the system

and the surroundings.) The second law of thermodynamics states that all

changes or processes occurring in an isolated system must satisfy

≥ 0 (2.2)

where S is the total entropy of the system and t is the time. The system is

said to be at an equilibrium state if

= 0

If we start with a binary mixture of A

and A

, A

will be produced

as the reaction proceeds. In accord with (2.2), S must either increase or

remain constant. The expected variation of the entropy S and the molar

concentration c

of A

with t is sketched in Fig. 2.1. The quantities S

and

represent the equilibrium values of S and C

, respectively.

It follows from the above discussion that S is a maximum at an

equilibrium state of an isolated system. In thermodynamics, an isolated

system is deﬁned as one that has a constant volume V and a constant

internal energy U. Hence the equilibrium state corresponds to one that

implies a maximum of S at constant U and V .

In reaction engineering, it is convenient to work with a closed system,

rather than an isolated system. (A closed system is one that does not

exchange mass with the surroundings.) We shall now derive the condition for

the equilibrium state of a closed system in terms of a suitable thermodynamic

quantities. The material below has been adapted from Denbigh (1971, p. 67-

69).

Consider a closed system in contact with a heat reservoir that is

maintained at a constant temperature T

. Treating the system and the

reservoir as an isolated compound system, the second law implies that

∆S + ∆S

≥ 0 (2.3)

where ∆S and ∆S

are the entropy changes of the system and the reservoir,

Fig. 2.1. Variation of the total entropy of the system S and the concentration of

species 1 c

with the time t in an isolated system. The quantities S

and c

represent the equilibrium values of S and c

, respectively.

respectively. If ∆Q is the heat absorbed by the system from the reservoir,

we have

∆S

= −(∆Q)/T

(2.4)

regardless of whether the heat transfer is reversible or irreversible. The ﬁrst

law of thermodynamics implies that

∆U = ∆Q − ∆W (2.5)

where ∆U is the change in internal energy of the system and ∆W is the

Review of background material 11

work done by the system on the surroundings. Equations (2.3)-(2.5) imply

that

∆S − (∆U + ∆W ) ≥ 0

or, adding and subtracting ∆(p V ), where p is the pressure of the ﬂuid

∆(U + p V − T

S) ≤ −∆W + ∆(p V ) (2.6)

Consider a special case where the initial state 1 and the ﬁnal state

2 of the system are such that (i) T

= T

≡ T , and (ii) p

= p

≡ p.

Noting that the Gibbs free energy is deﬁned by

G ≡ U + p V − T S (2.7)

(2.6) reduces to

∆G|

T,p

≤ −(∆W − p ∆V ) ≡ −∆W

(2.8)

where ∆W

is the work done by the system, excluding that due to volume

change. (For a solid phase, the work due to volume change is not given by

p ∆V . However, the ﬁnal result (2.9) is unaltered (Callen, 1985, p. 305).)

If ∆W

= 0, (2.8) reduces to

∆G|

T,p

≤ 0 (2.9)

Hence the Gibbs free energy G must either remain constant or decrease for

all changes in a closed system maintained at constant T, p, and G must be

a minimum at equilibrium.

In order to relate (2.9) to measurable quantities such as temperature,

pressure, and composition, we use the Gibbs equation. For a single-phase

system containing N species, the Gibbs equation is given by

dG = V dp −S dT +

i=1

(2.10)

where

≡



∂G

∂n



T, p, n

j6=i

(2.11)

is the chemical potential of species i, and n

is the number of moles of species

Consider a single reaction occurring in a closed system containing a

ﬂuid, and assume that there are no spatial gradients. Then the mass balances

are given by

= V ν

˙r, i = 1, N (2.12)

where V is the volume of the system and ˙r is the reaction rate for this

reaction. For ﬂuid-phase systems, the usual dimensions of ˙r are moles/unit

volume/unit time. Even though there are N equations of the form (2.12),

there is only one independent reaction. Hence, as suggested by de Donder

(1922) (cited in Laidler, 2007, p. 7), all the {n

} can be expressed in terms

of a variable ξ, called the extent of reaction. Let

= n

+ ν

ξ, i = 1, N (2.13)

where n

is the initial (i.e. at time t = 0) number of moles of i. Equations

(2.13) and (2.12) imply that

dξ

= V ˙r (2.14)

with the initial condition ξ(0) = 0.

Expressing the mole numbers in terms of the extent of reaction,

(2.10) can be written as

dG = V dp −S dT +

i=1

dξ (2.15)

As G must be a minimum at an equilibrium state of a closed system main-

tained at constant (T, p), we must have



∂G

∂ξ



T,p

= 0

i=1

= 0 (2.16)

and



∂

∂ξ



T,p

> 0

i=1



∂µ

∂ξ



T,p

> 0 (2.17)

Equation (2.16) represents the condition for reaction equilibrium. It

Review of background material 13

holds even if the system contains more than one phase, and the reaction

involves components in diﬀerent phases (Denbigh, 1971, p. 140).

The aﬃnity of the reaction is deﬁned by

A ≡ −

i=1

(2.18)

Equation (2.15) implies that



∂G

∂ξ



T,p

= −

A (2.19)

Hence if

A > 0, G decreases as the extent of reaction ξ increases, and the

reaction proceeds from left to right.

2.3 Models for the chemical potential

Equation (2.15) can be used to compute the equilibrium composition for

a single reaction in a closed system, provided a model is available for the

dependence of the chemical potentials {µ

} on temperature T , pressure p,

and composition. Some models are discussed brieﬂy below. For more details,

the reader is referred to Denbigh (1971, pp. 111-115, 125-126, 249, 270-271)

and Smith et al. (2001, pp. 384, 390, 577).

(a) The perfect gas mixture

The perfect gas mixture is deﬁned as one for which (Denbigh, 1971,

p. 115)

(T, p, y) ≡ µ

(T, p

) + R T ln



p y



, i = 1, N (2.20)

where y is the vector of N − 1 independent mole fractions y

, i = 1, N − 1,

is the chemical potential of pure i at a temperature T and a reference

pressure p

, R is the gas constant, y

is the mole fraction of species i, and

p is the total pressure of the mixture. The use of (2.20) along with suitable

thermodynamic relations leads to the following familiar results for a perfect

gas mixture

p V = n R T ; p

≡ p y

= n

R T/V, i = 1, N (2.21)

where V is the volume occupied by the mixture, p

is the partial pressure of

i, n

is the number of moles of species i, and

n ≡

i=1

is the total number of moles.

(b) The ideal solution

The ideal solution is deﬁned by (Denbigh, 1971, p. 249)

(T, p, y) ≡ µ

(T, p) + R T ln(y

), i = 1, N (2.22)

where µ

is the chemical potential of pure i at (T, p). Equation (2.22) can

be used for ideal gaseous, liquid, or solid solution.

To account for non-ideal behaviour, (2.22) is modiﬁed by introducing

a variable γ

, called the activity coeﬃcient, such that (Denbigh, 1971, p. 270)

(T, p, y) ≡ µ

(T, p) + R T ln(γ

), i = 1, N (2.23)

Note that µ

is independent of the composition, and the composition de-

pendence of µ

is accounted for solely by the term γ

. The value of µ

can be ﬁxed by choosing a convention for γ

. If all the species forming the

solution remain in the same phase as the solution in their pure states at

(T, p), the usual convention is

→ 1 as y

→ 1 (2.24)

In this case, µ

is the chemical potential of pure i at (T, p).

Consider a liquid solution, and let i = 1, m denote species that re-

main as liquids in their pure states at (T, p). The other species (i = m+1, N)

are either gases or solids in their pure states at (T, p). Convention (2.24)

applies to the ﬁrst m species, and hence

→ 1 as y

→ 1, i = 1, m (2.25)

For the other species, the usual convention is

→ 1 as y

→ 0, i = m + 1, N (2.26)

For i = m + 1, N , µ

is the chemical potential of pure i in a hypothetical

liquid state. It is a hypothetical state as pure i will not, by deﬁnition, be in

a liquid state at (T, p).

Equation (2.23) can be used for gaseous, liquid, or solid solutions.

Review of background material 15

Additional details regarding the determination and use of activity coeﬃ-

cients may be found in Denbigh (1971, pp. 281-288), Prausnitz et al. (1999,

pp. 222-236), and Sandler (2006, pp. 419-461).

Remarks

1. The activity of a species i is deﬁned by Denbigh (1971, p. 287)

i(y)

≡ γ

(2.27)

where the subscript y indicates that mole fractions are used as a measure of

the composition of the mixture, and γ

is the activity coeﬃcient based on

mole fractions. For some applications, it is convenient to replace y

in (2.27)

by some other measures of the composition, such as the molar concentration

or the molality ˆc

. Here ˆc

is the number of moles of i per kg of the solvent.

The molality is often used for electrolyte solutions (Prausnitz et al., 1999,

p. 218).

Thus we have

i(c)

≡

i(c)

i(ˆc)

≡

i(ˆc)

ˆc

(2.28)

As the activity is a dimensionless quantity, (2.28) involve a reference compo-

sition characterized by c

or ˆc

. For ions and molecules dissolved in water,

the usual reference composition is (Sawyer et al., 2003, p. 31, Sandler, 2006,

p. 712)

= 1 M (i.e. 1 mol/L), ˆc

= 1 mol/kg of water (2.29)

For the solvent, and for pure liquids and solids in equilibrium with an aque-

ous solution, the usual reference state is the concentration or molality of the

pure component

= c

i, pure

; ˆc

= ˆc

i, pure

(2.30)

The conventions adopted for γ

i(c)

and γ

i(ˆc)

are similar to those used for γ

For example, consider a liquid mixture and a solute i that is not a liquid at

the same (T, p) as the solution. Then the convention is that

i(c)

→ 1 as c

→ 0 (2.31)

In terms of the activities, (2.23) can be rewritten as

≡ µ

(T, p) + R T ln a

i(y)

(2.32)

≡ µ

i0(c)

(T, p) + R T ln a

i(c)

(2.33)

≡ µ

i0(ˆc)

(T, p) + R T ln a

i(ˆc)

(2.34)

2. For gas mixtures, it is common practice to use fugacities instead of

activity coeﬃcients. Thus (2.23) can be rewritten as Denbigh (1971, p. 125)

(T, p, y) ≡ µ

(T, p

) + R T ln





, i = 1, N (2.35)

where f

is the fugacity of species i and µ

and f

are the chemical potential

and fugacity, respectively, of pure i at a temperature T and a reference

pressure p

. A common choice for f

is f

(T, p

) = 1 atm, in which case

is the pressure for which the fugacity of pure i is 1 atm. As noted by

Denbigh (1971, p. 123), if p ≤ 1 atm, f

≈ p

for most gases. Thus the

choice f

= 1 atm implies that p

≈ 1 atm.

As in the case of (2.23), (2.35) can be used for gaseous, liquid, and

solid solutions.

3. In accord with the experimental observation that the mixture should

behave like a perfect gas mixture in the limit p → 0, (2.20) and (2.35) imply

that

lim

p→0

p y

(2.36)

Thus the fugacity is proportional to the partial pressure p

≡ p y

at low

pressures.

4. For an ideal solution, (2.22) and (2.35) imply that

R T ln





= µ

(T, p) −µ

(T, p

) (2.37)

As the right hand side is independent of the composition, its value remains

unchanged in the limit y

→ 1. Hence (2.37) implies that

= lim

→1

i,pure

= f

i,pure

(2.38)

Equation (2.38) is called the Lewis and Randall rule.

Review of background material 17

2.4 The equilibrium constant and the equilibrium composition

For a single reaction in a single phase system, the condition for reaction

equilibrium is (see (2.17))

i=1

= 0 (2.39)

For a perfect gas mixture, (2.39) and (2.20) imply that

i=1

= −R T ln

i=1





(2.40)

where p

is the equilibrium value of the partial pressure p

of species i. The

equilibrium constant is deﬁned by

≡

i=1





i=1



p y



(2.41)

where y

is equilibrium mole fraction of species i. Introducing the standard

Gibbs free energy change for the reaction

∆G

≡

i=1

(T, p

)

) ν

(2.42)

(2.40) can be rewritten as

∆G

−R T ln K

(2.43)

Equations (2.42) and (??) imply that K

is independent of the pressure p

and K

= K

(T ). This is true only for a perfect gas mixture. For a non-ideal

gas mixture, K

is still deﬁned by (2.41), but (2.43) is not valid.

The equilibrium composition in a closed system at constant (T, p)

can be calculated as follows. For ease of discussion, consider a perfect gas

mixture. Using tables of thermodynamic properties, values of ∆G

can be

calculated for most reactions. The value of K

then follows from (2.43). As

= n

(2.44)

where n

and n

are the number of moles of species i and the total number

of moles, respectively, at equilibrium, (2.41) can be rewritten as

i=1



p n



(2.45)

or using (2.13)

i=1



 

+ ν

+ (∆ν) ξ



(2.46)

Here ξ

is the extent of reaction at an equilibrium state, and n

≡

i=1

and ∆ν ≡

i=1

are the total number of moles at the initial state and

the change in the number of moles accompanying the reaction, respectively.

Equation (2.46) represents a nonlinear equation for the extent of reaction

. Except in simple cases, the equation must be solved iteratively.

2.5 The eﬀect of temperature on the equilibrium composition of

a perfect gas mixture

Taking the logarithm of (2.45) and using (2.43), we obtain

−

∆G

R T

= ln K

i=1



p n



(2.47)

Diﬀerentiating (2.47) with respect to T , and using (2.13), we obtain



−

∆G

R T



(ln K

) =

i=1



dξ

−

dξ



dξ

(2.48)

Substituting for ∆G

from (2.42), we obtain



−

∆G

R T



== −

i=1





∆H

R T

(2.49)

where

∆H

≡

i=1

(2.50)

is the standard enthaply change for the reaction and h

is the molar enthalpy

of pure i at (T, p

). Hence (2.48) reduces to van’t Hoﬀ’s equation

(ln K

) =

∆H

R T

(2.51)

As n

= n

(ξ), (2.13) and (2.48) imply that

(ln K

) =

i=1

−

(∆ν)

dξ

(2.52)

Review of background material 19

Using (2.52), (2.51) can be rewritten as

∆H

R T

i=1

−

(∆ν)

dξ

(2.53)

The Schwarz inequality (see, for example, Arfken and Weber, 2001, p. 607)

can be used to show that the quantity in square brackets on the right hand

side of (2.53) is always positive. Hence dξ

/dT has the same sign as ∆H

An exothermic reaction is deﬁned as one for which ∆H

< 0. In this

case, dξ

/dT < 0, and hence the equilibrium extent of reaction decreases as

T increases. If species i is a reactant, its conversion X

, deﬁned by

≡

−n

(2.54)

is directly proportional to the extent of reaction ξ. Hence X

also decreases

as T increases. Conversely, for an endothermic reaction, the equilibrium

conversion increases as T increases.

Remark Consider a chemical reaction in closed system, whose volume V

changes suitably to maintain constant (T, p). If the state of the system

changes from state 1 to state 2 as the reaction proceeds, the enthalpy change

of the system is given by

∆H = ∆U + ∆(p V ) = ∆U + p ∆V

or, using the ﬁrst law of thermodynamics, and assuming that work is asso-

ciated only with volume change

∆H = ∆Q (2.55)

where ∆Q is the heat absorbed by the system. Thus ∆H

is the heat

absorbed by the system when the reactants are taken in stoichiometric pro-

portions, with each reactant at (T, p

), and are completely converted to

products, with each product at (T, p

) (Denbigh, 1971, p. 142).

2.6 The eﬀect of pressure on the equilibrium composition of a

perfect gas mixture

Diﬀerentiating (2.47) with respect to p, we obtain

∂

∂p

(ln K

) = 0 =

i=1

−

(∆ν)

∂ξ

∂p

∆ν

∂ξ

∂p

= −

∆ν

i=1

−

(∆ν)

(2.56)

If the reaction is accompanied by an increase in the number of moles, i.e.,

∆ν > 0, (2.56) implies that ∂ξ

/∂p < 0. Hence the equilibrium conversion

decreases as p increases. Conversely, if ∆ν > 0, the conversion increases as

p increases.

Remark Equations (2.53) and (2.56) are quantitative expressions of Le Chate-

lier’s principle (Atkins and de Paula, 2002, p. 234): “A system at equilib-

rium, when subjected to a disturbance, responds in a way that tends to

minimize the eﬀect of the disturbance.”

2.7 Feasibility of reactions

As mentioned in section 2.2, a reaction proceeds from left to right if

A = −

i=1

> 0

The computation of

A requires a knowledge of T, p, and the composition.

A rough idea of the direction in which the reaction is likely to occur may

be obtained by calculating the standard free energy change for the reaction

∆G

i=1

(T, p

) ν

. Large negative values of ∆G

imply that the

reaction is promising, i.e. it is likely to proceed from left to right. On

the other hand, large positive values of ∆G

imply that the reaction is not

promising, i.e. it is likely to proceed from right to left.

Equation (2.43) implies that if ∆G

 0, K

is  1. To understand

the eﬀect of K

on the conversion, it is helpful to rewrite (2.46) as

ln K

= q ≡

i=1

[ln(n

+ ν

) −ln(n

+ (∆ν) ξ

)] + (∆ν) ln(p/p

) (2.57)

At a ﬁxed value of p, if K

increases, (2.57) implies that ξ

increases, as

∂q/∂ξ

> 0. Conversely, if K

 1, we may expect ξ

to be small. However,

if the initial mixture contains only the reactants, the equilibrium value of ξ

will be small but non-zero even if ∆G

 0.

Dodge (1944) has listed the following thumb rules: (a) if ∆G

(298

K, 1 atm) < 0, the reaction is promising, (b) if 0 < ∆G

< 40 kJ/mol, a

more detailed examination is warranted, and (c) if ∆G

> 40 kJ/mol, the

Review of background material 21

reaction is very unlikely. For example, consider the synthesis of NO from

and O

. For this reaction, ∆G

= 86.6 kJ/mol of NO, whereas for the

synthesis of NH

by the reaction

(g) +

(g)  NH

(g) (2.58)

∆G

= −17 kJ/mol of NH

. Hence the thumb rules suggest that very little

NO will be formed.

2.8 Reaction equilibrium in an imperfect gas mixture

Using the model (2.35) for the chemical potential and the condition for

reaction equilibrium (2.16), we obtain

i=1

= 0

R T ln K

= −∆G

(T, p

) (2.59)

where

≡

i=1





(2.60)

is the equilibrium constant based on fugacities and f

is the equilibrium

value of the fugacity of species i.

For an imperfect gas mixture, it is K

and not K

that is independent

of the pressure p. This point is illustrated by the data of Larson and Dodge

(cited in Denbigh, 1971, p. 152). As p increases from 10 atm to 300 atm,

increases by 34 % of the smaller value, whereas K

increases by only 1.5

%. The small variation of K

occurs because of the errors introduced by

using the Lewis and Randall rule (2.38) to estimate the fugacities {f

}. For

p > 300 atm, the variation of K

is more pronounced.

2.9 Reaction equilibrium in a liquid mixture

If (2.23) is used as a model for the chemical potential, (2.39)implies that

i=1

= −R T ln K

(2.61)

where

≡

i=1

; K

≡

i=1

(2.62)

Equation (2.61) can be used to calculate the equilibrium composition

provided the values of the {µ

} are known. For solutes that are not liquids

at the same (T, p) as the solution, (2.26), (2.27), and (2.32) imply that µ

the chemical potential of pure i in a hypothetical liquid state. However, the

values tabulated for such solutes often correspond to a hypothetical ideal

solution of unit molality. In such cases, activities deﬁned on the molality

scale may be used. Equations (2.34) and (2.29) imply that µ

i0(ˆc)

(T, p) is the

chemical potential of i in a hypothetical ideal solution of unit molality. Fur-

ther, as noted by Denbigh (1971, p. 294), the eﬀect of pressure on the liquid

and solid phases may usally be neglected, unless it is very high compared to

the standard pressure p

= 1 atm. Hence

i0(ˆc)

(T, p) ≈ µ

i0(ˆc)

(T, p

) (2.63)

and the tabulated value of the standard Gibbs free energy of formation of i

may be used for µ

i0(ˆc)

(T, p).

2.10 Reaction equilibrium in systems involving multiple phases

Consider the thermal decomposition of caclium carbonate (Denbigh, 1971,

p. 156)

CaCO

(s)  CaO (s) + CO

(g) (2.64)

The partial pressure of CO

, p

can be related to the standard Gibbs free

energy change for the reaction as follows.

The condition for reaction equilibrium is

−µ

CaCO

(s) + µ

CaO

(s) + µ

(g) = 0 (2.65)

Assuming that both CaCO

(s) and CaO (s) have negligible vapour pres-

sures, only CO

will be present in the vapour phase. Treating it as a perfect

gas, we have

(g) = µ

0,CO

(T, p

) + R T ln(p

) (2.66)

Using (2.66), (2.67) reduces to

−µ

CaCO

(s) + µ

CaO

(s) + µ

0,CO

(g) = −R T ln(p

) (2.67)

Review of background material 23

To proceed further, we assume that CaCO

(s) and CaO (s) form

immiscible solid phases, i.e. a phase contains either pure CaCO

or pure

CaO. (The case where two solids are miscible and form a mixture or solid

solution is discussed in Sandler (2006, pp. 679-687).) Then µ

CaCO

and µ

CaO

depend only on (T, p). For a pure phase, dµ = −s dT + v dp, where s and v

are the molar entropy and the molar volume, respectively, of the material.

Considering CaO and assuming that v

CaO

is approximately independent of

p, we obtain

CaO

(T, p) = µ

CaO

(T, p

) + v

CaO

(p −p

) (2.68)

Using a similar expression for µ

CaCO

0,CaO (s)

(T, p

) + µ

0,CO

(g)

(T, p

) − µ

0,CaCO

(s)

(T, p

)

+(v

CaO (s)

− v

CaCO

(s)

) (p − p

) = −R T ln(p

) (2.69)

∆G

= µ

0,CaO (s)

(T, p

) + µ

0,CO

(g)

(T, p

) − µ

0,CaCO

(s)

(T, p

) (2.70)

(2.69) reduces to

∆G

+ (v

CaO (s)

− v

CaCO

(s)

) (p − p

) = −R T ln(p

) (2.71)

The second term on the left hand side of (2.71) is usually negligible

compared to the ﬁrst term. For example, ∆G

(298K, 1atm) = 135 kJ/mol,

whereas (v

CaO (s)

− v

CaCO

(s)

) (p − p

) ≈ 0.2 kJ/mol for p − p

= 100 atm.

Hence (2.71) may be approximated as

∆G

≈ −R T ln(p

) (2.72)

By analogy with (2.41), we may deﬁne a partial equilibrium constant

≡

(2.73)

Equation (2.72) permits evaluation of the pressure of CO

(g) that is in

equilibrium with CaCO

(s) and CaO (s). As noted by Denbigh (1971,

p. 158), if p

> K

, CaO will be converted to CaCO

Another example involving gaseous and liquid phases is given by the

formation of urea by the reaction (Denbigh, 1971, p. 294)

(g) + 2 NH

 CO(NH

)

(aq) + H

O (2.74)

Numbering the species from left to right in the above equation as 1-4, the

values reported in the literature for the standard free energies of formation

are, in kJ/mol

= −394.4, µ

= −16.64, µ

30(ˆc)

= −203.8, µ

40(y)

= −237.2 (2.75)

Hence the standard free energy change for the reaction (2.75) is

∆G

≡ µ

40(y)

+ µ

30(ˆc)

− 2 µ

− µ

= −13.32 kJ/mol (2.76)

and the condition for reaction equilibrium is

∆G

= −R T ln



3(ˆc)

(ˆc

/ˆc

)



(2.77)

2.11 Systems with multiple reactions

The discussion will be conﬁned to a single phase system containing N species

participating in M reactions. For example. the following reactions occur

during the production of formaldehyde (HCHO) from methanol (CH

OH):

OH  HCHO + H

OH +

 HCHO + H

HCHO  CO + H

HCHO +

 CO + H

CO +

 CO

(2.78)

We could also add the reaction

OH  CO + 2 H

but it is not an independent reaction, as it can be obtained by adding the

ﬁrst and the third reactions in (2.78).

It is useful to calculate the number of stoichiometrically independent

reactions N

. The value of N

permits examination of a system involving

reactions, rather than the original system of M ≥ N

reactions. A

method for calculating N

is explained below.

Consider the system of reactions

j=1

= 0, i = 1, M (2.79)

Review of background material 25

where ν

is the stoichiometric coeﬃcient for the j

species in the i

reac-

tion. Let the stoichiometric matrix [ν ] be deﬁned by

[ν ] = [ν

] (2.80)

Thus the element in the i

row and j

column of [ν ] is ν

. The number

of stoichiometrically independent reactions is given by

= rank([ν ]) (2.81)

Here the rank of [ ν ] is the maximum number of linearly independent rows

or columns of [ ν ] (Kreyszig, 1993, p. 356-357); it can be found by Gaussian

elimination or Gauss elimination, as discussed in Aris (1999, p. 13) and in

books on numerical analysis (see, for example, Gerald and Wheatley, 1994,

pp. 113-115). The following example is taken from Aris (1999, pp. 13-14).

Consider the reactions involved in the formation of hydrogen bromide

(HBr):

→ 2 Br

Br + H

→ HBr + H

H + Br

→ HBr + Br

H + HBr → H

+ Br

2 Br → Br

(2.82)

The stoichiometric matrix is given by

ν =







−1 2 0 0 0

0 −1 −1 1 1

−1 1 0 1 −1

0 1 1 −1 −1

1 −2 0 0 0







The procedure is as follows.

(i) Rearrange the rows if necessary, so that the pivot ν

6= 0. In the

present example, rearrangement is not required.

(ii) Divide the ﬁrst row by ν

so that the ﬁrst row is now given by

= [1 − 2 0 0 0].

(iii) Using row operations, i.e. forming linear combinations of the other

rows with the ﬁrst row, make all the elements in the ﬁrst column

(except ν

) equal to zero. Thus we obtain







1 −2 0 0 0

0 −1 −1 1 1

0 −1 0 −1 1

0 1 1 −1 −1

0 0 0 0 0







(iv) Repeat steps 1-3, starting with ν

as the pivot, and reducing ν

, i =

3, N to zero. The result is

ν =







1 −2 0 0 0

0 1 1 −1 −1

0 0 1 −2 0

0 0 0 0 0







As all the elements in the 3rd and 4th rows are zero, the process can

be stopped at this stage. The rank of ν is 3, the number of non-zero rows.

Hence (2.81) implies that the number of stoichiometrically independent re-

actions is N

= 3 in the present example.

2.12 Degrees of freedom for reactive systems

Suppose there are N species, P phases, and N

independent chemical reac-

tions. At an equilibrium state, the following conditions hold (Denbigh, 1971,

p. 187):

= T

; p

= p

; µ

= µ

, j = 2, P ; i = 1, N (2.83)

Here T

and p

are the temperature and pressure, respectively, of phase j

and µ

is the chemical potential of species i in phase j. In addition, there

are N

conditions of reaction equilibrium, given by

i=1

= 0; l = 1, N

(2.84)

Equations (2.83) and (2.84) constrain the values of the independent

intensive variables T

, p

. i = 1, N and the mole fractions y

, j = 1, P, i =

1, N − 1. Here y

is the mole fraction of species i in phase j. Hence the

number of degrees of freedom F , i.e. the excess of the number of variables

over the number of constraints, is given by

F = (N − N

) − P + 2 (2.85)

Review of background material 27

In the absence of chemical reactions, N

= 0. Hence (2.85) reduces to the

familiar phase rule.

In a solution containing electrolytes, the requirement of electrical

neutrality leads to an equation relating the concentrations of the ions. Hence

the value of F is reduced by unity, as discussed in Denbigh (1971, p. 188).

2.13 Notation for reaction kinetics

Let ˙r

denote the molar reaction rate for species i, i.e. the moles of i produced

per unit volume per unit time. For a single reaction, it is convenient to

introduce an intrinsic molar reaction rate

β, such that

˙r

= ν

β (2.86)

Similarly, for multiple reactions we set

˙r

j=1

(2.87)

where ν

is the stoichiometric coeﬃcient for species i in reaction j,

the intrinsic molar reaction rate for reaction j , and N

is the number of

stoichiometrically independent reactions.

2.14 Elementary reactions and reaction mechanisms

The reaction

+ A

→ A

(2.88)

is said to be an elementary reaction if a molecule of A

is formed by direct

interaction or collision between one molecule of A

and one molecule of

. The molecularity of an elementary reaction is the number of reactant

molecules participating in the reaction. Thus, for an elementary reaction of

the form (2.88), the molecularity is 2.

If the molecules of A

and A

move independently, the probability of

ﬁnding one molecule of A

and one molecule of A

at a given (macroscopic)

location is proportional to c

, where c

is the molar concentration of A

Thus it may be expected that

β ∝ c

(2.89)

for the reaction (2.88). This assumption may break down if there are strong

long-range interactions between A

and A

. Even if such interactions are

ignored, (2.89) does not guarantee that (2.88) is an elementary reaction. For

example, consider the gas-phase reaction

(g) + I

(g)  2 HI(g) (2.90)

It was found that

β ∝ c

(2.91)

for the forward reaction and

β ∝ c

(2.92)

for the reverse reaction. Hence it was assumed till the early 1930’s that

(2.90) represented an elementary reaction. Subsequent work showed that

other mechanisms were also involved, depending on the temperature and

other conditions (Laidler, 2007, pp. 298-300). For example, iodine atoms

may be involved in some cases. Thus the reaction rate for (2.90) is consistent

with the assumption that it is elementary reaction, even though it is not.

A non-elementary reaction is one which proceeds by a sequence of

elementary reactions. The sequence is called the reaction mechanism. Given

a mechanism, the rate expression can be derived for a non-elementary reac-

tion. The success of the postulated mechanism depends partly on the degree

of agreement between the predicted and observed reaction rates. In some

cases, as for the reaction (2.90), several diﬀerent mechanisms may lead to

the same rate expression. More information is then required to discriminate

between rival mechanisms.

An example of a non-elementary reaction is provided by the reaction

(g) + Br

(g)  2 HBr (g) (2.93)

Bodenstein and Lind (1907) (cited in Laidler, 2007, p. 291 found that

β =

k c

√

1 + c

HBr

/(m c

)

(2.94)

where k and m are constants. The mechanism for this reaction was suggested

almost simultaneously by Christiansen (1919), Herzfeld (1919) and Polanyi

(1920) (all cited in Laidler, 2007, p. 291). The steps involved are

→ 2 Br initiation

Br + H

→ HBr + H propagation

H + Br

→ HBr + Br ”

H + HBr → H

+ Br ”

Review of background material 29

Br + Br → Br

termination (2.95)

Using suitable assumptions, the mechanism (2.95) leads to a rate expres-

sion that is identical in form to the observed rate expression (2.94). The

initiation, propagation, and termination steps are characteristic of chain re-

actions. Equation (2.95) involves the free radicals H and Br. (A free radical

is an atom or group of atoms having an unpaired electron (Morrison and

Boyd, 2002, p. 47).)

2.15 Rate expressions for homogeneous irreversible reactions

No reaction is truly irreversible, but under certain operating conditions, the

concentrations of the reactants may be negligible at an equilibrium state. In

such cases, the reaction is treated as an irreversible reaction. Some examples

of reactions which must be treated as reversible reactions under industrial

operating conditions are SO

oxidation and the synthesis of NH

For a homogeneous ﬂuid-phase reaction involving N species, the in-

trinsic reaction rate has the form

β =

β(T, p, x) (2.96)

where T is the absolute temperature, p is the total pressure, and x is the vec-

tor of independent mole fractions with components x

, i = 1, N −1. Equation

(2.96) can be rewritten as

β =

β(T, p, c

) (2.97)

β =

β(T, c) (2.98)

where c

is a vector of molar concentrations with components c

, i = 1, N −

1 and c is the vector with components c

, i = 1, N. As an equation of

state f (T, p, c) = 0 relates T , p, and c, only N + 1 of these variables are

independent.

2.15.1 Special cases

(a) Power law kinetics

The intrinsic reaction rate is given by

β = k(T )

i=1

(2.99)

where q

is the order of the reaction with respect to the ith species. The

overall order of the reaction is given by

q =

i=1

(2.100)

The function k(T ) is independent of the concentration, and hence is called

the rate constant.

As an example, consider the thermal cracking of propane (C

)

in the gas phase. A kinetic model proposed by Sundaram and Froment

(1977) involves nine reactions, three of which are listed below along with

rate expressions for the intrinsic reaction rates β

→ C

+ CH

;

β = k

(2.101)

 C

+ H

;

β = k

− k

(2.102)

2 C

→ 3 C

;

β = k

(2.103)

The above reactions are approximations to a more realistic scheme involv-

ing free radicals. Equations (2.102) and (2.103) are ﬁrst order irreversible

reactions, whereas (2.103) is a reversible reaction of mixed order. At 800

◦

C, the values of the rate constants in s

−1

are k

= 2.34 , k

= 2.12, and

= 0.72.

In (2.99), if q

< 0 for a product, the reaction is said to be product-

inhibited. This terminology is not restricted to power law kinetics, but is

used whenever an increase in the product concentration decreases the reac-

tion rate. For example, (2.94) shows that the forward reaction is inhibited

by HBr. If q

= 0 for a reactant A

, the reaction is said to be zero order in A

This condition is likely to hold when A

is present in a large excess compared

to the other reactants, but it will break down at low concentrations.

(b) Mass action kinetics

The reaction

+ q

→ q

(2.104)

is said to follow mass action kinetics if the intrinsic reaction rate is given by

(Guldberg and Waage, 1864, cited in Laidler, 2007, p. 2)

β = k(T ) c

(2.105)

These kinetics are often used to describe reactions catalyzed by en-

zymes. (An enzyme is usually a protein of high molecular weight, between

Review of background material 31

15,000 and several million daltons (Shuler and Kargi, 2004, p. 57). A dalton

is the mass of a H atom, i.e. 1.66 × 10

−27

kg.) For example, the enzyme

urease catalyzes the hydrolysis of urea (CO(NH

)

) (Laidler, 2007, p. 399)

CO(NH

)

+ H

O → CO

+ 2 NH

(2.106)

For a single reactant or substrate S (in addition to water) that is converted

to a product or products P by an enzyme E, the reaction is

→ P (2.107)

Henri (1902) (cited in Kooijman, 2001) and Michaelis and Menten (1913)

postulated the following mechanism (Shuler and Kargi, 2004, p.60)

E + S  ES → E + P (2.108)

where ES denotes an enzyme-substrate complex. Assuming that the re-

versible reaction was at equilibrium, Henri (1902) and Michaelis and Menten

(1913) showed that for an ideal batch reactor, the rate of production of the

product was given by

k c

+ c

(2.109)

where c

= c

+ c

is the initial concentration of the enzyme. Equation

(2.109) represents Michaelis-Menten kinetics, but it is not clear why Henri’s

name is usually not included.

Rate expressions similar to (2.109) are also used when gaseous re-

actants are catalyzed by a solid catalysts. These are discussed in section

2.18.5.

2.16 Rate expressions for homogeneous reversible reactions

Consider the reaction

+ m

 m

+ m

(2.110)

If (2.110) follows power law kinetics, the intrinsic reaction rate is given by

β = k

(T )

− k

(T )

(2.111)

where k

and k

are the rate constants for the forward and backward re-

actions and the {q

} and {q

} are constants. If (2.110) follows mass action

kinetics, the reaction rate is given by

β = k

(T ) c

− k

(T ) c

(2.112)

2.17 Thermodynamic consistency of the rate expressions for

reversible reactions

For the reaction (2.110), the condition for reaction equilibrium in a closed

system at constant (T, p) is

+ m

− m

= 0 (2.113)

Considering a perfect gas mixture, (2.20) and (2.113) imply that

(T ) ≡



˜p





˜c

 

∗

R T



∆m

(2.114)

where

˜p

≡ p

; ˜c

≡ c

∗

; ∆m ≡ m

+ m

− m

(2.115)

and c

are the equilibrium values of the partial pressure and molar

concentration, respectively, of species i, and p

and c

∗

are the reference

pressure and the reference concentration, respectively.

The equilibrium constant based on concentrations K

is deﬁned by

≡



˜c



(2.116)

or for a general reaction by

≡

i=1

˜c

(2.117)

Note that K

is dimensionless. In some books (see, for example, Schmidt,

2005, p. 35), K

appears to be a dimensional quantity, but is not, as c

∗

has

been chosen as unity in the same units as the {c

}. Equations (2.116) and

(2.117) imply that

= K



∗

R T



∆m

(2.118)

Hence K

= K

(T ).

Review of background material 33

If the reaction follows power law kinetics, (2.111) holds. As

β = 0

at equilibrium, we have

i=1

(2.119)

For the reaction (2.110), (2.119) reduces to

= ˜c

−q

˜c

−q

˜c

−q

˜c

−q

∆q

∗

(2.120)

where

∆q ≡

i=1

− q

) (2.121)

Equation (2.120) may be compared with (2.116), which can be rewrit-

ten as

= ˜c

˜c

−m

˜c

−m

(2.122)

As noted by Denbigh (1971, pp. 444-445), both k

and K

are functions of

the temperature T. Hence the right hand side of (2.120) must be a function

of the right hand side of (2.122), i.e.

˜c

−q

˜c

−q

˜c

−q

˜c

−q

∆q

∗

= f (˜c

˜c

−m

˜c

−m

) (2.123)

For a batch reactor, the equilibrium concentrations {c

} can be var-

ied by varying the initial composition of the reaction mixture. Hence (2.123)

must hold for arbitrary values of the {c

}. It can be shown that (2.123)

can be satisﬁed by requiring that f(x) = x

, where n is a positive constant

(Blum and Luus, 1964). Hence

− q

−q

− q

−m

− q

−m

= n (2.124)

If (2.124) holds, (2.116) and (2.120) imply that

= K

∆q

∗

(2.125)

For a perfect gas mixture involving a reaction following power law

kinetics, (2.124) ensures that the rate expression is thermodynamically con-

sistent. For example, consider the production of phosgene (COCl

) by the

reaction

CO (g) + Cl

(g)  COCl

(g) (2.126)

The reaction rate is given by (Denbigh and Turner, 1972, p. 21)

β = k

3/2

− k

COCl

1/2

(2.127)

Equation (2.127) implies that

COCl

1/2

3/2

˜c

COCl

˜c

−1

∗

= K

−1

∗

(2.128)

Hence (2.127) is thermodynamically consistent.

2.18 Theories for the rates of elementary reactions

The temperature dependence of the rate constant k is usually ﬁtted by the

Arrhenius equation (Arrhenius, 1899) (cited in Laidler, 2007, p. 42)

k = Ae

−

R T

(2.129)

where the pre-exponential factor A and the activation energy E are treated

as constants and T is the absolute temperature. van’t Hoﬀ (1884) (cited in

Laidler, 2007, p. 42) had proposed this equation earlier. He also considered

a more general case where E = B + D T

, and B and D are constants.

Kooij (1893) and Harcourt and Esson (1895) (cited in Laidler, 2007, p. 42)

suggested equations of the forms

k = A T

−

R T

(2.130)

k = A T

(2.131)

respectively. As noted by Laidler (2007, p. 46), current practice is to use

the Arrhenius equation where it is applicable. If a plot of ln k versus 1/T

shows a signiﬁcant curvature, (2.130) is the preferred choice. Transition

state theory provides some justiﬁcation for the use of this equation. On the

other hand, there is no theoretical basis for the use of (2.131), even in cases

where it ﬁts the data well.

Theories for estimating the pre-exponential factor are discussed brieﬂy

below.

Review of background material 35

2.18.1 The collision theory

This theory was proposed by Trautz (1916) and Lewis (1918) for gas phase

reactions. It is based on the kinetic theory of dilute gases. Consider an

elementary reaction of the form

+ A

→ products (2.132)

The reaction rate, i.e the number of molecules of A

consumed per unit

volume per unit time is assumed to be equal to the number of collisions per

unit volume per unit time between molecules of A

and A

. The latter may

be estimated as follows.

Using the Maxwell-Boltzmann velocity distribution for each species,

the mean relative velocity between molecules of A

and A

is given by (Frost

and Pearson, 1961, p. 60)

u =

8 k

π µ

(2.133)

where

≡

+ m

(2.134)

is the reduced mass of two molecules of masses m

and m

. As the molecules

are assumed to be rigid spheres of diameters d

and d

, the centres of

molecules of A

must lie in the volume π d

u∆t if a molecule of A

to collide with them in a time interval ∆t. Here

≡

+ d

(2.135)

is the mean diameter for molecules of A

and A

. If n

is the number of

molecules of A

per unit volume, the number of collisions per unit time

between a molecule of A

and molecules of A

= π d

u n

(2.136)

As there are n

molecules of A

per unit volume, the number of collisions

per unit volume per unit time between molecules of A

and A

is given by

Γ = π d

u n

(2.137)

or, in terms of the molar concentrations c

Γ = N

π d

u c

(2.138)

where N

= 6.022 × 10

molecules/mol is the Avogadro number.

Substituting for u from (2.133) and assuming that each collision

results in a reaction, i.e. the conversion of one molecule of A

and one

molecule of A

to products, the molar reaction rate is given by

β = N

8 π k

(2.139)

Consider gases at normal temperature and pressure (273 K, 1 atm), with

= c

= p/(R T ), where R is the gas constant. The reaction rate calculated

using (2.139) is of the order of 10

mol/m

-s (Frost and Pearson, 1961,

p. 61), which is several orders of magnitude higher than the observed rate

for many reactions.

The discrepancy can be reduced signiﬁcantly by assuming that only

“energetic” collisions result in a chemical reaction. An energetic collision is

one for which the kinetic energy associated with the relative velocity along

the line of centres exceeds a threshold value or activation energy E (Frost and

Pearson, 1961, p. 65). It can be shown (Frost and Pearson, 1961, pp. 65-67)

that the fraction of energetic collisions is exp(−E/(R T )), and hence (2.139)

may be replaced by

β = N

8 π k

−

R T

(2.140)

Thus the rate constant for (2.132) is given by

k = N

8 π k

−

R T

(2.141)

Comparison with the Arrhenius equation (2.129) shows that the pre-exponential

factor is now a weak function of the temperature T .

For a reaction of the form

2 A

→ products (2.142)

µ = m

/2 and the right hand side of (2.143) must be divided by a factor of

2 to avoid counting the collisions twice. Thus the rate constant for (2.142)

is given by

k = 2 N

π k

−

R T

(2.143)

For the reaction

2 HI → H

+ I

(2.144)

Lewis (1918) (cited in Laidler, 2007, p. 82) found that the pre-exponential

Review of background material 37

factor predicted by (2.143) was 3.5 × 10

−10

/mol-s. This value agrees

very well with the observed value, but the agreement is fortuitous. For many

reactions, the collision theory considerably overestimates the rate constants.

Extensions to the collision theory are described in Laidler (2007,

pp. 84-87).

2.18.2 Transition state theory or activated complex theory

This theory was developed by several workers, notably Pelzer and Wigner

(1932), Evans and Polanyi (1935), Eyring (1935), and Wynne-Jones and

Eyring (1935). Consider a reaction of the type

A + BC  AB + C (2.145)

where A, B, and C are atoms. For a system containing one atom each of A,

B, and C, the potential energy of interaction U

will vary with interatomic

distances, owing to the existence of attractive and repulsive forces. For ease

of analysis, consider a linear or one-dimensional conﬁguration of the atoms

(Fig. 2.2). Then

Fig. 2.2. A linear conﬁguration of three atoms A, B, and C

= U

, r

) (2.146)

where r

is the distance between the nuclei of atoms A and B.

Equation (2.146) may be regarded as a potential energy surface in a

space with Cartesian coordinates (U

, r

). For H atoms, the potential

energy surface has been calculated using quantum mechanics. The reaction

considered is

+ H

− H

→ H

− H

+ H

(2.147)

where the superscripts indicate diﬀerent H atoms. For a linear conﬁguration

of atoms, contours of constant U

can be plotted in the r(H

−H

)−r(H

−

) plane (Fig. 2.3).

The upper left hand corner of the ﬁgure represents a region where

r(H

−H

)  r(H

−H

), and hence it corresponds to the reactants shown

in (2.147). Similarly the lower right hand corner corresponds to the products

shown in (2.147). The broken curve ABC is the locus of local minima of the

potential energy U

. It is postulated that the reaction occurs along this path

of minimum energy, in the direction shown by the arrows. An examination

of the numerical values of the contours intersecting this locus shows that U

exhibits a maximum as the locus is traversed (Fig. 2.4). Thus the point B

in Fig. 2.3 represents a saddle point or col (Laidler, 2007, p. 59).

The distance measured along ABC is called the reaction coordinate

(Frost and Pearson, 1961, p. 81). The conﬁguration of the atoms in the

vicinity of the maximum B in Fig. 2.4 is referred to as an activated complex

or a transition state (Laidler, 2007, pp. 60-61). Once the reaction coordinate

and the location of the activated complex have been identiﬁed, the rate of

the reaction can be calculated as discussed below. The treatment has been

adapted from Laidler (2007, pp. 95-97).

The procedure consists of estimating (i) the concentration of ac-

tivated complexes, and (ii) the time required for a complex to cross the

transition state from left to right. The ratio of these two factors gives the

reaction rate.

The reaction (2.145) is assumed to occur through an intermediate

step involving the activated complex X

∗

A + BC  X

∗

→ AB + C (2.148)

In order to derive the reaction rate for the forward reaction, the key as-

sumption is that X

∗

is in equilibrium with the reactants A and BC. This

assumption is diﬃcult to justify. For reactions whose activation energy is

 R T , where R is the gas constant and T is the absolute temperature, it ap-

pears that its use may not cause large errors, unless the reactions are highly

exothermic (Denbigh, 1971, p. 455). For a perfect gas mixture containing

three species A, BC, X

∗

, it can be shown using statistical mechanics that

the molar concentrations c

are related to the molecular partition functions

Review of background material 39

Fig. 2.3. Contours of constant potential energy U

for a system containing three H

atoms H

, H

, and H

. Here r(H

−H

) denotes the distance between the centres of

and H

. The contours are only rough sketches, used to indicate the qualitative

features. The actual contours, representing the results of Truhlar and Horowitz

(1978), are given in their paper and also in Laidler (2007, p. 67). The numbers

on the contours show values of U

in kcal/mol. Adapted from Fig. 3.10 of Laidler

(2007, p.67).

ˆq

by (McQuarrie, 2003, pp. 142-144)

∗

(ˆq

∗

/V )

(ˆq

/V ) (ˆq

/V )

−E

/(R T )

(2.149)

where N

= 6.022 × 10

molecules/mol is Avogadro’s number, V is the

volume of the system, and E

is the diﬀerence in the molar zero point

Fig. 2.4. Locus of local minima of the potential energy for the potential energy

contours shown in Fig. 2.3).

energies of the activated complex and the reactants. For the ﬁrst of the

reactions (2.148)

≡ E

∗

− E

(2.150)

where E

∗

, E

and E

are the zero point energies per mole of X

∗

, A, and

BC, respectively, relative to an arbitrary zero of energy.

For a one-dimensional vibration of a molecule, the partition function

is given by (McQuarrie, 2003, p. 96)

ˆq =

1 − e

−h ν/(k

T )

(2.151)

Review of background material 41

where ν is the frequency of vibration, h = 6.626 × 10

−34

J-s is Planck’s

constant, and k

= 1.38 ×10

−23

J/K is Boltzmann’s constant. It is assumed

that the activated complex crosses the saddle point via a very loose vibration,

i.e. the frequency ν → 0 (Laidler, 2007, p. 95). In this limit, (2.151) reduces

ˆq =

h ν

(2.152)

Thus

ˆq

∗

h ν

ˆq

∗

(2.153)

where ˆq

∗

represents the partition function for all the degrees of freedom of

the activated complex, except the loose vibration near the saddle point.

Equations (2.149) and (2.153) imply that

∗

h ν

(ˆq

∗

/V )

(ˆq

/V ) (ˆq

/V )

−E

/(R T )

(2.154)

As the time required for the activated complex to cross the saddle region is

≈ 1/ν, the molar reaction rate for the forward reaction is given by

β =

∗

1/ν

R T

(ˆq

∗

/V )

(ˆq

/V ) (ˆq

/V )

−E

/(R T )

(2.155)

where R ≡ k

is the gas constant. Hence the rate constant for the forward

reaction is given by

k =

R T

(ˆq

∗

/V )

(ˆq

/V ) (ˆq

/V )

−E

/(R T )

(2.156)

Note that the pre-exponential factor is now a function of the temperature,

unlike in the Arrhenius equation.

2.18.3 Comparison of the theories

For the reaction

A + B → AB (2.157)

where A and B are atoms, both the theories give identical expressions for the

rate constant (Laidler, 2007, pp. 106-107). For reactions involving molecules,

the partition functions occurring the activated complex theory involve other

degrees of freedom such as rotation and vibration in addition to transla-

tion, whereas the collision theory considers only translation. As discussed

in Laidler (2007, pp. 107-109), these additional degrees of freedom cause the

pre-exponential factor predicted the activated complex theory to be signiﬁ-

cantly lower than than that predicted by the collision theory. Two examples

are given below.

Table 2.1 compares the measured and predicted pre-exponential fac-

tors for the dimerization of cyclopentadiene (Fig. 2.5)

2 C

(cyclopentadiene) → C

(2.158)

and

Fig. 2.5. Dimerization of cyclopentadiene.

2NO

→ 2 NO + O

(2.159)

In both the cases, the activated complex theory gives reasonable estimates

of the pre-exponential factors, whereas the collision theory overestimates

them considerably.

2.18.4 Reactions in solution

Some reactions occur in both the gas phase and also in a liquid solvent.

In such cases, the rate in solution is comparable to that in the gas phase

(Table 2.2), except for the dimerization reaction in C

. For reactions that

Review of background material 43

Table 2.1. Measured and predicted values of the pre-exponential factor A

for the reactions (2.158) and (2.159). Source: Laidler (1965, pp. 124-125).

A (m

/mol-s)

reaction (2.158) (2.159)

experiment 10

2 × 10

collision theory 10

4 × 10

activated complex theory 2 × 10

2.5 × 10

occur in solution, but not in the gas phase, the solvent may signiﬁcantly

aﬀect the rate (Table 2.3).

2.18.5 Catalytic reactions

A catalyst is a substance that increases the reaction rate without being

permanently aﬀected by the reaction. As it increases the rates of the forward

and backward reactions by the same amount, the equilibrium conversion is

unaﬀected.

As noted by Laidler (2007, p. 229), the word catalysis is derived

from the Greek words kata (wholly) and lyein (loosen). The word catalyst

was coined by Berzelius (1836), who studied several reactions, and is usually

credited with the discovery of catalysis. However, as noted by Laidler and

Cornish-Bowden (1979) (cited in Datta, 2005, p. 248-249), the phenomenon

of catalysis had been discovered forty years earlier by Elizabeth Fulhame.

Catalytic reactions may be broadly classiﬁed as homogeneous or het-

erogeneous. In the former, the reactants, products, and catalyst are all in

the same phase; in the latter, more than one phase is involved. Enzyme

catalysis is usually classiﬁed under homogeneous catalysis (Laidler, 2007,

p. 378).

An example of a homogeneously catalyzed gas phase reaction is pro-

vided by the decomposition of acetaldehyde in the presence of iodine (Lai-

dler, 2007, p. 414):

CHO → CH

+ CO (2.160)

Here I

acts as a catalyst, reducing the activation energy from about 200

Table 2.2. Comparison of pre-exponential factors A and activation energies

E for reactions in the gas phase and in liquids. Source: Laidler (2007,

pp. 184-185).

solvent log

A E

(A in m

/mol-s) (kJ/mol)

(i) decomposition of N

gas phase 7.6 103.3

CCl

7.6-7.8 101.3-106.7

CHCl

7.6-7.7 102.5-102.9

(ii) dimerization of cyclopentadiene

gas phase 0.1 69.9

CCl

-0.1 67.8

1.1 29.7

-0.3 74.1

kJ/mol to about 135 kJ/mol. There are many aqueous phase reactions that

are catalyzed by acids and bases, such as the reaction between acetone and

iodine (Laidler, 2007, p. 392).

Many industrial reactions such as the synthesis of NH

involve het-

erogeneous (solid) catalysts. Such reactions involve several steps, such as

the mass transfer between the bulk ﬂuid and the surface of the catalyst,

adsorption of reactants on the surface, either surface reactions between ad-

sorbed reactants or reactions between adsorbed and ﬂuid-phase reactants,

and desorption of reactants and products from the surface.

In the context of the transition state theory, the main eﬀect of a solid

catalyst is to lower the energy barrier (Fig. 2.6). For example, the activation

energy for the decomposition of NH

decreases from 340 kJ/mol to 134-180

kJ/mol when a molybdenum catalyst is used (Laidler, 2007, p. 267).

Consider the unimolecular reaction A

(g) → A

(g) which is cat-

Review of background material 45

Table 2.3. Eﬀect on the solvent on the rate of reaction between

triethylamine and ethyliodide. Here k is the second order rate constant,

evaluated as 100

◦

. Source: Laidler (2007, p. 185).

solvent 10

(A in m

/mol-s)

hexane 0.5

toluene 25.3

benzene 39.8

alyzed by a solid catalyst. Suppose it is assumed that the surface reaction

is the rate-limiting step, i.e. of adsorption and desorption can occur at rates

that are potentially much faster than the rate of surface reaction. Hence

these processes may be assumed to be at equilibrium, and the concentrations

of the adsorbed species may be calculated using the adsorption isotherms.

The latter are obtained by equating the rates of adsorption and desorption

of each reactant, in the absence of reactions. The above model leads to the

rate expression (Schmidt, 2005, p. 303)

β =

k K

1 + K

+ K

(2.161)

where

β is the intrinsic reaction rate, k is the rate constant, K

and p

are

the adsorption equilibrium constants and partial pressure, respectively, of

species A

. Equation (2.163) assumes that A

is ﬁrst formed as an adsorbed

species that then desorbs, i.e. the reactions involved are

1∗

→ A

2∗

→ A

g (2.162)

where A

1∗

represents adsorbed A

. If A

(g) is formed directly from A

1∗

, we

set K

= 0 in (2.163). Equation (2.163) represents an example of Lanmuir-

Hinshelwood, or equivalently, Hougen-Watson kinetics, named after Lang-

muir (1922), Hinshelwood (1926), and Hougen and Watson (1947) (cited in

Froment and Bischoﬀ, 1990, p. 71). Instead of assuming that the surface

reaction is rate-limiting, alternative assumptions may be used, leading to

diﬀerent rate expressions (Froment and Bischoﬀ, 1990, pp. 76-77).

Similarly, for the reaction A

(g) + A

(g) → A

(g), the reaction

rate is given by (Schmidt, 2005, p. 308)

β =

k K

(1 + K

+ K

)

(2.163)

As noted by Schmidt (2005, p. 309), the use of Langmuir-Hinshelwood kinet-

ics involves several assumptions that may not be justiﬁed in actual surface

catalyzed reactions. However, it provides relatively simple kinetic models

for complex heterogeneous catalytic reactions. Zhang et al. (2009) discuss

the use of such models for the Fischer-Tropsch synthesis of hydrocarbons

from water gas, which is a mixture of CO and H

2.18.6 Ideal reactors

As mentioned in Chapter 1, ideal reactors are based on simple models of

ﬂow patterns and mixing in the reaction vessel.

(a) The ideal batch reactor

In an ideal batch reactor, the concentration and temperature ﬁelds

are assumed to be spatially uniform. In practice, the condition can be ap-

proximately realized by vigorous agitation or stirring. In the absence of

stirring, beautiful spatial patterns, caused by an interaction between dif-

fusion and reactions, may develop in some systems (Epstein et al., 1983;

Epstein and Showalter, 1996).

All the elements of the ﬂuid spend the same amount of time in the

reactor, and hence have the same residence time. From the viewpoint of

thermodynamics, a batch reactor represents a closed system. The steady

states of the batch reactor correspond to states of reaction equilibria, which

satisfy (2.84).

Batch reactors are often used in the pharmaceutical industry, where

small volumes of high-value products are made.

(b) The ideal continuous stirred tank reactor (CSTR)

Like in an ideal batch reactor, the concentration and temperature

ﬁelds in an ideal CSTR are spatially uniform. As there are no spatial gra-

dients, the species concentrations in the exit stream are identical to the

corresponding values in the reactor. On the other hand, the species con-

centrations in the inlet stream are in general diﬀerent from those in the

reactor.

Unlike the batch reactor, the CSTR is an open system as it can

Review of background material 47

exchange heat and mass with the surroundings. Hence it operates away from

equilibrium, and steady states are usually not states of reaction equilibria.

On account of the assumption of perfect mixing, the sequence in

which ﬂuid elements leave the reactor is uncorrelated with the sequence in

which they enter. As shown later, this leads to a distribution of residence

times for the ﬂuid leaving the reactor.

These reactors are widely used for polymerization reactions such as

the polymerization of styrene, production of explosives, synthetic rubber,

etc. Compared to tubular reactors, CSTRs are easier to clean and permit

better control of the temperature.

The PFR is an idealization of a tubular reactor. The velocity, tem-

perature, and concentration ﬁelds are assumed to be uniform across the

cross section of the reactor. In practice, this situation can be approximately

realized for the case of turbulent ﬂow through a tube with a large ratio of

the length to the diameter. The latter condition ensures that axial mixing

has a negligible eﬀect on the conversion.

In a PFR, there is perfect mixing in the radial or transverse direction.

Further, there is no mixing or diﬀusion in the axial direction. Like a CSTR,

the PFR also represents an open system, and hence steady states are not

states of reaction equilibrium. Owing to the assumption of plug ﬂow, all

the ﬂuid elements have the same residence time. The velocity of the ﬂuid is

often treated as a constant, but this assumption must be relaxed when the

density of the ﬂuid changes signiﬁcantly along the length of the tube.

The steady state equations for a PFR are similar in form to the

dynamic equations for an ideal batch reactor. In many cases, the results

for the latter can be translated into results for a PFR operating at a steady

state.

Tubular reactors are used for many gas phase and liquid phase reac-

tions, such as the oxidation of NO and the synthesis of NH

. These reactors

are often modelled as PFRs, but more detailed models involving complica-

tions such as radial gradients, may be required in some cases.

Fig. 2.6. Variation of the potential energy U

along the reaction coordinate for an

uncatalyzed gas phase reaction (—) and a solid catalyzed heterogeneous reaction

(- - -). Here A, C, D, and G denote the gaseous reactants, gaseous products,

adsorbed reactants, and adsorbed products, respectively. and D and F denote the

activated complexes for the gas phase and heterogeneous reactions, respectively.

The quantities ∆U

and ∆U

represent the energy barriers for the homogeneous

and catalytic reactions, respectively. Adapted from Laidler (2007, p. 268).

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