3. Rate Laws and Stoichiometry^*

Rationale For Chapter 3

In chapter 2 we saw that if we had –r_A as a function of X, [–r_A= f(X)] we could size many reactors and reactor sequences and systems.

Topics

1. Relative Rates of Reaction
2. Power Law Model
3. Rate Constant, k
4. Elementary Reactions
5. Non-Elementary Rate Laws
6. Reversible Reactions

1. Batch System Stoichiometric Table
2. Flow System Stoichiometric Table

Part 1: Rate Laws

A rate law describes the behavior of a reaction. The rate of a reaction is a function of temperature (through the rate constant) and concentration.

aA + bBcC + dD

Relative Rates of Reaction

k is the specific reaction rate (constant) and is given by the Arrhenius Equation:

Where:
Where: E = activation energy (cal/mol) R = gas constant (cal/mol*K) T = temperature (K) A = frequency factor (units of        A, and k, depend on overall     reaction order)

The larger the activation energy, the more temperature sensitive k and thus the reaction rate.

Why is there an Activation Energy?

            (1) the molecules need energy to distort or stretch their bonds in order to break them and to thus form new bonds

            (2) as the reacting molecules come close together they must overcome both steric and electron repulsion forces in order to react

In our development of collision theory we assumed all molecules had the same average energy. However, all the molecules don’t have the same energy, rather there is distribution of energies where some molecules have more energy than others. The distraction function f(E,T) describes this distribution of the energies of the molecules. The distribution function is read in conjunction with dE

One such distribution of energies is in the following figure.

By increasing the temperature we increase the kinetic energy of the reactant molecules which can in turn be transferred to internal energy to increase the stretching and bending of the bonds causing them to be in an activated state, vulnerable to bond breaking and reaction.

We see that as the temperature is increased we have greater number of molecules have energies E A or greater and hence the reaction rate will be greater.

You can tell the overall reaction order by the units of k

The activation energy is a measure of the minimum energy a that the reacting molecules must have in order for the reaction to occur.

The reaction of AB + C to form A + BC is shown above along the reaction coordinate. One way to think of the reaction coordinate is the linear distance between the AB molecule for a fixed linear distance between the AC molecule. At the start of the reaction the AB distance is small and the BC distance is large. As the reaction proceeds, the A–C distance remains fixed but B moves away for A and closer to C and the energy of system increases. At the top of the barrier molecule B is equal distance from A and C. But as it crosses the barrier it moved close to C to form the BC molecule and the A molecule alone.

Activation Energy

The following movie was made by the students of Professor Alan Lane's chemical reaction engineering class at the University of Alabama Tuscaloosa

 Molecular Simulation 1 (external link)

Crickets

Activation Energies

 Excel regression analysis tutorial

 Polymath regression analysis tutorial

 Molecular Simulation 2a (external link)

 Molecular Simulation 2b (external link)

 For Molecular Simulation 2

 Molecular Simulation 3 (external link)

A reaction follows an elementary rate law if and only if the (iff) stoichiometric coefficients are the same as the individual reaction order of each species. For the reaction in the previous example (), the rate law would be:

if 2NO+O₂   2NO₂     then -r_NO = k_NO (C_NO)² C_O2   if elementary! 

See the example below for more examples of rate laws.

Rate Laws

Example: If the rate law for the non-elementary reaction

is found to be

then the reaction is said to be 2nd order in A, 1st order in B, and 3rd order overall. 

The net rate of formation of any species is equal to its rate of formation in the forward reaction plus its rate of formation in the reverse reaction:

rate_net = rate_forward + rate_reverse

At equilibrium, rate_net0 and the rate law must reduce to an equation that is thermodynamically consistent with the equilibrium constant for the reaction.

Example: Consider the exothermic, heterogeneous reaction

At low temperature, the rate law for the disappearance of A is

At high temperature, the exothermic reaction is significantly reversible:

What is the corresponding rate law? Let's see.

If the rate of formation of A for the forward reaction (A + BC) is

then we need to assume a form of the rate law for the reverse reaction that satisfies the equilibrium condition. If we assume the rate law for the reverse reaction (CA + B) is

then:                   

and:

 

Does this rate law satisfy our requirement that, at equilibrium, it must reduce to an equation that is thermodynamically consistent with K_P? Let's see.

From Appendix C we know that for a reaction at equilibrium:

At equilibrium, r_net 0, so:

Solving for K_P gives:

The conditions are satisfied.

Rate Law for Reversible Reactions

Rate Law Examples

Molecular Simulation 4

Part 2: Stoichiometry

We shall set up Stoichiometric Tables using A as our basis of calculation in the following reaction. We will use the stoichiometric tables to express the concentration as a function of conversion. We will combine C_i = f(X) with the appropriate rate law to obtain -r_A = f(X).

Species	Symbol	Initial	Change	Remaining
A	A
B	B
C	C
D	D
Inert	I	________	-------	____________

Where:

and

Constant Volume Batch:

Note:              if the reaction occurs in the liquid phase                                                or           if a gas phase reaction occurs in a rigid (e.g., steel) batch reactor
Then
	etc.

            if       then

            and we have -ra=f(x)
 

Writing -r_A soley as a function of X

Species	Symbol	Reactor Feed	Change	Reactor Effluent
A	A
B	B
C	C
D	D
Inert	I	________	-------	____________

Where:

and

etc.

 

If the rate of reaction were -r_A = kC_AC_B then we would have

This gives us -r_A = f(X). Consequently, we can use the methods discussed in Chapter 2 to size a large number of reactors, either alone or in series.

etc. Again, these equations give us information about -rA = f(X), which we can use to size reactors. For example if the gas phase reaction has the rate law  then

with

Rate Law in terms of Partial Pressures

Stoichiometry Table for Conversion

Oxidation of Naphthalene to Phtahalic Anhydride

Critical Thinking Questions for the Oxidation of Naphthalene

 

Calculating the equilibrium conversion for gas phase reaction

Consider the following elementary reaction with K_C and = 20 dm³/mol and C_A0 = 0.2 mol/dm³. Pure A fed. Calculate the equilibrium conversion, X_e, for both a batch reactor and a flow reactor.

Solution

At equilibrium

Stoichiometry

Batch

Species	Initial	Change	Remaining
A	NA0	-NA0X	NA = NA0(1-X)
B	0	+NA0X/2	NB = NA0X/2
	NT0 = NA0		NT = NA0 - NA0X/2

Constant Volume V = V₀

Solving

Flow

Species	Fed	Change	Remaining
A	FA0	-FA0X	FA = F A0(1-X)
B	0	+FA0X/2	FB = F A0X/2
	FT0 = F A0		FT = F A0 - F A0X/2

  Recall

Old Exam Questions

Matching C_A versus X curves

What is wrong with this solution?

Objective Assessment of Chapter 3

^* All chapter references are for the 4th Edition of the text Elements of Chemical Reaction Engineering .

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