# Chemical Kinetics First And Second Order Reaction Biology Essay

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'The branch of physical chemistry which deals with the study of the rates of the reaction, the factors which governs these rates and the mechanism by which the reaction proceed is called chemical kinetics.'

The rates of chemical reactions form the subject matter of chemical kinetics .Experimentally it is found that the rate of chemical reaction is dependent on the temperature, pressure and the concentrations of the species involved. The presence of a catalyst or inhibitor can change the rate by many powers of ten. From the study of the rate of the reaction and its dependence on all these factors, much can be learned about the detailed steps by which the reactants are transformed to products .It not only include factors affecting it or steps involved but also many other topics like rat law, stiochiometry of a reaction, order of reaction, molecularity, mechanism involved.

The reactions are divided into three categories:

Those reactions which take place almost instantaneously,

Those reactions which takes place at extremely slow pace,

Those reactions which takes place at measurable speed.

Figure 1

Basic chemical kinetics

The birth of chemical kinetics often is taken to have occurred in1850, when the German chemist Ludwig Ferdinand Wilhelmy (1812-1864) studied the rate of inversion of sucrose. The pioneering work is of special significance as being the first in which a quantitative approach was made to reaction rate. He interpreted the course of the reaction by the use of differential equation to express the temperature dependence of the rate. He gave 'The Law by Which the Action of Acids on Cane Sugar Occurs'

The early chemists were largely concerned with discovering new substances and not so much with interpreting chemical behaviour. It was only in the second half of the 19th century that the physical methods began to be applied to chemical problems and that investigations were carried out of science now known as physical chemistry.

A collaboration carried out between a chemist and a mathematician although independently was far more successful as far as chemical kinetics is concerned. It was during the years 1865 - 1867 that Augustus George Vernon Harcourt, who carried out an experimental investigation on the reaction between hydrogen peroxide and hydrogen iodide and between potassium permanganate and oxalic acid, paying more attention to the reactant concentration of the rate. The result was analysed mathematically, in terms of the integrated forms of differential equations, by William Esson (1839-1916), who's procedure were very similar to those that are used today.

Figure 2

William Esson

Harcourt and Esson paid no attention to the then very popular but nebulous "chemical affinity" and were not concerned about the equilibrium states; this was probably fortunate, since at the time these questions tended to confuse the kinetic problem.

Scope of Chemical Kinetics

Chemical Kinetics deals with the rates of chemical reactions and with how the rate depends on the factors such as concentration and temperature. Valuable evidence about mechanisms of reaction can be satisfactorily detected only after a careful kinetic investigation has been carried out.

A Kinetic study can disprove a mechanism but it cannot establish a mechanism with certainty.

The word "kinetics" originated from a Greek word "kinetikos" that, in turn, originated from Greek "kinetos" which means "moving".

Chemical kinetics is a branch of chemistry which is concerned with the rate of change in the concentration of reactants in a chemical reaction.

Chemical kinetics

Kinetics is the study of the rates of chemical processes in an effort to understand what it is that influences these rates and to develop theories which can be used to predict them. A knowledge of reaction rates has many practical applications, for example in designing an industrial process, in understanding the complex dynamics of the atmosphere and in understanding the intricate interplay of the chemical reactions that are the basis of life.

At a more fundamental level we want to understand what happens to the molecules in a chemical reaction - that is what happens in a single reactive encounter between two reagent molecules. By understanding this we may be able to develop theories that can be used to predict the outcome and rate of reactions.

â€¢Macroscopic kinetics describes the branch of kinetics, which results relate to the behaviour of a very large group of molecules in thermal equilibrium.

â€¢Microscopic kinetics is to investigate the molecules in well-defined states, which will provide information about the dynamic of both reactive and unreactive collisions.

"Why the speed of different reactions are different"

A reaction involves the breaking and making of the bonds. Since different bonds require different amount of energy for breakage and different amount of energies are evolved when different kind of new bonds are formed, the rates of different reactions are different. The instantaneous nature of ionic reactions is due to the fact that the se do not involve any breaking of bonds( as ions are already present in the solution).

Rate of reaction

The rate is defined as change in concentration, in time t

We can talk about the rate of formation or loss of any species - reactant, intermediate or product. It is, however, important to specify which species we are talking about. The rate can be positive or negative: a positive rate means that the concentration is increasing with time e.g. a product; a negative rate means that the concentration is falling with time e.g. a reactant.

The rate may vary with time (and concentration), so it is usual to define the rate over a very small time, dt. We think of the rate as the derivative of concentration with respect to time.

Figure 3

From its definition it is clear that the units of a rate are concentration per unit time, for example mol dm-3 s-1. There are other measures of concentration, for example in the gas phase pressure is proportional to concentration, so a rate can be expressed in torr min-1 (1 torr = 1 mm Hg, a

measure of pressure). It is also common to express concentration not in moles per unit volume but in molecules per unit volume, so the rate would be expressed in molecules dm-3 s-1 or molecules cm-3 s-1.

Symbolically, concentration is often indicated by square brackets. So [A] means the concentration of species A, and [Br2] means the concentration of bromine, and so on.

Figure 4

The rate of a chemical reactionÂ is the amount of substance reacted or produced per unit time. The rate law is an expression indicating how the rateÂ depends on the concentrations of the reactants. The power of concentrationÂ in the rate law expression is called theÂ orderÂ with respect to the reactant or catalyst.

Factors affecting the rate of reaction

Nature of the Reactants,

Concentration of the reactants,

Temperature of the reactants,

Presence of catalyst,

Nature of solvent,

Presence of catalyst.

Figure 5

Figure 6

Figure 7

Reaction stiochiometry

Consider the reaction whose stiochiometric equation is

The stoichiometric equation shows how the number of moles of reactants and products are related; it must be balanced.

This equation says that to form two moles of water, one mole of oxygen and two moles of hydrogen must react. It follows that the rate of consumption of hydrogen is twice that of oxygen.

Figure 8

The rate of formation of water is twice the rate of loss of oxygen, as two moles of water are formed from one mole of oxygen. As water is the product, the rate of change of its concentration is positive i.e. the concentration is increasing with time; however, the rates of change of the concentrations of both hydrogen and oxygen are negative as these concentrations are decreasing with time. In derivative form:

Rate =

With all these different "rates", different for each species, how can we define the "rate of the reaction"?

The usual procedure is to include the stoichiometric coefficients in the definition of the rate. These coefficients are the numbers in front of the species in the balanced chemical equation. So the stoichiometric coefficient is 1 for oxygen and 2 for both hydrogen and water.

The rate of reaction, r, is defined for any species A as

where is the stoichiometric coefficient of species A in the balanced chemical equation. In addition the stoichiometric coefficients are given negative signs for reagents and positive signs for products. This definition ensures that the rate is always positive and the same for a given reaction no

matter which species is considered.

Figure 9

Rate laws and rate constants

Experimentally it is found that depend on the concentration of the species involved in the reaction equation (and sometimes on the concentrations of species which do not appear at first sight to be involved!). The relation between the rate and these concentrations can often be expressed mathematically in the form of an equation called a rate law.

Some rate laws are very simple and some are very complicated. A rate law may be determined experimentally (Section 4) or may be the result of a theoretical prediction, or both.

Often reactions are found to have rate laws of the form

where k is the rate constant or rate coefficient.

Rate Law

â€¢One objective of chemical kinetics is to establish a relationship between the rate of reaction and the concentration of the reactants -this relationship is called the rate law, or rate equation â€¢For the general reaction,

aA+ bB â†’gG+ hH,

we can write

rate of reaction

in which [A] and [B] represent reactant molar concentrations and the exponents m and n are generally small, positive integers, but may be zero, fractional and/or negative .Note: The product are not involved in the equation since the reaction is not reversible (forward arrow only in the balanced equation).

Figure 10

Importance of Rate Law

If we know the rate law and the constants in it we can use this to predict the rate for any set of conditions (concentrations). The rate law is thus a very succinct and practical way of expressing the rate. You might use this, for example, in a model of the atmosphere or in predicting the rate of an enzyme catalysed reaction.

The form of the rate law can tell us something about the mechanism of the reaction. This is a point which we will consider in more detail below.

Knowing the rate law enables us to separate the concentration dependence from the underlying, fundamental effect which is the size of the rate constant.

Theories about reaction rates

We would like to be able to develop a theory which will enable us to calculate - almost certainly with the aid of a computer - the rate constant of any reaction we care to specify. To develop such a theory will be first and foremost a test of our understanding of the fundamental processes involved.

If the theory produces predictions which are in accord with experiment, then we may have some faith in the theory.

Then, if the theory proves to be successful and our calculations reliable, we could go on to use the theory to predict the rates of unknown or little studied reactions. We might want to do this to avoid experiments, which are not always easy or possible.

It turns out that although the theory is now quite well understood the

calculations needed to predict values of rate constants are very challenging.

With the currently available super-computers it is feasible to calculate rates for simple gas phase reactions (e.g. ). Reactions in solution present an even greater challenge as we have to consider the role of a large number of solvent molecules.

Even though we will not be able to actually make any calculations of

rate constants ourselves, a great deal of insight into chemical reactions is obtained by studying the model on which the theory is based and looking at the general features we expect from such a model. This will lead to an interpretation of why rate laws have the form they do, the factors that influence the size of the rate constant, and how the Arrhenius equation arises.

Order of reaction

Figure 11

: In order to understand what do you mean by the order of a reaction, consider the general equation

It has been observed experimentally that the rate of this reaction may not depend upon on all a concentration of A and the B concentration of B. Now let us suppose that the rate of reaction is found to depend upon the Î± concentration terms of A and Î² concentration terms of B.

Where [A] and [B] are the molar concentration of A and B respectively and k is the rate constant or velocity constant. If the concentrations are taken to be unity

[A]+ [B] = 1 mol/L

Rate= k

Figure 12

Hence the rate constant is defined as the rate of reaction when the concentration of each reactant is taken unity.

K is also called the specific rate constant.

The sum of concentration terms on which the rate of reaction actually depends as observed experimentally is called the order of reaction

Hence the order of reaction may also be defined as the sum of the exponents to which the concentration terms in the rate law is raised to express the observed rate of reaction.

Depending on the value of Î±+Î²= 0,1,2 or 3, the reactions are said to be zero, 1st order,2nd order 3rd order respectively.

First order Reactions

Consider a simple reaction of the type

Figure 13

Figure 14

Since substance A is the only reactant, we choose to balance the equation with the coefficient of equal to unity. Suppose that the reaction is of first order with respect to A and that the rate does not depend on the concentration of any product, then the rate law becomes

Integrate this equation at; when t=0 then

when t=t then

Thus for a first order reaction/ decomposition, the concentration of A decreases exponentially with the time. After measuring as a function of t, we can test whether the reaction is first order by plotting a graph between versus t. According to the equation 2, this plot should b straight line if the reaction is first order in A. If we find that our experimental points lie on a straight line we conclude that the reaction is a first order reaction with respect to A. The slope of the line is -k.

The half life , of the reaction is the time required for the concentration of A to reach one-half of its initial concentration of its initial value .Therefore,

When t=, Putting these values in equation 2

One way to evaluate the rate constant of a reaction is to determine the half-life for various initial concentrations of the reactant A. If the half life is independent of initial concentration, then the reaction is first order, and the rate constant is calculated using equation 4. It is only for first order reactions that the half-life is independent of initial concentration.

**The decomposition of is an example of a first order reaction. The stiochiometry is represented by

Figure 15

Figure 16

The rate law is

At the rate constant is 3.38 x . Note the absence of any relation between the order of reaction and the stiochiometric coefficient of in the chemical equation.

**Radioactive Decay

The radioactive decay is an unstable nucleus, which is an important example of a process that follows a first order rate law. If we choose as an example , we have the trans formation

=12.8hrs

The emission of a Î²-particle occurs with the formation of a stable isotope of zinc. The probability of this occurrence in the time interval dt, is simply proportional to dt. Therefore

When -dN is the number of copper nuclei that disintegrate in the interval dt. The given rate equation is a first order law, and can be integrated to the form

2

being the number of nuclei presented at t=0, N the number at any time t. The constant Î»is the decay constant and is related to the same half-life by

In contrast to the rate constant of a chemical reaction, the decay constant Î» is completely independent of any external influence such as temperature or pressure using the value of Î» from 3 in 2, we obtain[since the value of exp(ln2)=2]

It is clear that after the elapse of a period equals to two half-life's, of the substance remains. After three half-life's have elapsed, remains, after four half-life's, and so on.

**Bacterial growth

A bacterial colony grows most commonly by cell division. In an actively grown colony the probability of cell division in a time interval dt is proportional to dN; thus

Where dN is the number of cells that that divide in the time interval dt, and is a constant. The growth law is similar to the law of radioactive decay in equation , except that the negative sign is missing. Upon integrating we obtain

The generation time , is the time required for the population to double; that is, when ,;equation 2 becomes

The growth law, is not applicable during the entire history of the bacterial colony. A typical population curve, N versus t. There is an initial induction period, followed by a period between , during which the exponential growth occurs, as described. The population growth slows, then stops; in the final stage the population drops as the bacteria dies more rapidly than they are produced.

Figure 17

Equation 3 describes the growth only during the exponential phase in the interval from .The levelling off occurs as the supply of the nutrients is exhausted. Finally, if the environment becomes sufficiently hostile (due to lack of nutrients or increased concentration of toxic substances),the colony dies.

Figure 18

Second order reactions

**With one reactant

We return to the decomposition reaction,

## ,

Figure 19

Figure 20

But now assume that the reaction is second order. If [A] is the concentration of A at any time, the rate law is

Separating variables, we have

Integrating the given equation; when t=0, then [A] =

when t=t, then [A] =

This is the integrated rate law for the second order reaction. To discover whether the reaction is second order, we test the data by plotting versus t. Equation 3 requires that this plot be linear. If the data fall on straight line, this is evidence that the reaction n is second order. The slope of the reaction is equal to the rate constant.

The half-life is defined as the time required in which the concentration of the reaction becomes half the initial value. When t=, then . Using these values in 3, we obtain

For a second order reaction, the half-life depends on the initial concentration of reactant. If the in initial concentration is doubled, the time required for half of A to react will be reduced by one-half. If the half life for various initial concentrations is plotted againstthe rate constant is the reciprocal of the slope of the graph.