Kinetics– study of reaction rates and mechanisms
Factors affecting rate:
- Nature of reaction (miscibilities, surface area, molecular structure)
- Reaction Rates
- Concentration Changes
Chemical kinetics is the study of the speed with which a chemical reaction occurs and the factors that affect this speed. This information is especially useful for determining how a reaction occurs.
What is meant by the speed of a reaction? The speed of a reaction is the rate at which the concentrations of reactants and products change.
Consider the following hypothetical example. The letters A, B, and C represent chemical species (in this context, the letters do not represent elements). Suppose the following imaginary reaction occurs:
A + 2 B → 3 C
The simulation below illustrates how this reaction can be studied. The apparatus at the left is called a stopped-flow apparatus. Each syringe contains a solution filled with a different reactant (A or B). When the two solutions are forced out of the syringes, they are quickly mixed in a mixing block and the reaction starts. The reacting solution passes through the tube at the bottom. An analytical technique such as spectrophotometry is used to measure the concentrations of the species in the reaction mixture (which is in the tube at the bottom) and how those concentrations change with time. All these things we will discuss below in methodology part.
The rate of change in the concentrations of the reactants and products can be used to characterize the rate of a chemical reaction. The rate of change in the concentration corresponds with the slope of the concentration-time plot.
First order reaction
A first order reaction is a reaction whose rate depends on the reactant concentration raised to the first power. In a first-order reaction of the type
The rate is rate=-∆ [A] /∆t
From the rate law we also know that
To obtain the units of k for this rate law, we write
k=rate/[A]=M/s/M=1/s or s-1
Combining the first two equations for the rate we get
Usings calculus, we can show from above equation that
Ln [A]t/[A]0=-kt ……………(A)
Where ln is the natural logarithm, and [A]0 and [A]t are the concentrations of A at times t=0 and t= t, respectively. It should be understand that t=0 need not correspond to the beginning of the experiment; it can be any time when we choose to start monitoring the change in the concentration of A.
Equation A can be rearranged as follows:
Ln [A]t =-kt+ ln [A]0 ……………(B)
Equation B has the form of the linear equation y=mx+b, in which m is the slope of the line that the graph of the equation:
y= mx+ c
A plot of ln[A]t versus t.The slope of the line is equal to -k.
Consider figure. As we would expect during the course of a reaction, the concentration of the reactant A decreases with time. For a first-order reaction, if we plot ln[A]t versus time (y versus x), we obtain a straight line with a slope equal to -k and a y intercept equal to ln[A]0. Thus, we can calculate the rate constant from the slope of this plot.
There are many first -order reactions. An example is the decomposition of ethane (C2 H6) into highly active fragments called methyl radicals (CH3):
The decomposition of N2 O5 is also a first-order reaction
2N2 O5(g)—à 4NO2(g) +O2(g)
As a reaction proceeds, the concentration of the reactants(s) decreases. Another measure of the rate of a reaction, relating concentration to time, is the half-life, t1/2, which is the time required for the concentration of a reactant to decreases to half of its initial concentration. We can obtain an expression for t1/2 for a first-order reaction as follows. Equation A can be rearranged to give
t = 1/k ln [[A]0/[A]t]
By the definition of half-life, when t=t1/2, [A]t=[A]0/2, so
t1/2 = 1/k ln[[A]0/([A]0/2)]
t1/2 = 1/k ln2=0.693/k …………….(1)
Equation 1 tells us that the half-life of a first-order reaction is independent of the initial concentration of the reactant. Measuring the half-life of a reaction is one way to determine the rate constant of a first order reaction.
The following analogy may be helpful for understanding equation1. The duration of a college undergraduate’s career, assuming he student does not take off, is four years. Thus, the half-life of his/her stay at the college is two years. This half-life is not affected by how many others students are present. Similarly, the half-life of a first order reaction is concentration independent.
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The usefulness of t1/2 is that it gives us an estimate of the magnitude of the rate constant – the shorter the half-life, the larger the k. Consider, for example, two radioactive isotopes used in nuclear medicine: 24Na(t1/2=14.7h) and 60Co(t1/2=5.3 yr). It is obvious that the 24Na isotopes decay faster because it has a shorter half-life. If we started with 1mole each of the isotopes, most of the 24Na would be gone in a week while the 60Co sample would be mostly intact.
Second -Order Reactions
Second-order reaction is a reaction whose rate depends on the concentration of one reactant raised to be second power or on the concentrations of two different reactants, each raised to the first power. The simpler type involves only one kind of reactant molecule:
Where rate = -∆ [A]/∆t
From the rate law,
Rate = k [A]2
As before, we can determine the units of k by writing
k = rate/[A]2=M/s/M2 =1/M.s
Another type of second-order reaction is
A + B ——à product
And the rate law is given by
Rate =k [A][B]
The reaction is first order in A and first order in B, so it has an overall reaction order of 2.
Using calculus, we can obtain the following expressions for “A–à product” second-order reactions:
1/[A]t =kt +1/[A]0 ………………… (2)
Equation has the form of a linear equation. As figure shows a plot of 1/[A]t versus t gives a straight line with slope =k and y intercept=1/[A]0.
Half -life for second order reaction
We can obtain an equation for the half-life of a second-order reaction by setting [A]t=[A]0/2 in equation 2
1/[A]0/2 =kt1/2 + 1/[A]0
Solving for t1/2 we obtain
Note that the half-life of a second-order reaction is inversly proportional to the initial reactant concentration. This result makes sense because the half-life should be shorter in early stage of the reaction when more reactant molecules are present to collide with each other. Measuring the half-lives at different initial concentrations is one way to distinguish between a first-order and a second order reaction.
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Half lives and life times of reaction
A rate constant is a quantitative measure of how fast reactions proceed and therefore is an indicator of how long a given set of reactants will survive in the atmosphere under a particular set of reactant concentrations. However, reaction rate constant is not a parameter which by itself is readily related to the average length of the time a species will survive in atmosphere before reacting. A more intuitively meaningful parameter is half life timeT1/2 or the natural life time t of a pollutant with respect to reaction.
The half life time is defined as time required for the concentration of a reactant to fall to on half of its initial value where as life time is defined as the time it takes for reactant concentration to fall to ½ of its initial value. Both T1/2 and t are directly related to the reaction rate constant and to the concentration. The relationship is given in table:
Relationship between reaction rate constant and half lives for first, second order reactions
Examples of homogeneous gaseous first order reactions are the thermal dissociations of nitrous oxide, nitrogen pentoxide, acetone, propionic aldehyde, various aliphatic ethers, azo compounds, amines, and ethyl bromide, and the isomerization of d-pinene to dipentene. As typical of these may be taken the thermal decomposition of azoisopropane to hexane and nitrogen
(CH3)2CHN=NCH (CH3)2——-à N2 +C6 H14
Which was investigated by Ramsperger over the pressure range 0.0025 to 46 mm Hg and between 250 and 290 0C.? The rate of reaction was followed by pressure measurement with a McLeod gage. The only data necessary were the initial pressure of the reactant and the total pressure of the system at various stages of decomposition.
Examples of second order reactions
Homogeneous gas reactions of second order are very common. In this category are included various thermal dissociations, suh as those of hydrogen iodide, nitrogen di oxide, ozone , chloride monoxide, nitrosyl chloride , formaldehyde, and acetaldehyde; the combination of hydrogen and iodine to form hydrogen iodide; the polymerization of ethylene ; the hydrogenation of ethylene; and others. The behavior of all such reactions is exemplified by the thermal decomposition of acetaldehyde. This reaction, investigated by Hinshelwood and Hutchison, was found to be almost totally homogeneous and to proceed according to the equation
2CH3 CHO—à 2CH4 + 2CO
Since in the reaction there is an increase in pressure at constant volume on dissociation, the change in pressure observed on a manometer attached to the system may be employed to follow the reaction course. From these pressure measurements K2 is calculated.
Nature of first and second order of reaction
The first order nature of radioactive decay processes has been used for finding the age of objects that contain carbon. The concentration of 14C in the atmosphere has remained constant over the ages because of equilibrium between its decay and its formation from nitrogen through interaction with cosmic rays. Any living body that assimilates carbon di oxide from the atmosphere will therefore have 14C and 12 C in the ratio in which it is found in the atmosphere. This ratio has remained constant in the atmosphere. So the amount of 14C in a given weight of the body that is alive today is the same as the amount of 14C that would have been present in the same weight of the body at the time it lost its capability of assimilating carbon di oxide from the atmosphere. By analyzing a known weight of the living material, we will therefore know the amount (N0) of 14C that the dead material had at of its death. Analysis of the dead material at present will give the 14C content (N) as of date. These are related by equation t1/2=0.693//.
N=N0 exp (-0.693*t/ t1/2 )
Since t1/2, the half life of 14C is known, t the time that has elapsed since the material lost its capability of assimilating carbon di oxide from the atmosphere can be determined.
The protons and neutrons in the nucleus are crowded together into a very small sphere of dense matter. This involves attractive forces amounting to thousands of kilojoules per bond.So ordinary external forces do not affect nuclear behavior. Concentration, pressure and temperature do not affect the decay constant significantly.
Sanitary engineers find application for the cocepts of first order reactions in areas that do not involve decomposition reactions. For examples, the dissolution of oxygen in water under a given set of conditions is a first order reaction. The rate of death of microorganisms by disinfection is considered to be a first order reaction dependent upon the concentration of microorganisms remaining. The very complex reaction involved in the decomposition of organic matter by bacteria in the Biochemical Oxygen Demand (BOD) test is normally considered to be first order reaction dependent only upon the concentration of organic matter remaining.
First order reaction are analogous to physical process such as cooling of hot bodies according to Newtons’s law of cooling, or the loss of charge of an electrified body.
Oscillations given by a flexible spring decrease systematically and the decay in the amplitude of the movements can be described in mathematical terms analogous to those used for a first order chemical reaction. There are certain growth processes which follow similar behavior. Examples are the multiplication of bacteria, the growth of a crystal from a solution and the increase in the thickness of certain oxide films on the surface of metals. The law of oganic growth is such a way that the quantum by which the organism increases in size is proportional to the size of the organism itself. Hence, if we look at the growth and compare it with the decay, while in a given time, the amount is proportional to the size itself , for equal intervals of time , the size of the steps will not be equal .
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4) DISCUSSION AND SUMMARY:
Chemical kinetics is the branch of chemistry which addresses the question: “how fast do reactions go?” Chemistry can be thought of, at the simplest level, as the science that concerns itself with making new substances from other substances. Or, one could say, chemistry is taking molecules apart and putting the atoms and fragments back together to form new molecules. (OK, so once in a while one uses atoms or gets atoms, but that doesn’t change the argument.) All of this is to say that chemical reactions are the core of chemistry.
If Chemistry is making new substances out of old substances (i.e., chemical reactions), then there are two basic questions that must be answered:
- Does the reaction want to go? This is the subject of chemical thermodynamics.
- If the reaction wants to go, how fast will it go? This is the subject of chemical kinetics.
Differential Rate Laws
In many reactions, the rate of reaction changes as the reaction progresses. Initially the rate of reaction is relatively large, while at very long times the rate of reaction decreases to zero (at which point the reaction is complete). In order to characterize the kinetic behavior of a reaction, it is desirable to determine how the rate of reaction varies as the reaction progresses.
A rate law is a mathematical equation that describes the progress of the reaction. In general, rate laws must be determined experimentally. Unless a reaction is an elementary reaction, it is not possible to predict the rate law from the overall chemical equation. There are two forms of a rate law for chemical kinetics: the differential rate law and the integrated rate law.
The differential rate law relates the rate of reaction to the concentrations of the various species in the system.
Differential rate laws can take on many different forms, especially for complicated chemical reactions. However, most chemical reactions obey one of three differential rate laws. Each rate law contains a constant, k, called the rate constant. The units for the rate constant depend upon the rate law, because the rate always has units of mole L-1 sec-1 and the concentration always has units of mole L-1.
For a first-order reaction, the rate of reaction is directly proportional to the concentration of one of the reactants.
Differential Rate Law: r = k [A]
The rate constant, k, has units of sec-1.
For a second-order reaction, the rate of reaction is directly proportional to the square of the concentration of one of the reactants.
Differential Rate Law: r = k [A]2
The rate constant, k, has units of L mole-1 sec-1.
For a first-order reaction, the rate of reaction is directly proportional to the concentration. As the reactant is consumed during the reaction, the concentration drops and so does the rate of reaction.
For a second-order reaction, the rate of reaction increases with the square of the concentration, producing an upward curving line in the rate-concentration plot. For this type of reaction, the rate of reaction decreases rapidly (faster than linearly) as the concentration of the reactant decreases.
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