CHAPTER Five: Collection & Analysis of Rate Data In Chapter 4 we have shown that once rate law is known, it can be substituted into appropriate design equation, & though use of appropriate stoichiometric relationships, apply CRE algorithm to size any isothermal reaction system.

Focus on ways of obtaining & analyzing reaction rate data to obtain rate law for a specific reaction.

In particular, discuss two common types of reactors for obtaining rate data:

batch reactor, which is used primarily for homogeneous reactions, & differential reactor, which is used for solid-fluid heterogeneous reactions.

In batch reactor experiments, C, P, & or V are usually measured & recorded at different times during course of reaction.

Data are collected from batch reactor during transient operation, whereas measurements on differential reactor are made during steady-state operation.

In experiments with a differential reactor, product concentration is usually monitored for different sets of feed conditions.

Two techniques of data acquisition are presented: concentration-time measurements in a batch reactor & concentration measurements in a differential reactor.

Six different methods of analyzing data collected are used: differential method, integral method, method of half-lives, method of initial rates, linear, & nonlinear regression (least-squares analysis).

Differential & integral methods are used primarily in analyzing batch reactor data. Because a number of software packages (e.g., Polymath, MATLAB) are now available to analyze data, a rather extensive discussion of nonlinear regression is included.

5.1 Algorithm for Data Analysis For batch systems, usual procedure is to collect concentration time data, which then use to determine rate law. Table 5-1 gives procedure we will emphasize in analyzing reaction engineering data.

Data for homogeneous reactions is most often obtained in a batch reactor. After postulating a rate law & combining it with a mole balance, next use any or all of methods in Step 5 to process data & arrive at reaction orders & specific reaction rate constants.

Analysis of heterogeneous reactions is shown in Step 6 Analysis of heterogeneous reactions is shown in Step 6. For gas-solid heterogeneous reactions, need to have an understanding of reaction & possible mechanisms in order to postulate rate law in Step 6B. The procedure we should use to delineate rate law & rate law parameter is given in Table 5-1.

5.2 Batch Reactor Data Batch reactors are used primarily to determine rate law parameters for homogeneous reactions. This determination is usually achieved by measuring concentration as a function of time & then using either differential, integral or nonlinear regression method of data analysis to determine reaction order, α, & specific reaction rate constant, k.

When a reaction is irreversible, it is possible in many cases to determine reaction order a & specific rate constant by either nonlinear regression or by numerically differentiating concentration versus time data.

This later method is most applicable when reaction conditions are such that rate is essentially a function of concentration of only one reactant; for example if, for decomposition reaction, Then differential method may be used.

However, by utilizing method of excess, it is also possible to determine relationship between –rA , & concentration of other reactant. That is, for irreversible reaction

5.2.1 Differential Method of Analysis To outline procedure used in differential method of analysis, consider a reaction carried out isothermally in a constant-volume batch reactor & concentration recorded as a function of time.

By combining mole balance with rate law given by Eq. (5-1), obtain

To obtain derivative –dCA/dt used in this plot, differentiate concentration-time data either numerically or graphically. Describe three methods to determine derivative from data giving concentration as a function of time.

These methods are: Graphical differentiation Numerical differentiation formulas Differentiation of a polynomial fit to data

5.2.1A Graphical Method

5.2.1B Numerical Method

5.2.1C Polynomial Fit

5.2.1.D Finding the Rate Law Parameters

Example 5-1 Determining Rate Law

5.2.2 Integral Method To determine reaction order by integral method, guess reaction order & integrate differential equation used to model batch system.

If order we assume is correct, appropriate plot (determined from this integration) of concentration-time data should be linear.

Integral method is used most often when reaction order is known & it is desired to evaluate specific reaction rate constants at different temperatures to determine activation energy.

In integral method of analysis of rate data, looking for appropriate function of concentration corresponding to a particular rate law that is linear with time.

You should be thoroughly familiar with methods of obtaining these linear plots for reactions of zero, first, & second order.

It is important to restate that, given a reaction rate law, you should be able to choose quickly appropriate function of concentration or conversion that yields a straight line when plotted against time or space time.

5.2.3 Nonlinear Regression In nonlinear regression analysis, we search for those parameter values that minimize sum of the squares of differences between measured values & calculated values for all data points.

Many software programs are available to find these parameter values so that all one has to do is enter data.

Polymath software will be used to illustrate this technique.

In order to carry out search efficiently, in some cases one has to enter initial estimates of parameter values close to actual values.

A number of software packages are available to carry out procedure to determine best estimates of parameter values & corresponding confidence limits.

All one has to do is to type experimental values in computer, specify model, enter initial guesses of parameters, & then push computer button, & best estimates of parameter values along with 95% confidence limits appear.

Concentration-Time Data Concentration-Time Data. We will now use nonlinear regression to determine rate law parameters from concentration-time data obtained in batch experiments. We recall that combined rate law-stoichiometry-mole balance for a constant-volume batch reactor is

5.4 Method of Half-lives The half-life of a reaction, t1/2, is defined as time it takes for concentration of reactant to fall to half of its initial value.

By determining half-life of a reaction as a function of initial concentration, reaction order & specific reaction rate can be determined.

if two reactants are involved in chemical reaction, experimenter will use method excess in conjunction with method of half-lives to arrange rate law in form For the irreversible reaction

There is nothing special but using time required for concentration to drop to one-half of its initial value. We could just as well use time required for concentration to fall to 1/n of initial value, in which case