Chemical Dynamics – I

The temperature of the system shows a very marked effect on the overall rate of the reaction. In fact, it has been observed that the rate of a chemical reaction typically gets doubled with every 10°C rise in the temperature. However, this ratio may differ considerably and may reach up to 3 for different reactions. Besides, this ratio also varies as the temperature of the reaction increases gradually. The ratio of rate constant at two different temperatures is called as “temperature coefficient” of the reaction. Although we can determine the temperature coefficient between any two temperatures for any chemical reaction, generally it is calculated for 10°C difference.

Where and +10 are rate constants at temperature T and T+10, respectively. Now, if once the temperature coefficient is known, you can determine the relative increase or decrease in the overall reaction-rate by using the following relation.
Where R2 and R1 are the reaction-rates at temperatures T2 and T1, respectively. The ∆ is the temperature range for the temperature coefficient.
In order to illustrate the dominance of the effect of temperature change on the reaction rate, consider a reaction in which the temperature of the system is raised from 310°C to 400°C. Now, if the temperature coefficient for 10°C temperature-rise is 2, the relative increase in the rate constant or rate will be The physical significance of k is that it represents the number of collisions that result in a reaction per second; A is the number of collisions (leading to a reaction or not) per second occurring with the proper orientation to react. The exponential factor is the probability that any given collision will result in a reaction. It can also be seen that either increasing the temperature or decreasing the activation energy (for example through the use of catalysts) will result in an increase in the rate of reaction. Taking the natural logarithm of both side of equation (8), we get ln = ln + ln − / (10) Rearrange the above equation, we get ln = − + ln The equation (12) has the same form as the equation of straight line i.e. = + ; which means that if we plot "ln k" vs 1/T, the slope and intercept will yield "−Ea/R" and "ln A", respectively.

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In addition to the equation (12), one of the more popular forms of the Arrhenius equation can be derived by converting it to the common logarithm as given below.
2.303 log = − + 2.303 log (13) or log = − 2.303 + log The equation (14) also has the same form as the equation of straight line i.e. = + ; which means that if we plot "log k" vs 1/T, the slope and intercept will yield "−Ea/2.303R" and "log A", respectively.
The above equation can be used to find the activation energy if rate constants are known at two different temperatures.
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 Rate Law for Opposing Reactions of Ist Order and IInd Order
A reaction will be called as the opposing or reversible reaction if the reactants react together to form a product and the products also react to yield the reactants simultaneously under the same conditions.
In a simple context, we can say these reactions proceed not only in the forward direction but also in the backward direction. These reactions can be classified into the following categories based upon the kinetic order of the reactions involved.

 First Order Opposed by First Order
In order to understand the kinetic profile of first-order reactions opposed by the first order, consider a general reaction in which the reactant A forms product B i.e.

⇌ (17)
Now, if kf ⋙ kb, kb can be neglected. However, kf and kb have comparable values, a rate law depending upon both the constants can be written. To do so, suppose that a is the initial concentration of the reactant A and x is the decrease in the concentration of A after ʻtʼ time. The concentration of the product after the same time would also be equal to x. Hence, the rates of forward reaction (Rf) and backward reaction (Rb) can be given as: The net reaction rate i.e. rate of formation of the product can be given as However, when the equilibrium is attained, the rate of forward reaction will be equal to the rate of backward reaction i.e. Rf = Rb. Therefore, the will take the form Where xeq is the concentration of product B or the decrease in the concentration of reactant A at equilibrium. Now putting the value of kb from equation (20) into equation (19), we get or or Integrating equation (25), we get − ln( − ) = + Where C is the constant of integration. When t = 0, x = 0; putting these values in equation (26), we get − ln = Using the value of C form equation (27) Using equation (30), the rate constant for the forward reaction can easily be determined by measuring simple quantities like a, t, xeq and x. Now rearranging equation (20) for kb Now putting the value kf from equation (30) in equation (32), we get Hence, the value of the rate constant for backward reaction can also be obtained just by measuring t, xeq and x; or in other words, the kf eventually yields the kb also.
Also rearranging equation (20), we have Equating the right-hand sides of equation (34) and (35), we get or Equation (37) and (38)   Now, finding kf +kb from slope and kf/kb from equilibrium constant, kf and kb can easily be obtained from the elimination method. It should also be noted that if plot ln − and log − , the slopes will become negative.
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 First Order Opposed by Second Order
In order to understand the kinetic profile of first-order reactions opposed by second-order, consider a general reaction in which the reactant A forms product B and C i.e.

⇌ +
Now, if kf ⋙ kb, kb can be neglected. However, kf and kb have comparable values, a rate law depending upon both the constants can be written. To do so, suppose that a is the initial concentration of the reactant A and x is the decrease in the concentration of A after ʻtʼ time. The concentration of both the products after same time would also be equal to x. Hence, the rates of forward reaction (Rf) and backward reaction (Rb) can be given as: The net reaction rate i.e. rate of formation of the product can be given as However, when the equilibrium is attained, the rate of forward reaction will be equal to the rate of backward reaction i.e. Rf = Rb. Therefore, the equation (42) will take the form Where xeq is the concentration of product B and C or the decrease in the concentration of reactant A at equilibrium. Now putting the value of kb from equation (43) into equation (42), we get or Buy the complete book with TOC navigation, high resolution images and no watermark.
Integrating equation (48), and then rearranging Using equation (49), the rate constant for the forward reaction can easily be determined by measuring simple quantities like a, t, xeq and x. Now, we know that the equilibrium constant for the first order opposed by secondorder will be Now putting the value of kf from equation (49) in equation (51) and the rearranging for kb, we get Hence, the value of the rate constant for backward reaction can also be obtained just by measuring t, xeq and x and the equilibrium constant from equation (50).

 Second Order Opposed by First Order
In order to understand the kinetic profile of second-order reactions opposed by first order, consider a general reaction in which two reactants A and B form product C i.e.
Now, if kf ⋙ kb, kb can be neglected. However, kf and kb have comparable values, a rate law depending upon both the constants can be written. To do so, suppose that a is the initial concentration of both the reactant A and B; while x is the decrease in the concentrations of both reactants after ʻtʼ time. The concentration of the product after the same time would also be equal to x. Hence, the rates of forward reaction (Rf) and backward reaction (Rb) can be given as: Buy the complete book with TOC navigation, high resolution images and no watermark.

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The net reaction rate i.e. rate of formation of the product can be given as However, when the equilibrium is attained, the rate of forward reaction will be equal to the rate of backward reaction i.e. Rf = Rb. Therefore, the equation (56) will take the form Where xeq is the concentration of the product C or the decrease in the concentration of reactant A or B at equilibrium. Now putting the value of kb from equation (57) into equation (56), we get or Integrating equation (64), and then rearranging Buy the complete book with TOC navigation, high resolution images and no watermark.

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Using equation (66), the rate constant for the forward reaction can easily be determined by measuring simple quantities like a, t, xeq and x. Now, we know that the equilibrium constant for a second-order reaction opposed by first order will be Now putting the value of kf from equation (65) in equation (67) and the rearranging for kb, we get Hence, the value of the rate constant for backward reaction can also be obtained just by measuring t, xeq and x and the equilibrium constant from equation (66).

 Second Order Opposed by Second Order
In order to understand the kinetic profile of second-order reactions opposed by second-order, consider a general reaction in which two reactants A and B form product C and D i.e.
Now, if kf ⋙ kb, kb can be neglected. However, kf and kb have comparable values, a rate law depending upon both the constants can be written. To do so, suppose that a is the initial concentration of both the reactant A and B; while x is the decrease in the concentrations of both reactants after ʻtʼ time. The concentration of the products after the same time would also be equal to x. Hence, the rates of forward reaction (Rf) and backward reaction (Rb) can be given as: The net reaction rate i.e. rate of formation of the product can be given as However, when the equilibrium is attained, the rate of forward reaction will be equal to the rate of backward reaction i.e. Rf = Rb. Therefore, the equation (72) will take the form

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Where xeq is the concentration of the product C and D or the decrease in the concentration of reacant A or B at equilibrium. Now putting the value of kb from equation (73)  (76) Integrating equation (80), and then rearranging Using equation (81), the rate constant for the forward reaction can easily be determined by measuring simple quantities like a, t, xeq and x. Now, we know that the equilibrium constant for a second-order reaction opposed by second-order will be Now putting the value of kf from equation (81) in equation (83) and the rearranging for kb, we get Hence, the value of the rate constant for backward reaction can also be obtained just by measuring t, xeq and x and the equilibrium constant from equation (84).

 Consecutive Reactions
In many complex reactions, the order of the reaction has not been found equal to the molecularity noted from the stoichiometry. So, these reactions must take place in multiple steps rather than a single step. These multiple steps are individually labeled as consecutive reactions.
The consecutive reactions may be defined as the single-step reactions which can be written to represent an overall reaction.
In order to understand the kinetic profile of consecutive reactions, consider two first-order reactions in which reactant A converts to B which in turn converts to product C.
Where k1 and k2 are the rate constants for the first and second steps, respectively. In other words, A is the reactant, B is simply the intermediate and C is the final product.
However, kf and kb have comparable values, a rate law depending upon both the constants can be written. Now suppose that the initial concentrations of reactant A is C0; while the concentrations of A, B and C after time t are CA, CB and CC, respectively. So, we can say that Now, the rate can be deduced in terms of CA, CB and CC as given below.

Rate law in terms of CA:
The rate of disappearance reactant of A in the given reaction can be given by the following relation.
Integrating both sides, we get − ln = 1 + I Where I is the constant of integration. However, when t = 0, CA = C0, the equation (89)  [ ] After putting the value of CA from equation (96) in equation (98), we get a linear differential equation of first order i.e.
[ ] Integrating and then rearranging equation (99), both side, we get Buy the complete book with TOC navigation, high resolution images and no watermark.

Rate law in terms of CC:
The overall rate of formation of the product C in the given reaction can be given by the following relation.
[ ] = 2 (101) After putting the value of CA and CB from equation (96) and equation (100) into equation (86), we get the following result.
The equation (96), (100) and (109) can be used to plot the time-dependent variation of CA, CB and CC, respectively.

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A Textbook of Physical Chemistry -Volume I It can be clearly seen that the concentration of A decreases exponentially, while the concentration of B increases first and then declines. The concentration of C increases continuously and finally becomes equal to the concentration of A.

Maxima in the concentration of B:
In addition to the time-dependent concentration variation of different species, one more important parameter to measure is the maximum B concentration. Since, for this the value of [dCB]/dt = 0, the differentiation of equation (100) and then putting equal to zero gives Taking logarithm both side, we get Buy the complete book with TOC navigation, high resolution images and no watermark.

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Now putting the value of t from equation (117) in equation (100), we get Simplifying and then rearranging the above equation, we get Rate law in special cases: In addition to the typical consecutive reaction i.e. k1 = k2, two special cases also arise from the nature of the step reactions discussed below.
i) When k2 ⋙ k1: In these types of reactions, the value of k1 can be neglected. Therefore, the equation (109) takes the form Graphically, Figure 8. The plot CA, CB and CC vs time in a typical consecutive reaction when k2 ⋙ k1.
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It can be clearly seen that the concentration of the intermediate practically remains constant, and therefore, the steady-state approximation can be applied in this case.
ii) When k1 ⋙ k2: In these types of reactions, the value of k2 can be neglected. Therefore, the equation (109) takes the form Graphically, Figure 9. The plot CA, CB and CC vs time in a typical consecutive reaction when k1 ⋙ k2.

 Parallel Reactions
In many reactions, the reactant reacts to form more than one product simultaneously. If the amount of one the reaction product is very large in comparison to the others, then we can simply neglect these other reactions. However, if the amount of the product formed by other reactions are significant, we must refine the overall rate equation to represent this.
The parallel or side reactions may simply be defined as the reactions in which initial species react to give multiple products simultaneously.
In order to understand the kinetics of parallel reactions of the first order, suppose that a reactant A reacts to form product B and C simultaneously. A typical depiction of the parallel or side reaction with two pathways is given below.

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Where k1 and k2 are the rate contents. Now suppose that a is the initial concentration of the reactant A, while x is the decrease in the concentrations of the reactant after ʻtʼ time. Hence, the rates of first (R1) and second reaction (R2) can be given as: The overall reaction rate can be obtained by adding equation (122) and equation (123) as or Integrating the equation (126) and then rearranging, we get Also, dividing equation (122) by (123), we get Which implies that Buy the complete book with TOC navigation, high resolution images and no watermark.

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Hence, the value of rate constants involved, i.e., k1 and k2 can easily be obtained from the use of equation (127) and equation (129). It should also be noted from the equation (129) that the ratio of the concentration of products remains the same with time. Furthermore, the percentage of both products can also be obtained from the knowledge of rate constants using the relations given below.
The percentage is obtained by multiplying corresponding fractional quantum yields by 100. Similarly, parts per thousand (ppt) and parts per million (ppm) can be obtained by multiplying equations (130) and (131) by 10 3 and 10 6 , respectively.
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 Collision Theory of Reaction Rates and Its Limitations
In 1916, a German chemist Max Trautz proposed a theory based on the collisions of reacting molecules to explain reaction kinetics. Two years later, a British chemist William Lewis published similar results, however, he was completely unaware of Trautz's work. The remarkable work of these two gentlemen was extremely beneficial in explaining the rate of many chemical reactions.
The collision theory states that when the right reactant particles strike each other, only a definite fraction of the collisions induce any significant or noticeable chemical change; these successful changes are called successful collisions and are possible only if reacting molecules have sufficient energy at the moment of impact to break the pre-existing bonds and form all new bonds.
The minimum energy required to make a collision successful is called as the activation energy, and these types of collisions result in the products of the reaction. The rise in reactant concentration or increasing the temperature, both result in more collisions and hence more successful collisions, and therefore, increase the reaction rate. Sometimes, a catalyst is involved in the collision between the reactant molecules that decreases the energy required for the chemical change to take place, and so more collisions would have sufficient energy for the reaction to happen. In this section, we will discuss the collision theory of bimolecular and unimolecular reactions in the gaseous phase.

 Collision Theory for Bimolecular Reactions
In order to understand the collision theory for bimolecular reactions, we must understand the cause of a reaction itself first. The primary requirement for a reaction to occur is the collision between the reacting molecules. Therefore, if we assume that every collision results in the formation of the product, the rate of reaction should simply be equal to collision frequency (Z) of the reacting system i.e. the number of collisions occurring in the container per unit volume per unit time. Mathematically, we can say that = However, the actual rate would be much less than what is predicted by the equation (132); which is obviously due to the fact that all the collisions are not effective. Therefore, equation (132) must be modified to represent this factor. If f is the fraction of the molecules which are activated, the rate expression can be written as given below.

= ×
Now, according to the Maxwell-Boltzmann distribution of energies, the fraction of the molecules having energy greater than a particular energy E is Where N is the total number of molecules while ΔN represents the number of molecules having energy greater than E. However, if E = Ea, the fraction of activated molecules can be written as Where R is the gas constant and T is the reaction temperature. After putting the value of f from equation (135) into equation (133), we get At this point, two possibilities arise; one, when the colliding molecules are similar and other, is when the colliding molecules are dissimilar. We will discuss these cases one by one.
1. Rate of reaction when the colliding molecules are dissimilar: Consider a bimolecular reaction between different molecules A and B yielding product P as The number of collisions between A and B occurring in the container per unit volume per unit time can be given by the following relation.
Where nA and nA are the number densities (in the units of m −3 ) of particles A and B, respectively. The term σAB is simply the average collision diameter i.e. σAB = (σA + σB)/2. kB is the Boltzmann's constant (m 2 kg s −2 K −1 ). T represents the temperature of the system. The term μAB represents the reduced mass of the reactants A and B i.e. μAB = mAmB/mA+mB.
The equation (138) can also be expressed in terms of molar masses by putting mA = MA/N, mB = MB/N and k = R/N; where MA and MB are molar masses of the reactants, R is the gas constant and N represents to Avogadro number. Therefore, equation (138) takes the form or Also, as we know that the reaction rate can be written in terms of molecules of reactants reacting per cm 3 per second as Now, in order to express the rate in terms of molar concentrations, we need to recall some typical relations like Using the results of equation (143) and (144) in equation (142) Comparing equation (147) with Arrhenius rate constant i.e. = − / , we get 2. Rate of reaction when the colliding molecules are similar: Consider a bimolecular reaction between similar molecules A and A yielding product P as Buy the complete book with TOC navigation, high resolution images and no watermark.

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The number of collisions between A and A occurring in the container per unit volume per unit time can be given by the following relation.
Where nA is the number density (in the units of m −3 ) of particle A. The term σ is simply the average collision diameter. kB is the Boltzmann's constant (m 2 kg s −2 K −1 ). T represents the temperature of the system. The term mA represents the mass of the reactants A.
The equation (150) can also be expressed in terms of molar masses by putting mA = MA/N and k = R/N; where MA is the molar mass of the reactant, R is the gas constant and N represents to Avogadro number. Therefore, equation (150) takes the form Also, as we know that the reaction rate can be written in terms of molecules of reactants reacting per cm 3 per second as Now, in order to express the rate in terms of molar concentrations, we need to recall some typical relations like Buy the complete book with TOC navigation, high resolution images and no watermark.

 Collision Theory for Unimolecular Reactions
In order to understand the collision theory for unimolecular reactions, we must understand the root cause of these reactions. In a typical unimolecular reaction, a single molecule converts into the product by simply rearranging itself. However, the question that arises here is how these molecules get activated. The mystery was solved by a British physicist, Frederick Alexander Lindemann, who proposed a time-leg between activation and actual reaction. In other words, when ordinary molecules collide with each other, some of them get activated, and the rate depends only upon these molecules but not the ordinary ones i.e.
Where k1, k2 and k3 are the rate constants for the different processes; while A and A * are the ordinary and activated molecule. The overall rate of formation of the product can be given as Equation (165) gives rise to two possibilities discussed below.

If the concentration of reactant A is very high: In this situation, k2[A]
⋙ k3, and k3 can be neglected, therefore, the equation (165) Where ko is the overall rate constant. It is clear from the above result that the unimolecular reactions follow first-order kinetics in such cases.

If the concentration of reactant A is very low: In this situation, k3 ⋙ k2[A] and k2[A]
can be neglected, therefore, the equation (165) Where k1 is the overall rate constant. It is clear from the above result that the unimolecular reactions follow second-order kinetics in such cases.
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 Steric Factor
One of the most glaring limitations of the collision is that the predicted values of rate constants for many reactions were found to be considerably different from the values obtained experimentally. Moreover, it was also noticed that more the complexity, the higher was the deviation. This happened because the collision theory supposed that the particles participating in the chemical reaction are completely spherical, and thus, are able to react in every direction. However, this is far from the truth since the orientation of the collisions is not always appropriate to result in the chemical change. For instance, in the hydrogenation of ethylene, the dihydrogen molecule must approach the bonding zone between the atoms, and not all the possible collisions would be able to satisfy this requirement. For more clear view, consider the formation of CO2 as shown below.
To solve this problem, the concept of steric factor (ρ) was introduced, which is simply the ratio of experimental value to the predicted value of the rate constant. In other words, the steric factor may be defined as the ratio between the frequency factor and the collision frequency i.e.

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It is worthy to note that the value of the steric factor most of the cases is less than unity. Typically, it has been seen that more the complex the reactant molecules are, the lower is the steric factor. However, some reactions do have steric factors higher than unity; for instance, the harpoon reactions in which atoms involved exchange electrons generating ions. The deviation from unity may arise due to different reasons such as non-spherical shape of reacting molecules, or the partial delivery of kinetic energy, the presence of a solvent when applied to solutions.
In order to derive the expression for modified collision theory that does consider the reactants steric, recall the rate constant calculated from simple collision theory first i.e.
For experimental rate, multiply the equation (172) Now considering both the possibilities i.e. whether the reacting species are the same or different, we can simplify the above equation in two ways:

i) For dissimilar molecules:
If the colliding molecules are not the same, the exponential part in equation (173) takes the form Substituting the values of different constants, we get = 2.753 × 10 29 × × 2 √ ( + ) ii) For similar molecules: If the colliding molecules are the same, the exponential part in equation (173) This modified collision theory can account for probability factors up to 10 −4 but not less than that. This limitation can be overcome by "transition state theory" discussed in the next section.
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 Activated Complex Theory
In 1935, an American chemist Henry Eyring; alongside two British chemists, Meredith Gwynne Evans and Michael Polanyi; proposed a new theory to rationalize the rate of different chemical reactions which was based upon the formation of an activated intermediate complex. This theory is also known as the "transition state theory", "theory of absolute reaction rates", and "absolute-rate theory".
The activated complex theory states that the rates of various elementary chemical reactions can be explained by assuming a special type of chemical equilibria (quasi-equilibrium) between reactants and activated complexes.
Before the development of activated complex theory, the Arrhenius rate law was popularly used to determine energies for the potential barrier. However, the Arrhenius equation was based on empirical observations rather than mechanistic investigations as if one or more intermediates are involved in the conversion or not. For that reason, more development was essential to know the two factors present in the Arrhenius equation, the activation energy (Ea) and the pre-exponential factor (A). The Eyring equation from transition state theory successfully addresses these two issues and therefore contributed significantly to the conceptual understanding of reaction kinetics. Figure 11. The variation of free energy as the reaction proceeds.
To explore the concept mathematically, consider a reaction between reactant A and B forming a product P via the activated complex X * as: The equilibrium constant for reactants to activated complex conversion is * =

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Henry Eyring showed that the rate constant for a chemical reaction with any order or molecularity can be given by the following relation.

= ℎ * (180)
Where K * is the equilibrium constant for reactants to activated complex conversion at temperature T. Whereas, R, N and h represent the gas constant, Avogadro number and Planck's constant, respectively. Now, as we know from thermodynamics * = − ln * (181) Where ΔG * is the free energy of activation for reactants to activated complex conversion step. Using the value of K * from equation (183) into equation (180), we get If we put the thermodynamic value of free energy i.e. ΔG * = ΔH * − TΔS * in equation (184), we get Where ΔH * and ΔS * are enthalpy change and the entropy change of the activation step. Equation (186) is popularly known as the Eyring equation. Now, since the equation (186) contains very fundamental factors of the reacting species, that is why this theory got its name of "theory of absolute reaction rates".

 Significance of Entropy of Activation and Enthalpy of Activation
As far as the equation (184) is concerned, it can easily be seen that as the free energy change of the activation step increases, the rate constant would decrease. However, if we look at the simplified form i.e. equation (186), we find three factors; one is RT/Nh which is constant if the temperature is kept constant. The second factor involves ΔS * , and therefore, we can conclude that the reaction rate would show exponential increase if the entropy of activation increases. The third factor includes ΔH * , and therefore, we can conclude that the reaction rate would show exponential decrease if the enthalpy of activation increases. It is also worthy to note that the first two terms collectively make the frequency factor.

 Comparison with Arrhenius Rate Constant
Like the collision theory, the validity of the "activated complex theory" must also be checked against the results of the Arrhenius rate equation. In order to do so, recall the Arrhenius equation i.e.
Now we can use equation (188) and (189) to determine the value of entropy of activation and enthalpy of activation.

Calculation of entropy of activation:
In order to determine the entropy change of the activation step, take the natural logarithm of the equation (188) Thus, higher is the value of the frequency factor, larger will be the entropy of activation.

Calculation of enthalpy of activation:
In order to determine the entropy change of the activation step, we must look at the equation (189) which suggests that the enthalpy change of the activation step as exactly equal to the activation energy of the reaction dictating the rate considerably i.e. ΔH * = Ea. However, it has been observed that the exact value of activation energy is slightly different than the enthalpy of activation. In order to prove the aforementioned statement, rewrite the equation (186) as: 194) or Buy the complete book with TOC navigation, high resolution images and no watermark.

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A Textbook of Physical Chemistry -Volume I Where C is another constant. Now taking natural logarithm both side, equation (195) Comparing (198) and (202), we get If n is the change in the number of moles of gas in going from reactant to activated complex, the result of equation (205) takes the form * = − .
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 Ionic Reactions: Single and Double Sphere Models
It is a quite well-known fact that the rate of ionic reactions is generally small, which is obviously due to the larger magnitude of activation energies arising from the very strong nature of electrostatic interactions. The magnitude of the frequency factor in ionic reactions is a function of ionic charges. The frequency factors have larger values if the charges on the participating ions are opposite, while smaller values are obtained in the case of like-charged ions. This behavior can be explained in terms of the kinetic theory of gases; which suggests that oppositely charged ions are more prone to collision due to attraction than the ions colliding with same charges (repulsive forces). Besides the collision theory, the activated complex theory also provides an alternate explanation for the ionic reactions. In this section, we will discuss the rationalization of ionic reactions on the basis of the single-sphere model and the double-sphere model in detail.

 Double Sphere Model
Before we discuss the double sphere model of the ionic reactions, a simplified surrounding must be assumed. Although it would be an oversimplification of the actual situation, it is highly beneficial as far as conceptual and quantitative understanding is concerned. To do so, the solvent is considered as continuous surrounding with a ε as the dielectric constant. According to this model, two ions, which can same or opposite charges, combine together to form an activated complex. In the initial state, the ions are considered as discrete; while in the final state, they assumed to form a dumbbell like coordination with ʻrʼ as the distance of separation between their centers.  Where ε0 and ε are permittivities of the vacuum (8.854 × 10 −12 C 2 N −1 m −2 ) and the dielectric constant of the solvent used, respectively. The symbol e represents the elementary charge and has a value equal to 1.6 × 10 −19 C. The value of parameter varies from ∞ to r with the mutual approach of two ions. The amount of work done in moving the two ions closer by an extant dx will be The negative sign is an indicator of decreasing separation i.e. distance is reduced by dx. The total amount of work done in moving the two ions from x = ∞ to x = r will be The work given the above equation is actually the potential energy of the system which would have a negative sign for oppositely charged ions and positive sign if the ions have same charges. Furthermore, we can also say that this work is the free energy change due to electrostatic interactions, therefore, multiplying it by Avogadro number (N) would give the value of the corresponding molar free energy change ( * ) i.e. * = 2 4 0 (212) Correcting the above equation for non-electrostatic contribution * , the total molar free energy change for the whole process can be given by the following relation. * = * Also, from the activated complex theory, we know that After putting the value of ΔG * from equation (213) Where k0 represents the magnitude of the rate constant for the ionic reaction carried out in a solvent of infinite dielectric constant so that the electrostatic interactions become zero.

 Single Sphere Model
Besides the double-sphere model, another theoretical model that is quite rationalizing is a singlesphere model. Just like the double-sphere model, the solvent is also considered as a continuum with a ε as the dielectric constant. However, the primary differentiating aspect of this model is that it considers the two ions, which can same or opposite charges, to form a single-sphere activated complex. Buy the complete book with TOC navigation, high resolution images and no watermark.

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A Textbook of Physical Chemistry -Volume I

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In the initial state, the ions are considered as discrete; while in the final state, they assumed to form a singlesphere activated complex with ʻr * ʼ as the overall radius. The rate law for this case was derived by Born by considering the energy required to charge an ion in solution. Now suppose that we need to charge a conducting sphere of radius r from an initial value of zero to the final value Ze. This can be visualized as a process in which a very small charge is e.dλ (λ = 0 − Z) is carried from infinite to this sphere. Now, if ZA and ZB are the charge numbers of the participating ions and x as the distance of separation between the sphere and the "increment" at any time, the force of electrostatic interaction (dF) between them can be given from the Coulomb's law as: The total amount of work done can be obtained by carrying out the double integration with respect to x = ∞r and λ = 0 -Z i.e.
Correcting the above equation for non-electrostatic contribution * , the total molar free energy change for the whole process can be given by the following relation. * = * Also, from the activated complex theory, we know that After putting the value of ΔG * from equation (230) Taking natural logarithm both side of equation (232) and rearranging, we get Which can also be expressed as Where k0 represents the magnitude of the rate constant for the ionic reaction carried out in a solvent of infinite dielectric constant so that the electrostatic interactions become zero.

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A Textbook of Physical Chemistry -Volume I

 Influence of Solvent and Ionic Strength
The rate of reaction in the case of ionic reactions is strongly dependent upon the nature of the solvent used and the ionic strength. The single and double-sphere treatment of these reactions enables us to study their effect in detail. In this section, we discuss the application and validity of solvent influence and ionic strength on the reaction rate.

 Influence of the Solvent
In order to study the influence of solvent on the rate of ionic reactions, recall the rate equation derived using the double sphere model i.e.
Where ε0 and ε are permittivities of the vacuum (8.854 × 10 −12 C 2 N −1 m −2 ) and the dielectric constant of the solvent used, respectively. The symbol e represents the elementary charge and has a value equal to 1.6 × 10 −19 C. k0 represents the magnitude of the rate constant for the ionic reaction carried out in a solvent of infinite dielectric constant so that the electrostatic interactions become zero. ZA and ZB are the charge numbers of the participating ions. The symbol N and R represents the Avogadro number and gas constant, respectively. Rearranging equation (235) Which is clearly the equation of the straight line (y = mx + c) with a negative slope and positive intercept. Therefore, it is obvious that the logarithm of the rate constant shows a linear variation with the reciprocal of dielectric constant. Figure 13. The plot of ln k vs 1/ε for a typical ionic reaction.

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It is quite obvious from the plot that equation (236) holds very good over a wide range of dielectric contents; however, as the large deviations are observed at lower values of ε. Moreover, if ʻmʼ is the experimental slope then from equation (236) Every term in the above equation is known apart from r, suggesting its straight forward determination from the slope of ln k vs 1/ε. The values of r obtained from equation (237) are found to be quite comparable to other methods, which in turn suggests its practical application.
Besides the calculation of r, the influence of dielectric constant of solvent can also be used to explain the entropy of activation. In order to do so, recall from the principles of thermodynamics Also, the electrostatic contribution to the Gibbs free energy using the double sphere model is * = 2 4 0 However, the only quantity which is temperature-dependent in the above equation is ε. Therefore, differentiating equation (239) Therefore, knowing the dielectric constant of the solvent and r, the entropy of activation can be obtained. Moreover, it is also worthy to note that the entropy of activation is negative and decreases with an increase in ZAZB.
One more factor that affects the entropy of activation is the phenomena of "electrostriction" or the solvent binding. This can be explained by considering the combination of two ions as of same and opposite charges. If the ions forming activated complex are having one-unit positive charge each, the double-sphere will have a total of two-unit positive charge. This would result in a very strong interaction between the activated Buy the complete book with TOC navigation, high resolution images and no watermark.

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complex and the surrounding solvent molecules. This would eventually result in a restriction of free movement and hence decreased entropy. On the other hand, If the ions forming activated complex possess opposite charges, the double-sphere would have less charge resulting in decreased electrostriction, and therefore, increased entropy. Figure 14. The dependence of entropy of activation on the solvent electrostriction in case of (left) same charges and (right) opposite charges.

 Influence of Ionic Strength
In order to study the influence of ionic strength (I) on the rate of ionic reactions, we need to recall the quantity itself first i.e.
Where mi and zi are the molarity and charge number of ith species, respectively. For instance, the value of z for Ca 2+ and Clin CaCl2 are +2 and −1, respectively. It has been found that an increase the ionic strength increases the rate of reaction if charges on the reacting species are of the same sign. On the other hand, the reaction rate has been found to follow a declining trend with increasing ionic strength if reaction ions are of opposite sign. The mathematical treatment of the abovementioned statement is discussed below.
To rationalize the effect of ionic strength of the solution on the rate of reaction in case of ionic reactions, consider a typical case i.e.
A Danish physical chemist, J. N. Brønsted, proposed the rate equation relating reaction-rate (R) and activity coefficient as Buy the complete book with TOC navigation, high resolution images and no watermark.

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Where yA, yB and yX are the activity coefficients for the reactant A, B and the activated complex X, respectively. Brønsted collectively labeled the term yAyB/yX as the "kinetic activity factor", and it was found be quite accurate with experimental data. Now, rearranging equation (245), we get Since the left-hand side simply equals to a second-order rate constant, the above equation takes the form = 0 Taking logarithm both side, we get log = log 0 + log (248) or log = log 0 + log + log − log Now the correlation of mean ionic activity coefficient with the ionic strength is given by famous Debye-Huckel theory i.e.

log = − 2 √
Where B is the Debye-Huckel constant. Using the concept of equation (250) Where k0 is the rate constant at zero ionic strength and can be obtained by the extrapolation of log k0 vs square root of the ionic strength. Figure 15. The plot of log (k/k0) vs (I) 1/2 for different ionic reactions in aqueous solution at 25°C.
The above equation can also be extended to explain the dependence of reaction-rate on ionic strength for third-order reactions. To do so, consider Following the same route as in second-order reactions, we will get log = log 0 + [( + + ) 2 − 2 − 2 − 2 ]√ (260) log = log 0 + 2 ( + + )√ Therefore, a negative slope will be observed if ( + + ) is negative while a positive slope is expected for a positive value of ( + + ). Table 1. The side-by-side comparison between the collision theory and transition state theory.
Collision Theory Activated Complex Theory 1. According to the collision theory, the chemical reactions occur when the reactant molecules collide with a sufficient amount of kinetic energy.
1. According to the transition state theory, the primary cause of the reaction is actually the formation of an activated complex or the transition state, which in turn, converts to the final product.