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## Theory of Unimolecular Reactions, Chemistry tutorial

Introduction:The elementary reaction in which just one molecule or a radical reacts is termed as the Unimolecular reaction.

Theory of Unimolecular Reactions:The Unimolecular reaction is an elementary reaction in which just one molecule or a radical reacts. The Unimolecular reactions obey first order kinetics. A number of gas phase reactions obey first order kinetics. Such reactions are supposed to carry on via Unimolecular rate-determining step. Though how does the reactant molecule acquire the activation energy? The activation energy is the minimum energy required for the reactant molecules to react and result products. If the molecules acquire their activation energy via collisions, it is hard to describe first order kinetics. A collision procedure requires at least two molecules and therefore, second order kinetics could be expected however not first order kinetics. In the year 1992, Lindemann and Hinshelwood introduced a mechanism which could describe the Unimolecular reactions in which molecules acquire their activation energy via collision.

Suppose two molecules of the reactant gas (X) collide resulting an activate molecule (X*) and a normal molecule (X). Such a collision is known as an activating collision.

X + X → (K

_{a}) → X* + XRate of activation of X = k

_{a}[X][X]The activated molecule, X*, can experience either of the given reactions:

a) X* can experience collision by the other molecule X and lose its surplus energy. Such kind of collision is known as a deactivating collision.

X* + X → (K'

_{a}) → X + XRate of decay of X* = k'a [X*][X]

b) On the other hand, if X* can decay to result the product, Y.

X* → (K

_{b}) → YRate of decay of X* = Rate of product formation = d[Y]/dt = K

_{b}[X*]It will be noted that decay of X* is a Unimolecular reaction.

Whenever the decay of X* resultant products is the rate-determining step, then the total rate of the reaction is provided by the given expression.

Rate = K

_{b}[X*]In order to deduce the concentration of X*, an active species, in terms of concentrations of reactants (or products) in the ground state, steady-state approximation is utilized. According to this method, it is supposed that a steady-state is reached after a reaction begins in such a way that the concentration of the activated species is more or less a constant and doesn't change with respect to time.

That is, d[X*]/dt = 0

According to the steady-state approximation, the concentrations of all reactive intermediates are constant and small throughout the main part of the reaction.

This signifies that the activated species, X*, is consumed as soon as it is made. As X* is made and consumed {equations, [X + X → (K

_{a}) → X* + X] and [X* → (K_{b}) → Y]}d[X*] = 0 = {K

_{a}[X]^{2}/dt} - K_{a}[X][X*] - K_{b}[X*]That is, [X*](K'

_{a}[X] + K_{b}) = K_{a}[X]^{2}Or [X*] = K

_{a}[X]^{2}/(K'_{a}[X] + K_{b})By employing the equation Rate = K

_{b}[X*] and [X*] = K_{a}[X]^{2}/(K'_{a}[X] + K_{b})Rate = K

_{a}K_{b}[X]^{2}/(K'_{a}[X] + K_{b})From p = cRT or p α c. Therefore, if the pressure of a gas is high, its concentration is high and as a result, there will be big number of collisions.

A) At High Pressures:

At high pressures, the no. of collisions is very large and the probability of deactivating collisions taking place is high that is, the rate of deactivation is bigger than the rate of product formation (during decay); the Unimolecular decay of X* is the rate-determining step at high pressures; that is,

k'

_{a}[X*][X] >> k_{b}[X*]Or k'

_{a}[X] >> k_{b}In another words, k'

_{a}[X] + k_{b}≈ k'_{a}[X]By using the above in equation Rate = K

_{a}K_{b}[X]^{2}/(K'_{a}[X] + K_{b}), we obtainRate = K

_{a}K_{b}[X]^{2}/K_{a}[X] = K_{a}K_{b}[X]/K_{a}This means that the rate is first order at high pressures.

B) At Low Pressures:

At low pressures, the no. of collisions reduces. This signifies that the activated molecule results the product as soon it is made and there is not much time left for deactivating collision to take place. In another words, the bimolecular formation of X* is the rate-determining step. Moreover, the rate of deactivating collisions is very small as compared to the rate of product formation.

k

_{b}[X*] >> k'_{a}[X*][X]Or k

_{b}>> k'_{a}[X]Or k

_{a}[X] + k_{b}≈ k_{b}By employing the above equation in Rate = K

_{a}K_{b}[X]^{2}/K_{a}[X] = K_{a}K_{b}[X]/K_{a}Rate = K

_{a}K_{b}[X]^{2}/K_{b}= K_{a}[X]^{2}Therefore, the reaction obeys the second order kinetics at low pressures.

By employing this Lindemann-Hinshelwood theory, we could describe the Unimolecular decomposition of N

_{2}O_{5}at high pressures.Theories of Reaction Rates:The rates of numerous reactions increase with the increase in temperature. Arrhenius stated the given empirical relationship between the rate constant, 'k', and temperature, 'T'.

ln k = ln A - E

_{a}/RTOr log k = log A - E

_{a}/2.303RTHere 'A' is termed as the Arrhenius factor or frequency factor or pre-exponential factor and 'E

_{a}' is the activation energy. The Activation energy is a threshold energy which the reactant molecules should encompass in order to react. Whenever log k is plotted against 1/T, a straight line (figure shown below) is acquired for numerous reactions. In these cases, the T slope of the line is - E_{a}/2.303R and the intercept as 1 = 0 provides log A.The equation ln k = ln A - E

_{a}/RT is as well written in the exponential form as follows:Fig: Plot of log k against 1/T

K = Ae-

^{EaRT}A feasible reason for the deviation from Arrhenius equation in several reactions is that A and E

_{a}might differ with temperature. The temperature dependence of Arrhenius factor will be illustrated in collision theory. Here, we consider that A and E_{a}are constant for a reaction. Whenever the activation energy is high for a reaction, it signifies that the temperature dependence of the reaction rate is as well high. In these cases, even a small change of the temperature yields in a large change in rate constant.However, activation energy of a reaction can be computed Wm log k versus l/T plot, the other mode of acquiring it is to compute rate constants (k

_{1}and k_{2}) at two temperatures (T_{1}and T_{2}). Supposing E_{a}and A to be constant and utilizing the equation log k = log A - E_{a}/2.303RT, we obtain,log K

_{1}= log A - E_{a}/2.303RT_{1}And log K

_{2 }= log A - E_{a}/2.303 RT_{2}Subtracting the first equation from the second, we obtain

log (K

_{2}/K_{1}) = (- E_{a}/2.303R)(1/T_{2}- 1/T_{1})= (- E

_{a}/2.303R) (T_{1}-T_{2}/T_{1}T_{2})That is, log (K

_{2}/K_{1}) = (E_{a}/2.303R) (T_{2}-T_{1}/T_{1}T_{2})It will be noted that the unit of A depends on the unit of k. For the first order reactions, A consists of s

^{-1}unit which is similar as the unit for frequency. This could be a cause for its name, frequency factor. A is as well termed as the pre-exponential factor as it precedes the exponential term in equation K = Ae^{-EaRT}Collision Theory:Collision theory is mainly applicable to bimolecular reactions in gaseous phase. By some modifications, this can be applied to Unimolecular and as well termolecular reactions. We describe collision theory by employing a gas-phase bimolecular elementary reaction of the following.

X + Y → Product

With respect to the collision theory, the rate of a bimolecular reaction based on:

However, the Stearic factor is too to be considered while computing the reaction rate, it will be treated in the refinement of collision theory.

Total Collision Frequency:

Total collision frequency (ZXY) is basically the number of collision between the molecules of X and the molecule of Y in unit time in unit volume. Merely X-Y collision are counted however not X-X or Y-Y collisions, as only X-Y collisions are possible for the reaction pointed in the equation X + Y → Product

The total collision frequency (Z) in common can be derived by using the given relationship:

Z = [π × (collision diameter)

^{2}= [× (average relative speed of gas molecules)

= [× (number density)

= [× (number density)

= [× (correction factor)

For computing the total collision frequency (ZXY) among the molecules of X and Y, in respect to the above equation, we make use of the given relationship:

a) Collision diameter = σ

_{XY}= 1/2 (σ_{X}+ σ_{Y})Here σ

_{X}and σ_{Y }are the diameters of the molecules, X and Y correspondingly. The collision diameter σ_{XY}is the distance of closest approach between the molecule of X and molecule of Y.b) Average relative speed of X and Y molecules = [8K

_{b}T/πμ]^{1/2}Here k

_{b}is the Boltzmann constant (that is, subscript b is added to k to differentiate it from the rate constant), T is temperature and µ is the reduced mass.It will be noted that (1/μ) = 1/m

_{X}+ 1/m_{Y}Or μ = (m

_{X}m_{Y})/(m_{X}+ m_{Y})Here, m

_{X}and m_{Y}are the masses of one molecule of X and Y, symbolically.c) Let us now compute the factor, (number density) x (number density). As we have two kinds of molecules, X and Y, we have to take number densities of X and Y both.

In case of collision between the molecules of X and Y (that is, between molecules of various gases), there is no need for the correction factor. This is due to the reason that we compute the collisions between; each and every molecule of X and each and every molecule of Y. Each and every collision is counted just once. Therefore omitting the correction factor and utilizing the above equations, we obtain,

Z

_{XY}= πσXY^{2}[8K_{b}T/πμ]^{1/2}N_{A}[X][Y]^{2}Therefore we have got a relationship helpful in computing the total collision frequency for the collision between each and every molecule of X and each molecule of Y.

Boltzmann Factor:

You should realize that not all collisions between the molecules of X and Y would yield in the product formation. Just those collisions, in which, the energy of the colliding molecules equivalents or exceeds certain critical value E

_{a}(termed as activation energy as per Arrhenius equation), are efficient in bringing regarding the reaction between X and Y. Whenever E_{a}>>RT, then the Boltzmann factor, e^{-Ea/RT}provides the fraction of the collisions in which the colliding molecules possess energy equivalent to or more than the activation energy.Boltzmann factor = e

^{-Ea/RT}Calculation of Reaction Rate:

The product of total collision frequency and the Boltzmann factor provides the number of molecules of X or Y in unit volume reacting per unit time. This obeys from the definitions of the terms, total collision frequency and the Boltzmann factor. In order to get the reaction rate in terms of concentrations of X or Y (or the number of moles of X or Y) consumed per unit time, we have to divide the product, Z

_{XY}e^{-Ea/RT}by means of Avogadro constant;Reaction rate = -d[X]/dt = -d[Y]/dt

= Z

_{XY}e^{-Ea/RT}/N_{A}By utilizing the equations Z

_{XY}= πσXY^{2}[8K_{b}T/πμ]^{1/2}N_{A}[X][Y]^{2}and reaction rate = ZXY e^{-Ea/RT}/N_{A}Reaction rate = πσXY

^{2}[8K_{b}T/πμ]^{1/2}N_{A}[X][Y]^{2}e^{-Ea/RT}/NA= πσXY

^{2}[8K_{b}T/πμ]^{1/2}N_{A}[X][Y] e^{-Ea/RT}By statement, the reaction rate for a bimolecular elementary reaction with respect to the equation X + Y → Product is as shown:

Reaction rate = k [X] [Y]

On comparing the equations, we obtain

k = π σXY

^{2}[8K_{b}T/πμ]^{1/2}N_{A }e^{-Ea/RT}The equation above provides the theoretical value of the rate constant for a bimolecular reaction as per collision theory; π σXY is known as the mean collision cross-section.

Activated Complex Theory:The activated complex theory or simply the absolute theory of reaction rates shows the formation of activated complex (A‡) from the reactants (X and Y) as a preceding step for the preparation of the product, P.

X + Y → A

^{‡}A

^{‡}→ PThe main characteristics of the activated complex theory are provided here:

The reactant molecules come to contact with one other. In this method, some of the bonds get distorted; as few bonds begin forming by the exchange or discharge of atom or groups. The composite molecule so formed from the reactants prior to the formation of the product is known as the activated complex. The activated complex then decomposes to provide the product. The reaction series could be symbolized as represented in the figure shown below.

Fig: Change in Potential Energy as a function of reaction coordinates

The total potential energy of the system is based in the y-axis and the reaction coordinate in the x-axis. Reaction coordinate is the series of concurrent changes in bond distances and bond angles. These changes result throughout the formation of the products from the reactants.

Let consider the reaction between the molecule of H

_{2 }and a molecule of I_{2}. To begin with, let us suppose that the two molecules are far apart and the total potential energy of the system is the total sum of the potential energies of H_{2}and I_{2}. This portion of the reaction course is symbolized via the horizontal part AB of the curve as shown in the figure above. As the two molecules approach one other to such a level that the orbital start to overlap (appoint B in the curve), H-H and I-I bonds start to stretch and H-I bond starts to form. The total potential energy begins rising and this is symbolized by increasing portion of the curve BC. As the extension of H-H and I-I bond breakage and H-I bond formation increase, a point is reached if the potential energy is maximum (point C). The activated complex, symbolized below as a composite molecule, consists of the maximum potential energy.The Activated complex is a configuration of the atoms which the reactant molecules have close to the top of the energy barrier which separates the reactants from the products. Transition state is the highest point in the potential energy curve.

The bond-breaking is an energy demanding procedure and the bond-making is an energy discharging process. The total energy need for the formation of the activated complex and its decomposition to products should be available via the translational or the vibrational energy of the reactants.

Fig: Activated Complex

The maxima point in the potential energy curve is known as the transition state. Even a slight distortion of the bonds in the form of compression of H-I bond and stretching of H-H and I-I bonds allows the activated complex pass via the transition state. The path all along CD in the figure shown above symbolizes the course of the events that yield in the complete breakage of H-H and I-I bonds all along by the formation of H-I bond. The horizontal part DE symbolizes the net potential energy of two H-I molecules. However a fraction of the activated complex molecules could form the reactants (all along the path CB) the formation of the products is approximately a certainty, once the activated complex is at the transition state. The fraction of the activated complex transformed to products is known as the transmission coefficient and in the majority of cases, it is unity.

Energy Requirement for the reaction:

Let us now consider the energy criterion for the reaction. The energy need for the reactants to cross the energy barrier is to be met from translational or the vibrational energy of the molecules. At transition state, the activated complex consists of some complex vibration like motion of all atoms. The activated complex consists of one specific mode of vibration all along which it is unstable. Whenever the activated complex vibrates by the frequency corresponding to this vibrational mode, the activated complex decomposes to products.

Rate Constant calculation using Activated complex theory:

Based on the statistical thermodynamics, Eyring prepared the activated complex theory. The fundamental postulate of the theory is that there exists equilibrium between the activated complex and the reactants. Let us now consider the bimolecular gas phase reaction

X + Y → A

^{‡ }Here, X and Y are the reactants and A

^{‡}is the activated complex: The activated complex then decomposes to provide the product, PA

^{‡ }→ PThe rate of formation of the product mainly dependent on:

a) The concentration of activated complex and

b) The frequency which is being transformed into the product. This is the frequency of one of the vibrational nodes with respect to which the activated complex is not stable.

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