# 5. Collection and Analysis of Rate Data*

## Topics

1. Integral Method
2. Differential Method
3. Non-Linear Least-Squares Analysis

#### Finding the Rate Law

Consider the following reaction that occurs in a constant volume batch reactor: (We will withdraw samples and record the concentration of A as a function of time.)

 Mole Balance: Rate Law: Stoichiometry: Combine:

 Integral Method top

We could integrate the combined mole balance and rate law to plot reaction rate data in terms of concentration vs. time for 0, 1st, and 2nd order reactions.

Table CD5-1- Derivation Equations used to Plot 0, 1st, and 2nd order reactions.

These types of plots are usually used to determine the values k for runs at various temperatures and then used to determine the activation energy.

Table CD 5-1
 Zero Order First Order Second Order

If the data do not fall on a straight line for α=0,1, or 2 such as α=2;

then we should stop guessing reaction orders and proceed to either the differential method of analysis or to regression.

 Differential Method (p. 224-232) top
 Taking the natural log of The reaction order can be found from a ln-ln plot of:

Methods for finding the slope of log-log and semi-log graph papers may be found at http://www.physics.uoguelph.ca/tutorials/GLP/.

However, we are usually given concentration as a function of time from batch reactor experiments.

 time (s) 0 t1 t2 t3 concentration (mol/dm3) CAo CA1 CA2 CA3

#### Three Ways to Determine (-dCA/dt) from Concentration-Time Data (Graphical,    Polynomial, Finite Difference, Non-Linear Least Squares Analysis)

2A. Graphical

This method accentuates measurement error!

2B. Polynomial (using Polymath)

CA = ao + a1t + a2t2 + a3t3 +a4t4

2C. Finite Difference

 Non-Linear Least-Squares Analysis (p. 252) top

We want to find the parameter values (alpha, k, E) for which the sum of the squares of the differences, the measured parameter (Pm), and the calculated parameter (Pc) is a minimum.

That is we want to be a minimum.

 time (s) 0 t1 t2 t3 concentration (mol/dm3) CAo CA1 CA2 CA3

For concentration-time data, the measured parameter P has concentration CA. We can integrate the combined mole balance equation and rate law

to obtain

We now guess k and alpha and calculate each CACi at the times shown in the above table and then compare it with the measured concentration by taking the difference and squaring it.

We then sum up the differences for all the data points.

We continue to guess k and alpha until we find the values of alpha and k which minimize S2 (actually we let the computer find these values.)

"What is Wrong with This Solution?"
Object Assessment of Chapter 5
Polymath regression analysis tutorial
Polymath non-linear regression analysis tutorial
Polymath non-linear equation Solver Tutorial
Excel non-linear regression analysis tutorial

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