3.4 The Search for a Rate Equation

3.1 Constant-volume Batch Reactor

3.2 Varying-volume Batch Reactor

3.3 Temperature and Reaction Rate

3.4 The Search for a Rate Equation

3.1 Constant Volume Batch Reactor

Constant volume of reaction mixture (Not the volume of reactor)

Meaning  the rate of reaction of any component is given by

the rate of change of its concentration or partial pressure

For liquid phase system, it is easy to measure the concentration of each component i However,

for gas phase system where the numbers of moles change,

it is impossible to measure the partial pressure of each component i directly.

Solution : calculation of partial pressure using total pressure () of the system which can be measured directly

How to calculate partial pressure of each component

General stoichiometric equation for a reaction in which the numbers of moles change

At any time t : total number =

Ideal gas law (pV = nRT)  Thus, the unknown variable

where  = total pressure of the system By measuring total pressure,

we can get partial pressure Conversion fraction

New terminology : Conversion (X)

Suppose that N_Ao is the initial amount of A in the reactor at time t = 0, and that N_A is the amount present at time t.

Then the conversion of A in the constant volume system

Two Procedures for Analyzing Kinetic Data

1) Integral method

2) Differential method

In the integral method of analysis

we guess a particular form of rate equation and,

after appropriate integration and mathematical manipulation,

predict that the plot of a certain concentration function versus time should yield a straight line.

The data are plotted, and if a reasonably good straight line is obtained, then the rate equation is said to satisfactorily fit the data.

In the differential method of analysis

we test the fit of the rate expression to the data directly and without any integration.

However, since the rate expression is a differential equation,

we must first find the value of (l/V)(dNldt) from the data before attempting the fitting procedure.

The integral method is easy to use and is recommended when testing specific mechanisms, or relatively simple rate expressions

The differential method is useful in more complicated situations but requires more accurate or larger amounts of data.

In general, integral analysis is attempted first, and, if not successful, the differential method is tried.

1) Integral Method of Analysis of Experimental Data

General Procedure :

The integral method of analysis always puts a particular rate equation to the test by integrating and comparing the predicted C versus t curve

with the experimental C versus t data.

If the fit is unsatisfactory, another rate equation is guessed and tested.

Integral method is especially useful for fitting simple reaction types corresponding to elementary reactions.

a) Irreversible unimolecular- type first-order reaction

b) Irreversible bimolecular- type second-order reaction #1 c) Irreversible bimolecular- type second-order reaction #2 d) Irreversible bimolecular- type second-order reaction #3 e) Irreversible trimolecular- type third-order reaction

f) N th order reactions g) Zero-order reactions

h) Irreversible reactions in parallel i) Homogeneous Catalyzed Reactions j) Autocatalytic reactions

k) Irreversible series reactions l) First-order reversible reactions

m) Second-order reversible reactions

a) Irreversible unimolecular- type first-order reaction

b) Irreversible bimolecular- type second-order reaction #1

c) Irreversible bimolecular- type second-order reaction #2

d) Irreversible bimolecular- type second-order reaction #3

e) Irreversible trimolecular- type third-order reaction

Other irreversible trimolecular- type third-order reactions

f) Empirical rate equations of nth order

When the mechanism of reaction is not known,

we often attempt to fit the data with an nth-order rate equation of the form

The order n cannot be found explicitly from the equation, so a trial-and-error solution must be made.

Select a value for n and calculate k.

 the value of n which minimizes the variation in k is the desired value of n.

g) Zero-order reactions

A reaction is of zero order

when the rate of reaction is independent of the concentration of materials

h) Irreversible reactions in parallel (multiple reaction)

i) Homogeneous catalyzed reactions (multiple reaction)

Catalyst concentration = constant

Making a series of runs with different catalyst concentrations allows us to find k₁ and k₂.

reaction would proceed even without a catalyst present and reaction rate is directly proportional to the catalyst concentration.

j) Autocatalytic reactions

One of the products of reaction acts as a catalyst

Because the total number of moles of A and R remain unchanged as A is consumed

k) Irreversible reactions in series

: consecutive unimolecular-type first-order reaction

if k₂ is much larger than k₁ if k₁ is much larger than k₂

maximum concentration of R

l) First-order reversible reactions

m) Second-order reversible reactions

for the bimolecular-type second-order reactions

Example 3.1 : Find a Rate Equation using the Integral Method

First-order kinetics Second-order kinetics nth order kinetics

Trial 1 ) First-order kinetics

Trial 2 ) Second-order kinetics

Trial 3 ) n th-order kinetics

REMIND) Two Procedures for Analyzing Kinetic Data

1) Integral method

2) Differential method

In the integral method of analysis

we guess a particular form of rate equation and,

after appropriate integration and mathematical manipulation,

predict that the plot of a certain concentration function versus time should yield a straight line.

The data are plotted, and if a reasonably good straight line is obtained, then the rate equation is said to satisfactorily fit the data.

In the differential method of analysis

we test the fit of the rate expression to the data directly and without any integration.

However, since the rate expression is a differential equation,

we must first find the value of (l/V)(dNldt) from the data before attempting the fitting procedure.

The integral method is easy to use and is recommended when testing specific mechanisms, or relatively simple rate expressions

The differential method is useful in more complicated situations but requires more accurate or larger amounts of data.

In general, integral analysis is attempted first, and, if not successful, the differential method is tried.

2) Differential Method of Analysis of Experimental Data

Example : If the curve pattern (-r

versus t) follow the M-M equation

n th order kinetic : Solve it using differential method (without integration)

3.2 Varying- Volume Batch Reactor

For isothermal constant pressure &

single stoichiometric reaction system, the volume (V) is linearly related to the conversion(X)

= the fractional change in volume of the system between no conversion and complete conversion of reactant A Example )

3.3 Temperature and Reaction Rate

Arrhenius’ Law

[J/mol]

Activation Energy and Temperature Dependency

Natural logarithm scale

Example 3.4 : Correct and Incorrect E Values

1) Incorrect E value (using Pressure Units)

2) Correct E value (using Composition Units)

At 400 K :

At 500 K :