Determining order of reaction and rate constant.
Contents
On completion of this section, you should be able to
Apply rate equations to given systems
Describe methods of determining reaction order
Introduction
The rate equations and various forms derived from them can be used in a variety of ways both to determine the order of reaction and the rate constant. A common method of following the progress of a reaction is to monitor either the concentration of a reactant or of a property proportional to concentration, such as conductivity, visible/UV light absorption or fluorescence. Such a process will lead to a curve typically of the form
The rate of reaction is the gradient of the tangent of the line representing concentration vs time. This leads us to two methods of determining the properties of a reaction.
For a reaction of the form

nA ® Products 

Assuming the reaction is of order, n
The rate equation may be expressed as


Taking logs gives


Which is the equation of a straight line, slope n & intercept log k.
Thus by taking tangents at various points on the graph and plotting log d[A]/dt vs [A] values of k and n may be found.
This method is useful as a single graph gives the order of the reaction, but the process of taking tangents to a line is liable to error as is the extrapolation of the graph to the yaxis to find k.
The differential method relies on taking tangents to the curve representing concentration vs time. The graph of the tangents represents the rate equation. Hence by integrating the rate equation, we get a function which represents concentration vs time. The kinetic properties of the equation may then be deduced from this graph. To determine order and the rate constant it is convenient to linearise the integrated rate equation. For the simpler rate equations, the appropriate linear plots are summarised below.
Table 1. Determination of kinetic parameters from concentration vs time data
Order 
Rate eq^{n} 
Integrated form 
Linear plot 
slope 
intercept 
0 
[A] vs t 
k 
[A]_{0} 

1 
ln[A] vs t 
k 
ln[A]_{0} 

2 
vs t 
k 

n 
vs t 
(n1)k 
Click here for the derivation of the integrated rate equations.
The integral method is more accurate than the differential method as the kinetic parameters are determined directly from a relationship between time and a function of concentration. In addition, it is not necessary to measure concentration as such. A property such as as conductivity, visible/UV light absorption or fluorescence will suffice. On the other hand, to determine order requires that a succession of graphs must be plotted until a linear relationship is found.
The halflife method involves measuring the time taken to reduce the concentration to half its initial value for a variety of values of initial concentration, then plotting a graph of concentration vs halflife  suitably linearised. The table below summarises the appropriate linear plots
Table 2. Determination of kinetic parameters from halflife
Order 
Rate eq^{n} 
Halflife t_{½} = 
Linear plot

slope 
0 
t_{½} vs [A]_{0} 

1 
Half life is constant for first order reactions 

2 
t_{½} vs 

n 
log t_{½} vs log[A]_{0} 
(n1) 
The method for reactions of order n is a general one. The value of the rate constant, k may be found from the intercept, which is equal to log
The half life method combines some of the advantages of the differential method; order may be found from a single plot but, as with the integral method; there is no need to determine tangents. For reactions of order zero, one and two, the rate constant is best determined as indicated in table 2.
The three methods described so far are really best suited to reactions with one reactant. If more than one reactant is involved, the integrated rate equation quickly becomes very complex and a different procedure is involved.
The isolation method involves arranging for all but one of the reactants to be in such large excess, that their concentrations are essentially constant during the progress of the reaction. In practice this means an excess of at least 10[A]_{0} and preferably >100[A]_{0}. ( [A]_{0} represents the concentration of the limiting reactant)
As an example, let us consider a second order reaction of the form

A + B ® Products 

For such a reaction, the second order rate equation would be


If [B] were in large excess, the rate equation would simplify to


The rate equation is thus, simplified to a first order equation. This is called a pseudofirst order reaction. The pseudo first order rate constant is defined from


The second order rate constant may be found by determining the value of k’ for various values of [B]_{0} and plotting a graph of k’ vs [B]_{0}. This method may be extended to reactions of order higher than two and with more than two reactants.
1. 
The reaction A ® B + C is found to be firstorder in A. If half of the starting quantity of A is used up after 56 s, calculate the fraction that will be used up after 6.0 min.


2. 
The halflife of the firstorder decay of radioactive ^{14}C is about 5720 years. 





a. 
Calculate the rate constant for the reaction 


b. 
The natural abundance of ^{14}C isotope is 1.1 x 10^{11} mol % in living matter. Radiochemical analysis of an object obtained in an archaeological excavation shows that the "C isotope content is 0.89 x 10^{14} Mol %. Calculate the age of the object.


3. 
If the halflife for the reaction 


C_{2}H_{5}Cl ® C_{2}H_{4} + HCl 


is the same when the initial concentration of C_{2}H_{5}Cl is 0.0050 M and 0.0078 M. What is the rate law?

4. 
When the concentration of A in the reaction A ® B was changed from 1.20 M to 0.60 M, the halflife increased from 2.0 min to 4.0 min at 25°C. Calculate the order of the reaction and the rate constant.

5. 
The progress of a reaction in the aqueous phase was monitored by following the absorbance of a reactant at various times: 

Time/s 
0 
54 
171 
390 
720 
1010 
1190 

Absorbance 
1.67 
1.51 
1.24 
0.847 
0.478 
0.301 
0.216 

Make appropriate plots of these data to test them for fitting zero, first, and secondorder rate laws. Test all three even if you happen to guess the correct rate law on the first trial.

6. 
For the reaction A ® products, the following data were obtained. 

Time (hrs) 
[A], M 
Time (hrs) 
[A], M 

0 
1.24 
6 
0.442 

1 
0.960 
7 
0.402 

2 
0.775 
8 
0.365 

3 
0.655 
9 
0.335 

4 
0.560 
10 
0.310 

5 
0.502 



(a) 
Determine the order of the reaction and the rate constant. 

(b) 
Determine the rate constant for the reaction. 

(c) 
Using the rate law that you have determined, calculate the halflife for the reaction. 

(d) 
At what time will the concentration of A be 0.380 M? 2.

7. 
For the reaction X ® Y, the following data were obtained 

Time (min) 
[X], M 
Time (min) 
[X], M 

0 
0.500 
60 
0.240 

10 
0.443 
70 
0.212 

20 
0.395 
80 
0.190 

30 
0.348 
90 
0.171 

40 
0.310 
100 
0.164 

50 
0.274 



(a) 
Make appropriate plots of these data to determine the reaction order. 

(b) 
Determine the rate constant for the reaction. 

(c) 
Using the rate law you have determined, calculate the halflife for the reaction. 

(d) 
Calculate how long it will take for the concentration of X to be 0.330 M. 
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Produced
by Geoff Walker
Last Modified November 29, 1998