1. Chapter 3: RATE LAWS & STOICHIOMETRY
PART 1: RATE LAWS
3.1 Basic Definitions
A homogeneous reaction is one that involves only one phase.
A heterogeneous reaction involves more than one phase, & reaction usually occurs at interface between phases.

2. An irreversible reaction is one that proceeds in only one direction & continues in that direction until reactants are exhausted.

3. A reversible reaction, can proceed in either direction, depending on concentrations of reactants & products relative to corresponding equilibrium concentrations.

4. 3.1.1 Relative Rates of Reaction
Relative rates of reaction of various species involved in a reaction can be obtained from ratio of stoichiometric coefficients.
For Reaction (2-2),

5. Every mole of A that consumed,
c/a moles of C appear.

6. For example

7. 3.2 Reaction Order & Rate Law
In chemical reactions considered in following paragraphs, we take as basis of calculation a species A, which is one of reactants that is disappearing as a result of reaction.
Limiting reactant is usually chosen as our basis for calculation.

8. Rate of disappearance of A, -rA depends on temperature & composition.
For many reactions, it can be written as product of a reaction rate constant kA & a function of concentrations (activities) of various species involved in reaction:

9. The algebraic equation that relates -rA to species concentrations is called the kinetic expression or rate law.
Specific rate of reaction (also called rate constant), kA , like reaction rate -rA always refers to a particular species in reaction & normally should be subscripted with respect to that species.

10. However, for reactions in which stoichiometric coefficient is 1 for all species involved in reaction, for example,

11. 3.2.1 Power Law Models & Elementary Rate Laws
Dependence of reaction rate, -rA, on concentrations of species present, fn(Cj), is almost without exception determined by experimental observation.

12. Although functional dependence on concentration may be postulated from theory, experiments are necessary to confirm proposed form.
One of most common general forms of this dependence is power law model.

13. Here rate law is product of concentrations of individual reacting species, each of which is raised to a power, for example,

14. Order of a reaction refers to powers to which concentrations are raised in kinetic rate law.
In Equation (3-3), reaction is α order with respect to reactant A, & β order with respect to reactant B. Overall order of the reaction, n, is

15. with a reaction order n. Units of specific reaction rate constant are
Units of -rA are always in terms of C per unit time while units of specific reaction rate, kA, will vary with order of reaction.
Consider a reaction involving only one reactant, such as
with a reaction order n. Units of specific reaction rate constant are

16. Consequently, rate laws corresponding to a zero-, first-, second-, & third -order reaction, together with typical units for the corresponding rate constants, are:

17. An elementary reaction is one that evolves a single step such as the bimolecular reaction between oxygen & methanol

18. Sstoichiometric coefficients in this reaction are identical to powers in rate law. Consequently, rate law for disappearance of molecular oxygen is

19. Reaction is first order in molecular oxygen & first order in methanol; therefore, we say both reaction & rate law are elementary.

20. Many reactions where stoichiometric coefficients in reaction are identical to reaction orders, but reactions are not elementary owing to such things as pathways involving active intermediates & series reactions.

21. These reactions that are not elementary but whose stoichiometric coefficients are identical to reaction orders in rate law, we say reaction follows an elementary rate law. For example, oxidation reaction of nitric oxide discussed earlier,
is not elementary but follows the elementary rate law

22. Another nonelementary reaction that follows an elementary rate law is gas-phase reaction between hydrogen & iodine

23. 3.2.3 Reversible Reactions
All rate laws for reversible reactions must reduce to thermodynamic relationship relating reacting species concentrations at equilibrium.

24. That is, for general reaction
At equilibrium, rate of reaction is identically zero for all species (i.e., -rA= 0).
That is, for general reaction
concentrations at equilibrium are related by thermodynamic relationship for equilibrium constant Kc.

25. units of thermodynamic equilibrium constant, Kc , are (mol/dm3)d+c-b-a.
To illustrate how to write rate laws for reversible reactions, we will use combination of two benzene molecules to form one molecule of hydrogen & one of diphenyl.

26. In this discussion, we shall consider this gas-phase reaction to be elementary & reversible:
Forward & reverse specific reaction rate constants, kB & k-B, respectively, will be defined with respect to benzene.
Benzene (B) is being depleted by the forward reaction

27. in which rate of disappearance of benzene is
If we multiply both sides of this equation by -1, we obtain expression for rate of formation of benzene for forward reaction:
For reverse reaction between diphenyl (D) & hydrogen (H2),

28. rate of formation of benzene is given as
Again, both rate constants kB and k-B are defined with respect to benzene!!!
Net rate of formation of benzene is sum of rates of formation from forward reaction [i.e., Equation (3-11)] & reverse reaction [i.e., Equation (3-12)]:

29. Multiplying both sides of Equation (3-13) by -1, we obtain rate law for rate of disappearance of benzene, rA :
Replacing ratio of reverse to forward rate law constants by equilibrium constant, we obtain

30. Equilibrium constant decreases with increasing temperature for exothermic reactions & increases with increasing temperature for endothermic reactions.

31. Write rate of formation of diphenyl, rD, in terms of concentrations of hydrogen,H2 ,diphenyl, D, & benzene, B.
Rate of formation of diphenyl, rD, must have same functional dependence on reacting species concentrations as does rate of disappearance of benzene, -rB.
Rate of formation of diphenyl is

32. Using relationship given by Equation (3-1) for general reaction
Obtain relationship between various specific reaction rates, kB, kD:

33. Comparing Equations (3-15) & (3-16), relationship between specific reaction rate with respect to diphenyl & specific reaction rate with respect to benzene is

34. Consequently, need to define rate constant, k, wrt a particular species.

35. Finally, we need to check to see if rate law given by Equation (3-14) is thermodynamically consistent at equilibrium.
Applying Equation (3-10) to diphenyl reaction & substituting appropriate species concentration & exponents, thermodynamics tells us that

36. Look at rate law. At equilibrium, -rB = 0, & rate law given by Equation (3-14) becomes
Rearranging, we obtain, as expected, equilibrium expression

37. From Appendix C, Equation (C-9), we know that when there is no change in the total number of moles & the heat capacity term, ∆Cp =0 the temperature dependence of concentration equilibrium constant is

38. Therefore, if we know equilibrium constant at one temperature, T1, [i
Therefore, if we know equilibrium constant at one temperature, T1, [i.e., Kc(T1)] & heat of reaction, HRx , we can calculate equilibrium constant at any other temperature T.

39. For endothermic reactions, equilibrium constant, Kc , increases with increasing temperature; for exothermic reactions, Kc decreases with increasing temperature.
A further discussion of equilibrium constant & its thermodynamic relationship is given in Appendix C.

40. 3.3 The Reaction Rate Constant
Reaction rate constant k is not truly a constant; it is merely independent of concentrations of species involved in reaction.

41. It is almost always strongly dependent on temperature
It is almost always strongly dependent on temperature. It depends on whether or not a catalyst is present, & in gas-phase reactions, it may be a function of total pressure.

42. In liquid systems it can also be a function of other parameters, such as ionic strength & choice of solvent.

43. These other variables normally exhibit much less effect on specific reaction rate than temperature does with exception of supercritical solvents, such as super critical water.

44. Consequently, for purposes of material presented here, it will be assumed that kA depends only on temperature.
This assumption is valid in most laboratory & industrial reactions & seems to work quite well.

45. It was great Swedish chemist Arrhenius who first suggested that temperature dependence of specific reaction rate, kA, could be correlated by an equation of type

46. Eq. (3-18), known as Arrhenius equation, verified empirically to give temperature behavior of most reaction rate constants within experimental accuracy over fairly large temperature ranges.

47. Why is there an activation energy?
If reactants are free radicals that essentially react immediately on collision, there usually isn’t an activation energy.
However, for most atoms & molecules undergoing reaction, there is an activation energy. A couple of reasons are that in order to react,

48. 1-Molecules need energy to distort or stretch their bonds so that they break them & thus form new bonds.
2-Steric & electron repulsion forces must be overcome as reacting molecules come close together.

49. Activation energy can be thought of as a barrier to energy transfer (from kinetic energy to potential energy) between reacting molecules that must be overcome.

50. One way to view barrier to a reaction is through use of reaction coordinates.
These coordinates denote potential energy of system as a function of progress along reaction path as we go from reactants to an intermediate to products.

51. reaction coordinate is shown in Fig. (3-1).
For reaction
reaction coordinate is shown in Fig. (3-1).

52. Fig. 3-l(a) shows potential energy of three atom (or molecule) system, A, B, & C, as well as reaction progress as we go from reactant species A & BC to products AB & C.
Initially A & BC are far apart & system energy is just bond energy BC.

53. At end of reaction, products AB & C are far apart, & system energy is the bond energy AB.
As we move along reaction coordinate (x-axis) to right in Figure 3-1(a), reactants A & BC approach each other, BC bond begins to break, & energy of reaction pair increases until top of barrier is reached.

54. At top, transition state is reached where intermolecular distances between AB & between BC are equal (i.e., A-B-C).
As a result, potential energy of initial three atoms (molecules) is high.

55. As reaction proceeds further, distance between A & B decreases, & AB bond begins to form.
As we proceed further, distance between AB & C increases & energy of reacting pair decreases to that of AB bond energy.

56. We see that for reaction to occur, reactants must overcome an energy barrier, EB , shown in Fig. 3-1.
Energy barrier, EB, is related to activation energy, E.

58. Energy barrier height, EB, calculated from differences in energies of formation of transition state molecule & energy of formation of reactants, that is,

59. Postulation of Arrhenius equation, Eq
Postulation of Arrhenius equation, Eq. (3-18), remains greatest single step in chemical kinetics, & retains its usefulness today, nearly a century later.
Activation energy, E, is determined experimentally by carrying out reaction at several different temperatures. After taking natural logarithm of Eq. (3-18):

60. & see that activation energy can be found from a plot of In kA as a function of (1/T).

65. There is a rule of thumb that states that rate of reaction doubles for every 10°C increase in temperature. However, this is true only for a specific combination of activation energy & temperature.

66. For example, if the activation energy is 53
For example, if the activation energy is 53.6 kJ/mol, rate will double only if temperature is raised from 300 K to 310 K.
If activation energy is 147 kJ/mol, rule will be valid only if temperature is raised from 500 K to 510 K.

67. The larger the activation energy, the more temperature-sensitive is the rate of reaction.
While there are no typical values of frequency factor & activation energy for a first-order gas-phase reaction, if one were forced to make a guess, values of A and E might be 1013 s-1 & 200 kJ/mol.

68. One final comment on Arrhenius equation, Equation (3-18).
It can be put in a most useful form by finding specific reaction rate at a temperature To , that is,

69. This equation says that if we know specific reaction rate ko(To) at a temperature, To , & we know activation energy, E, we can find specific reaction rate k(T) at any other temperature, T, for that reaction.

70. 3.4 Present Status of Our Approach to Reactor Sizing & Design
In Chapter 2, we showed how it was possible to size CSTRs, PFRs, & PBRs using design equations in Table 3-2 (page 99) if rate of disappearance of A is known as a function of X:

72. In Part 2, Sections 3.5 & 3.6. we show how concentration of reacting species may be written in terms of conversion X.

73. PART 2 STOICHIOMETRY
If rate law depends on more than one species, we must relate concentrations of different species to each other.
This relationship is most easily established with aid of a stoichiometric table.

74. This table presents stoichiometric relationships between reacting molecules for a single reaction.
It tells us how many molecules of one species will be formed during a chemical reaction when a given number of molecules of another species disappears.

75. These relationships will be developed for general reaction
already used stoichiometry to relate relative rates of reaction for Eq. (2-1):
In formulating our stoichiometric table, take species A as our basis of calculation (i.e., limiting reactant) & then divide through by stoichiometric coefficient of A,

76. in order to put everything on a basis of “per mole of A.”
Next, develop stoichiometric relationships for reacting species that give change in number of moles of each species (i.e., A, B, C, & D ).

77. 3.5 Batch Systems
Batch reactors are primarily used for production of specialty chemicals & to obtain reaction rate data in order to determine reaction rate laws & rate law parameters such as k, specific reaction rate.

78. Fig. (3-4) shows batch system in which we will carry out reaction given by Eq. (2-2).
At time t = 0, open reactor & place a number of moles of species A, B, C, D, & I (NA0, NB0, NC0 , ND0, & NI0 , respectively) into reactor.

80. Species A is our basis of calculation, & NA0 is number of moles of A initially present in reactor.
Of these, NA0X moles of A are consumed in system as a result of chemical reaction, leaving (NA0-NA0 X) moles of A in system.

81. Number of moles of A remaining in reactor after a X has been achieved is

82. To determine number of moles of each species remaining after NA0X moles of A have reacted, form stoichiometric table (Table 3-3).
This stoichiometric table presents following information:

84. Column 1: particular species
Column 2: number of moles of each species initially present
Column 3: change in number of moles brought about by reaction
Column 4: number of moles remaining in system at time t

85. Let’s take a look at totals in last column of Table 3-3.
Stoichiometric coefficients in parentheses (d/a+c/a–b/a-1) represent increase in total number of moles per mole of A reacted.
Because this term occurs so often in our calculations, it is given symbol δ

86. Parameter δ tells us change in total number of moles per mole of A reacted. Total number of moles can now be calculated from equation:

87. 3.5.1 Equations for Batch Concentrations
Concentration of A is number of moles of A per unit volume:
After writing similar equations for B, C, & D, we use stoichiometric table to express concentration of each component in terms of conversion X:

88. We further simplify these equations by defining parameter Θi which allows us to factor NA0 in each of expressions for concentration:

89. Now need only to find volume as a function of X to obtain species concentration as a function of X.

90. 3.5.2 Constant-Volume Batch Reaction Systems
Some significant simplifications in reactor design equations are possible when reacting system undergoes no change in volume as reaction progresses. These systems are called constant-volume, or constant-density.

91. This situation may arise from several causes
This situation may arise from several causes. In gas -phase batch systems, reactor is usually a sealed constant-volume vessel with appropriate instruments to measure pressure & temperature within reactor.

92. Volume within this vessel is fixed & will not change, & is therefore a constant-volume system (V=V0).
Laboratory bomb calorimeter reactor is a typical example of this type of reactor.

93. Another example of a constant-volume gas-phase isothermal reaction occurs when number of moles of products equals number of moles of reactants.

94. For constant-volume systems described earlier, Eq
For constant-volume systems described earlier, Eq. (3-25) can be simplified to give following expressions relating concentration & conversion:

95. To summarize for liquid-phase reactions, use a rate law for reaction (2-2) such as –rA= kACACB to obtain –rA= f(X), that is,
Substituting for given parameters k. Cao, & ΦB , now use techniques in Chapter 2 to size CSTRs & PFRs for liquid-phase reactions.

100. 3.6 Flow Systems
Form of stoichiometric table for a continuous-flow system (see Fig. 3-5) is virtually identical to that for a batch system (Table 3-3) except that we replace Nj0 by Fj0 & Nj by Fj (Table 3-4).

103. Taking A as basis, divide Eq
Taking A as basis, divide Eq. (2-1) through by stoichiometric coefficient of A to obtain

104. 3.6.1 Equations for Concentrations in Flow Systems
For a flow system, concentration CA at a given point can be determined from molar flow rate FA & volumetric flow rate ν at that point:

105. Now, can write concentrations of A, B, C, & D for general reaction given by Eq. (2-2) in terms of their respective entering molar flow rates (FA0, FB0, FC0, FD0), conversion X, & volumetric flow rate , ν.

107. 3.6.2 Liquid-Phase Concentrations
For liquids, volume change with reaction is negligible when no phase changes are taking place. Consequently, we can take

108. Consequently, using any one of rate laws in Part 1 of this chapter, we can now find –rA = f(X) for liquid-phase reactions.

109. However, for gas-phase reactions, volumetric flow rate most often changes during course of reaction because of a change in total number of moles or in temperature or pressure.

110. Hence, one cannot always use Eq
Hence, one cannot always use Eq. (3-29) to express concentration as a function of conversion for gas-phase reactions.

111. 3.6.3 Change in the Total Number of Moles with Reaction in Gas Phase
In our previous discussions, we considered primarily systems in which reaction volume or volumetric flow rate did not vary as the reaction progressed.
Most batch & liquid-phase & some gas-phase systems fall into this category.

112. There are other systems, though, in which either V or ν do vary, & these will now be considered.
A situation where one encounters a varying flow rate occurs quite frequently in gas-phase reactions that do not have an equal number of product & reactant moles.
For example, in synthesis of ammonia,

113. 4 mol of reactants gives 2 mol of product
4 mol of reactants gives 2 mol of product. In flow systems where this type of reaction occurs, molar flow rate will be changing as reaction progresses.
Because equal numbers of moles occupy equal volumes in gas phase at same temperature & pressure, volumetric flow rate will also change.

114. In stoichiometric tables presented on preceding pages, it was not necessary to make assumptions concerning a volume change in first four columns of table

115. (i.e., species,
initial number of moles or molar feed rate,
change within the reactor, & remaining number of moles or molar effluent rate).

116. All of these columns of stoichiometric table are independent of volume or density, & they are identical for constant-volume (constant-density) & varying-volume (varying-density) situations.
Only when concentration is expressed as a function of conversion does variable density enter picture.

117. Batch Reactors with Variable Volume
Although variable volume batch reactors are seldom encountered because they are usually solid steel containers, we will develop concentrations as a function of conversion because

118. (1) They have been used to collect reaction data for gas-phase reactions, &
(2) development of equations that express volume as a function of conversion will facilitate analyzing flow systems with variable volumetric flow rates.

119. Individual concentrations can be determined by expressing volume V for a batch system, or volumetric flow rate υ for a flow system, as a function of conversion using following equation of state:

120. Dividing Equation (3-30) by Equation (3-31) & rearranging yields
This equation is valid at any point in the system at any time t. At time t = 0 (i.e., when reaction is initiated), Eq. (3-30) becomes
Dividing Equation (3-30) by Equation (3-31) & rearranging yields
Now want to express volume V as a function of conversion X.

121. Recalling equation for total number of moles in Table 3-3,

124. In gas-phase systems that we shall be studying, temperatures & pressures are such that compressibility factor will not change significantly during course of reaction: hence Zo≈Z. For a batch system, volume of gas at any time t is:

125. Eq. (3-38) applies only to a variable-volume batch reactor, where one can now substitute Eq. (3-38) into Eq. (3-25) to express rA=f(X). However, if reactor is a rigid steel container of constant volume, then of course V=Vo.

126. Flow Reactors with Variable Volumetric Flow Rate
An expression similar to Eq. (3-38) for a variable-volume batch reactor exists for a variable-volume flow system.

127. To derive concentrations of each species in terms of conversion for a variable-volume flow system, use relationships for total concentration.

128. Total concentration, CT, at any point in reactor is total molar flow rate, FT divided by volumetric flow rate υ.

129. In gas phase, total concentration is also found from gas law, CT = P/ZRT.
Equating these two relationships gives

130. At entrance to reactor,
Taking ratio of Eq. (3-40) to Eq. (3-39) & assuming negligible changes in compressibility factor,

131. now express concentration of species j for a flow system in terms of its flow rate, Fj, temperature, T, & total pressure, P.
total molar flow rate is just sum of the molar flaw rates of each of species:

132. One of major objectives of this chapter is to learn how to express any given rate law -rA as a function of conversion.
Schematic diagram in Fig. 3-6 helps to summarize our discussion on this point.

134. Concentration of key reactant, A, is expressed as a function of conversion in both flow & batch systems, for various conditions of T, P, & V.

135. Now let’s express concentration in terms of conversion for gas flow systems. From Table 3-4 total molar flow rate can be written in terms of conversion:
Substituting for FT , in Eq. (3-41) gives