Determining order of reaction and rate constant.

Contents

Outcomes & Introduction

Differential method

Integral method

Half-life method

Isolation method

Problems

Outcomes

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.


The differential method

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 y-axis to find k.


The Integral Method

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 eqn

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

(n-1)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 Half-life Method

The half-life 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 half-life - suitably linearised. The table below summarises the appropriate linear plots

Table 2. Determination of kinetic parameters from half-life

Order

Rate eqn

Half-life

t =

Linear plot

 

slope

0

t vs [A]0

1

Half life is constant for first order reactions

2

t vs

n

log t vs log[A]0

-(n-1)

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.


Isolation method

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 pseudo-first 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.


Problems

1.

The reaction A B + C is found to be first-order 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 half-life of the first-order decay of radioactive 14C is about 5720 years.

 

 

 

a.

Calculate the rate constant for the reaction

 

b.

The natural abundance of 14C 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 half-life for the reaction

 

C2H5Cl C2H4 + HCl

 

is the same when the initial concentration of C2H5Cl 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 half-life increased from 2.0 min to 4.0 min at 25C. 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 second-order 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 half-life 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 half-life 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