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CHAPTER 3, Chemical Dynamics – I,  Effect of Temperature on Reaction Rates, 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., 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 =, , 𝑘 𝑇+10, =2−3, 𝑘𝑇, , (1), , Where 𝑘 𝑇 and 𝑘 𝑇+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., 𝑇2 −𝑇1, 𝑅2 𝑘2, =, = (𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡) ∆𝑇𝑡𝑐, 𝑅1 𝑘1, , (2), , Where R2 and R1 are the reaction-rates at temperatures T2 and T1, respectively. The ∆𝑇𝑡𝑐 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, 400−310, 𝑅2 𝑘2, =, = (2) 10, 𝑅1 𝑘1, , (3), , 𝑅2 𝑘2, =, = (2)9 = 512, 𝑅1 𝑘1, , (4), , 𝑅2, = 512, 𝑅1, , (5), , Hence, a 90°C rise in temperature increases the rate of reaction 512 times, which is definitely huge. Now the, question arises, why is it so? What did the temperature do that made this happen? In this section, we will, answer these questions., , Copyright © Mandeep Dalal

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A Textbook of Physical Chemistry – Volume I, , 114, , , Fundamentals of Temperature-Rate Correlation, , Before we discuss the effect of temperature on the reaction rate, 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. In other words, the reactant molecules must collide with each other to form the product. Therefore,, if we assume that every collision results in the formation of the product, the rate of reaction should simply be, equal to the collision frequency of the reacting system., For a reaction between A and B, the collision frequency (Z) is the number of collisions between A and, B occurring in the container per unit volume per unit time., 8𝑘𝐵 𝑇, 𝑍 = 𝑛𝐴 𝑛𝐵 𝛩𝐴𝐵 √, 𝜋𝜇𝐴𝐵, , (6), , Where nA and nA are the number densities (in the units of m−3) of particles A and B, respectively. The term ΘAB, is the reaction cross-section (in m2) when particle A with radius rA and B with radius rB collide with each other, i.e. ΘAB = π (σAB)2 = π(rA + rB)2 = π(σA/2 + σB/2)2. kB is the Boltzmann's constant (m2 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. From equation (6), it follows that when we heat the substance, the particles collide more, frequently and hence increase the collision frequency. Now one may think that this collision frequency would, result in a larger rate of reaction, and therefore, the mystery is solved. However, this isn’t sufficient to, rationalize the experimental observations. For instance, if we increase the temperature from 300 K to 310 K,, the relative increase in the collision frequency (nZ), and hence in reaction rate, from equation (6) can be, determined as given below., 310, 𝑛𝑍 = √, = 1.0165, 300, , (7), , This is only 1.65% increase for a 10° rise in temperature. This is pretty far from the reality i.e. reaction-rate, almost gets double (100% increase). So, the actual mechanism is still behind the scene and must be understood., At this point we must introduce the concept of activation energy, otherwise, the concept cannot be, discussed further. The collision of reacting molecules would result in the chemical reaction only if they possess, a certain amount of minimum energy i.e. threshold energy. Since every molecule does have some energy, the, energy it needs to reach the threshold is less than the actual threshold energy. The energy required by reactant, molecules to cross the barrier is called the activation energy or the enthalpy of activation for the reaction. A, simple equation can be used to deduce their relationships as given below., Activation energy = Threshold energy – Energy actually possessed by the molecules, The rate of a chemical reaction is inversely proportional to the magnitude of the activation energy i.e. larger, the activation energy, slower will be the reaction and vice-versa., Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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CHAPTER 3 Chemical Dynamics – I, , 115, , Figure 1. The reaction coordinate diagram for a typical chemical reaction., , Hence, we can say that only effective collisions would result in the chemical reactions, but how can we find, the number of molecules having energy high enough to react with each other. For this, we need to go into the, basics of energy distribution among a large number of particles i.e. Maxwell’s distribution of energies., , Figure 2. The Maxwell-Boltzmann distribution of energies at temperature T., , After marking the activation energy on the Maxwell-Boltzmann distribution curve, the particle with sufficient, energy to react can easily be found from the area under the corresponding curve i.e. dashed area. The undashed, area at a particular temperature is quite large, and therefore, represents the number particle whose collision, would not result in any reaction chemical change. It can be clearly seen that most of the particles don't have, Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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A Textbook of Physical Chemistry – Volume I, , 116, , enough energy, and hence, are unable to yield the product. The reaction-rate will be very small If there are, very few particles with enough energy at any time., However, if the temperature is raised, the maxima of the Maxwell-Boltzmann distribution curve shifts, towards higher energy. This makes the number of “efficient particles” to increase and thereby increases the, number of effective collisions too. Consider the Maxwell-Boltzmann energy distributions at temperature T and, T+10., , Figure 3. The Maxwell-Boltzmann distribution of energies at temperature T and T+10°C., , Now although the area under the whole curve remains the same, the dashed area is doubled. Thus,, the primary reason for almost 100% rise in the overall rate of reaction for every 10°C is the 100% increase in, the number of effective collisions.,  The Arrhenius Equation, In 1884, the famous Dutch chemist Jacobus Henricus Van't Hoff realized that his equation (Van't, Hoff equation) could also be used to suggests a formula for the rates of both forward and backward reactions., In 1889, Svante Arrhenius immediately noticed the importance of this invention and proposed an empirical, equation based on Van't Hoff’s work. This equation is extremely useful in the modeling of the temperature, variation of many chemical reactions. The equation proposed by Arrhenius is, 𝑘 = 𝐴 𝑒 −𝐸𝑎/𝑅𝑇, , (8), , Where the symbol k, R and T represent rate constant, gas constant and temperature, respectively. A is popularly, known as the pre-exponential factor or Arrhenius constant with the units identical to those of the rate constant, used, and therefore, will vary depending on the order of the reaction. The term Ea represents the activation, energy measured in joule mole−1., Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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CHAPTER 3 Chemical Dynamics – I, , 117, , Another popular form of the Arrhenius equation is, 𝑘 = 𝐴 𝑒 −𝐸𝑎/𝑘𝐵 𝑇, , (9), , The only difference in the equation (8) and equation (9) is the energy units of Ea; the former one uses energy, per mole, which is more common in chemistry, while the latter form uses energy per molecule directly, which, is common in physics. The different units are accounted for in using either the gas constant, R, or the Boltzmann, constant, kB, as the multiplier of temperature T. If the reaction is first order, A will have the units of s−1 and can, be called as collision frequency or frequency factor., 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 𝑘 = ln 𝐴 + ln 𝑒 −𝐸𝑎/𝑅𝑇, ln 𝑘 = ln 𝐴 −, , 𝐸𝑎, 𝑅𝑇, , (10), (11), , Rearrange the above equation, we get, ln 𝑘 = −, , 𝐸𝑎, + ln 𝐴, 𝑅𝑇, , (12), , The equation (12) has the same form as the equation of straight line i.e. 𝑦 = 𝑚𝑥 + 𝑐; which means that if we, plot “ln k” vs 1/T, the slope and intercept will yield “−Ea/R” and “ln A”, respectively., , Figure 4. The Arrhenius plot of ln k vs 1/T., , Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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A Textbook of Physical Chemistry – Volume I, , 118, , 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 𝑘 = −, , 𝐸𝑎, + 2.303 log 𝐴, 𝑅𝑇, , (13), , or, log 𝑘 = −, , 𝐸𝑎, + log 𝐴, 2.303 𝑅𝑇, , (14), , The equation (14) also has the same form as the equation of straight line i.e. 𝑦 = 𝑚𝑥 + 𝑐; which means that if, we plot “log k” vs 1/T, the slope and intercept will yield “−Ea/2.303R” and “log A”, respectively., , Figure 5. The Arrhenius plot of log k vs 1/T., , To obtain the integrated form of the Arrhenius equation, differentiate the equation (12) as given below., 𝑑 ln 𝑘, 𝐸𝑎, =, 𝑑𝑇, 𝑅𝑇 2, , (15), , Now integrating the above equation between temperature T1 and T2, we get, ln, , 𝑘2 𝐸𝑎 1, 1, =, [ − ], 𝑘1, 𝑅 𝑇1 𝑇2, , 𝑜𝑟, , log, , 𝑘2, 𝐸𝑎, 1, 1, =, [ − ], 𝑘1 2.303𝑅 𝑇1 𝑇2, , (16), , The above equation can be used to find the activation energy if rate constants are known at two different, temperatures., , Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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CHAPTER 3 Chemical Dynamics – I, , 119, ,  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., 𝐴, , 𝑘𝑓, ⇌, 𝑘𝑏, , (17), 𝐵, , 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:, 𝑅𝑓 = 𝑘𝑓 [𝐴] = 𝑘𝑓 (𝑎 − 𝑥), , (17), , 𝑅𝑏 = 𝑘𝑏 [𝐵] = 𝑘𝑏 𝑥, , (18), , The net reaction rate i.e. rate of formation of the product can be given as, 𝑑𝑥, = 𝑘𝑓 (𝑎 − 𝑥) − 𝑘𝑏 𝑥, 𝑑𝑡, , (19), , 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, 𝑘𝑓 (𝑎 − 𝑥𝑒𝑞 ) = 𝑘𝑏 𝑥𝑒𝑞, , (20), , 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, 𝑘𝑓 (𝑎 − 𝑥𝑒𝑞 ), 𝑑𝑥, = 𝑘𝑓 (𝑎 − 𝑥) −, 𝑥, 𝑑𝑡, 𝑥𝑒𝑞, , (21), , (𝑎 − 𝑥𝑒𝑞 ), 𝑑𝑥, = 𝑘𝑓 [(𝑎 − 𝑥) −, 𝑥], 𝑑𝑡, 𝑥𝑒𝑞, , (22), , or, Copyright © Mandeep Dalal

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CHAPTER 3 Chemical Dynamics – I, , 125, , 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, 𝐾=, , [𝐶], [𝐴][𝐵], , (66), , 𝑘𝑓, 𝑘𝑏, , (67), , 𝐾=, , Now putting the value of kf from equation (65) in equation (67) and the rearranging for kb, we get, 𝑘𝑏 =, , 𝑥𝑒𝑞, 𝑥𝑒𝑞 (𝑎2 − 𝑥𝑒𝑞 𝑥), ln, 2 )𝐾, 𝑎2 (𝑥𝑒𝑞 − 𝑥), 𝑡(𝑎2 − 𝑥𝑒𝑞, , (68), , 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., 𝐴+𝐵, , 𝑘𝑓, ⇌, 𝑘𝑏, , (69), 𝐶+𝐷, , 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:, 𝑅𝑓 = 𝑘𝑓 [𝐴][𝐵] = 𝑘𝑓 (𝑎 − 𝑥)2, , (70), , 𝑅𝑏 = 𝑘𝑏 [𝐶][𝐷] = 𝑘𝑏 𝑥 2, , (71), , The net reaction rate i.e. rate of formation of the product can be given as, 𝑑𝑥, = 𝑘𝑓 (𝑎 − 𝑥)2 − 𝑘𝑏 𝑥 2, 𝑑𝑡, , (72), , 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, 2, , 2, 𝑘𝑓 (𝑎 − 𝑥𝑒𝑞 ) = 𝑘𝑏 𝑥𝑒𝑞, , Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal, , (73)

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CHAPTER 3 Chemical Dynamics – I, , 127, ,  Rate Law for Consecutive & Parallel Reactions of Ist Order Reactions, In addition to the opposing or reversible reactions, two other types of simultaneous reactions are, consecutive and parallel reactions. In this section, we will discuss the kinetic profiles of these two types of, reactions up to the first order only.,  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., 𝐴→, , 𝑘1, , 𝐵, , →, , 𝑘2, , 𝐶, , (85), , 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, 𝐶0 = 𝐶𝐴 + 𝐶𝐵 + 𝐶𝐶, , (86), , Now, the rate can be deduced in terms of CA, CB and CC as given below., 1. Rate law in terms of CA: The rate of disappearance reactant of A in the given reaction can be given by the, following relation., −, , 𝑑[𝐶𝐴 ], = 𝑘1 𝐶𝐴, 𝑑𝑡, , (87), , −, , 𝑑[𝐶𝐴 ], = 𝑘1 𝑑𝑡, 𝐶𝐴, , (88), , or, , Integrating both sides, we get, − ln 𝐶𝐴 = 𝑘1 𝑡 + I, Where I is the constant of integration. However, when t = 0, CA = C0, the equation (89) takes the form, , Copyright © Mandeep Dalal, , (89)

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A Textbook of Physical Chemistry – Volume I, , 132, , 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, 𝐶𝐶 = 𝐶0 (1 − 𝑒 −𝑘2 𝑡 ), , (121), , 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., , Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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A Textbook of Physical Chemistry – Volume I, , 134, , 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)., , Figure 10. The variation of the concentrations of reactants and products as a function of time in a typical, parallel reaction., , 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., 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑄𝑢𝑎𝑛𝑡𝑢𝑚 𝑌𝑖𝑒𝑙𝑑 𝑜𝑓 𝐴 =, , 𝑘1, 𝑘1 + 𝑘2, , (130), , 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑄𝑢𝑎𝑛𝑡𝑢𝑚 𝑌𝑖𝑒𝑙𝑑 𝑜𝑓 𝐵 =, , 𝑘2, 𝑘1 + 𝑘2, , (131), , also, , 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, 103 and 106, respectively., , Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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CHAPTER 3 Chemical Dynamics – I, , 135, ,  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, 𝑅𝑎𝑡𝑒 = 𝑍, , (132), , 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., 𝑅𝑎𝑡𝑒 = 𝑍 × 𝑓, , (133), , Now, according to the Maxwell-Boltzmann distribution of energies, the fraction of the molecules having, energy greater than a particular energy E is, 𝑓=, , 𝛥𝑁, = 𝑒 −𝐸/𝑅𝑇, 𝑁, , (134), , 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, Copyright © Mandeep Dalal

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A Textbook of Physical Chemistry – Volume I, , 136, , 𝑓=, , (135), , 𝛥𝑁, = 𝑒 −𝐸𝑎/𝑅𝑇, 𝑁, , 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, (136), , 𝑅𝑎𝑡𝑒 = 𝑍𝑒 −𝐸𝑎/𝑅𝑇, , 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, (137), , 𝐴+𝐵 →𝑃, , 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., 𝑍=, , (138), , 8𝜋𝑘𝐵 𝑇, 𝜇𝐴𝐵, , 2 √, 𝑛𝐴 𝑛𝐵 𝜎𝐴𝐵, , 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 (m2 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, , 2 √, 𝑍 = 𝜎𝐴𝐵, , 𝑅, 𝑀, 𝑀, 8𝜋 (𝑁) 𝑇 ( 𝑁𝐴 + 𝑁𝐵 ), 𝑀𝐴 𝑀𝐵, 𝑁 × 𝑁, , (139), 𝑛𝐴 𝑛𝐵, , or, 𝑍=, , 8𝜋𝑅𝑇(𝑀𝐴 + 𝑀𝐵 ), 𝑛𝐴 𝑛𝐵, 𝑀𝐴 𝑀𝐵, , (140), , 2 √, 𝜎𝐴𝐵, , Also, as we know that the reaction rate can be written in terms of molecules of reactants reacting per cm3 per, second as, , Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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CHAPTER 3 Chemical Dynamics – I, , 141, ,  Limitations of Collision Theory, The collision theory of reaction rate is extremely successful in rationalizing the kinetics of many, reactions, however, it does suffer from some serious limitations discussed below., 1. This theory finds application only to reactions occurring in the gas phase and solution having simple reactant, molecules., 2. The rate constants obtained by employing collision theory are found to be comparable to what has been, obtained from the Arrhenius equation only for the simple reactions but not for complex reactions., 3. This theory tells nothing about the exact mechanism behind the chemical reaction i.e. making and breaking, of chemical bonds., 4. The collision theory considers only the kinetic energy of reacting molecules and just ignored rotational and, vibrational energy which also plays an important role in reaction rate., 5. This theory did not consider the steric factor at all i.e. the proper orientation of the colliding molecules, needed to result in the chemical change., ,  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., 𝜌=, , 𝐴𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑, 𝑍𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑, , Copyright © Mandeep Dalal, , (171)

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A Textbook of Physical Chemistry – Volume I, , 142, , 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., 𝑅𝑎𝑡𝑒 = 𝑍𝑒 −𝐸𝑎/𝑅𝑇, , (172), , For experimental rate, multiply the equation (172) by the probability or steric factor i.e., 𝑅𝑎𝑡𝑒 = 𝜌𝑍𝑒 −𝐸𝑎/𝑅𝑇, , (173), , 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, 𝑁 2 8𝜋𝑅𝑇(𝑀𝐴 + 𝑀𝐵 ), √, 𝐴 = 𝜌 3 𝜎𝐴𝐵, 10, 𝑀𝐴 𝑀𝐵, , (174), , Substituting the values of different constants, we get, 𝑇(𝑀𝐴 + 𝑀𝐵 ), 2 √, 𝐴 = 2.753 × 1029 × 𝜌 × 𝜎𝐴𝐵, 𝑀𝐴 𝑀𝐵, , (175), , ii) For similar molecules:, If the colliding molecules are the same, the exponential part in equation (173) takes the form, 𝑁, 𝜋𝑅𝑇, 𝐴 = 𝜌 3 4𝜎 2 √, 10, 𝑀𝐴, 𝑇, 𝐴 = 3.893 × 1029 × 𝜌 × 𝜎 2 √, 𝑀𝐴, , (176), , (177), , 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., , Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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CHAPTER 3 Chemical Dynamics – I, , 143, ,  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:, 𝐾∗, 𝑘, 𝐴 + 𝐵 ⇌ 𝑋∗ →, 𝑃, , (178), , The equilibrium constant for reactants to activated complex conversion is, [𝑋 ∗ ], 𝐾 =, [𝐴][𝐵], ∗, , Copyright © Mandeep Dalal, , (179)

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A Textbook of Physical Chemistry – Volume I, , 144, , Henry Eyring showed that the rate constant for a chemical reaction with any order or molecularity can be given, by the following relation., 𝑘=, , 𝑅𝑇 ∗, 𝐾, 𝑁ℎ, , (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, 𝛥𝐺 ∗ = −𝑅𝑇 ln 𝐾 ∗, −, , 𝛥𝐺 ∗, = ln 𝐾 ∗, 𝑅𝑇, 𝛥𝐺 ∗, , 𝐾 ∗ = 𝑒 − 𝑅𝑇, , (181), (182), (183), , 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, 𝑘=, , 𝑅𝑇 −𝛥𝐺∗, 𝑒 𝑅𝑇, 𝑁ℎ, , (184), , If we put the thermodynamic value of free energy i.e. ΔG* = ΔH* − TΔS* in equation (184), we get, 𝑅𝑇 −(𝛥𝐻∗ −𝑇𝛥𝑆 ∗ ), 𝑅𝑇, 𝑒, 𝑁ℎ, , (185), , 𝛥𝑆 ∗, 𝛥𝐻 ∗, 𝑅𝑇, × 𝑒 𝑅 × 𝑒 − 𝑅𝑇, 𝑁ℎ, , (186), , 𝑘=, or, 𝑘=, , 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., Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal

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CHAPTER 3 Chemical Dynamics – I, , 147, ,  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., , Figure 12. The pictorial depiction of the double-sphere model of ionic reactions., , Now, if ZA and ZB are the charge numbers of the participating ions and x as the distance of separation, the force, of electrostatic interaction (FAB) between them can be given from the Coulomb’s law as:, Copyright © Mandeep Dalal

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A Textbook of Physical Chemistry – Volume I, , 150, , 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:, 𝑑𝐹 =, , 𝜆𝑒 2 𝑑𝜆, 4𝜋𝜀0 𝜀𝑥 2, , (220), , Where ε0 and ε are permittivities of the vacuum (8.854 × 10−12 C2 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 amount of work done in moving the “increment” closer by an extant dx will be, 𝑑𝑤 = 𝑑𝐹 × 𝑑𝑥, 𝑑𝑤 =, , 𝜆𝑒 2 𝑑𝜆, 𝑑𝑥, 4𝜋𝜀0 𝜀𝑥 2, , (221), (222), , 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., 𝑍 𝑟, , 𝑒2, 𝜆, 𝑤=, ∫ ∫ 2 𝑑𝜆 𝑑𝑥, 4𝜋𝜀0 𝜀, 𝑥, , (223), , 0 ∞, , 𝑤=, , 𝑍2𝑒2, 8𝜋𝜀0 𝜀 𝑟, , (224), , The work given the above equation is actually the contribution of the electrostatic interactions to the Gibbs, energy of the ion i.e., 𝐺𝐸𝐼, , 𝑍2𝑒 2, =, 8𝜋𝜀0 𝜀 𝑟, , (225), , In the light of the above correlation, the electrostatic contribution to the Gibbs free energy of discrete ions and, activated complex can be written as, 𝑍𝐴2 𝑒 2, 𝐺𝐸𝐼 (𝐴) =, 8𝜋𝜀0 𝜀 𝑟𝐴, , Buy the complete book with TOC navigation,, high resolution images and, no watermark., , Copyright © Mandeep Dalal, , (225)

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A Textbook of Physical Chemistry – Volume I, , 152, ,  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., ln 𝑘 = ln 𝑘0 −, , 𝑁𝑍𝐴 𝑍𝐵 𝑒 2, 𝑅𝑇4𝜋𝜀0 𝜀 𝑟, , (235), , Where ε0 and ε are permittivities of the vacuum (8.854 × 10−12 C2 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), we get, ln 𝑘 = −, , 𝑁𝑍𝐴 𝑍𝐵 𝑒 2 1, + ln 𝑘0, 𝑅𝑇4𝜋𝜀0 𝑟 𝜀, , (236), , 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., , Copyright © Mandeep Dalal

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CHAPTER 3 Chemical Dynamics – I, , 153, , 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), we have, 𝑚=−, , 𝑁𝑍𝐴 𝑍𝐵 𝑒 2, 𝑅𝑇4𝜋𝜀0 𝑟, , (237), , 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, 𝜕𝐺, ( ) = −𝑆, 𝜕𝑇 𝑃, , (238), , Also, the electrostatic contribution to the Gibbs free energy using the double sphere model is, ∗, 𝛥𝐺𝐸𝐼, =, , 𝑁𝑍𝐴 𝑍𝐵 𝑒 2, 4𝜋𝜀0 𝜀 𝑟, , (239), , However, the only quantity which is temperature-dependent in the above equation is ε. Therefore,, differentiating equation (239) with respect to temperature at constant pressure gives, ∗, 𝛥𝑆𝐸𝐼, , ∗ ), 𝜕(𝛥𝐺𝐸𝐼, 𝑁𝑍𝐴 𝑍𝐵 𝑒 2 𝜕(1/𝜀), = −(, (, ), ) =−, 𝜕𝑇, 4𝜋𝜀0 𝑟, 𝜕𝑇 𝑃, 𝑃, , (240), , or, 𝑁𝑍𝐴 𝑍𝐵 𝑒 2 𝜕𝜀), ( ), 4𝜋𝜀0 𝜀 2 𝑟 𝜕𝑇 𝑃, , (241), , 𝑁𝑍𝐴 𝑍𝐵 𝑒 2 𝜕 ln 𝜀), (, ), 4𝜋𝜀0 𝜀 𝑟, 𝜕𝑇 𝑃, , (242), , ∗, 𝛥𝑆𝐸𝐼, =, , or, ∗, 𝛥𝑆𝐸𝐼, =, , 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., , Copyright © Mandeep Dalal

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A Textbook of Physical Chemistry – Volume I, , 154, , 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., 𝑖=𝑛, , 1, 𝐼 = ∑ 𝑚𝑖 𝑧𝑖2, 2, , (243), , 𝑖=1, , Where mi and zi are the molarity and charge number of ith species, respectively. For instance, the value of z, for Ca2+ and Cl– in 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 𝑍𝐴 + 𝐵 𝑍𝐵 → 𝑋𝑍𝐴 +𝑍𝐵 → 𝑃, , (244), , A Danish physical chemist, J. N. Brønsted, proposed the rate equation relating reaction-rate (R) and activity, coefficient as, 𝑅 = 𝑘0 [𝐴][𝐵], Buy the complete book with TOC navigation,, high resolution images and, no watermark., , 𝑦𝐴 𝑦𝐵, 𝑦𝑋, , Copyright © Mandeep Dalal, , (245)

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CHAPTER 3 Chemical Dynamics – I, , 157, ,  The Comparison of Collision and Activated Complex Theory, After studying the collision as well as the transition state theories in detail, it is time to highlight the, key points of similarities and differences between the two. A comparative analysis of both theories is quite, beneficial as far as the practicality is concerned., 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 1. According to the transition state theory, the, reactions occur when the reactant molecules collide primary cause of the reaction is actually the, formation of an activated complex or the transition, with a sufficient amount of kinetic energy., state, which in turn, converts to the final product., 2. It is based upon the kinetic theory of gases., , 2. It is derived from the fundamentals of, thermodynamics., , 3. This theory considers the activation energy as the 3. This theory assumes the activation energy as the, minimum energy required to make the collision difference between the energy of the reacting, effective., molecules and the energy of the activated complex., 4. This theory tells nothing about the entropy of 4. The transition state theory enables us to measure, activation., the entropy of activation., 5. Collision theory is applicable to simple chemical 5. This theory provided reasonable predictions even, reactions and large deviations with experimental for the complex reactions., results are observed as the complexity increases., 6. The incorporation of the correction factor in 6. The incorporation of correction factor was, modified collision theory was arbitrary., justified in terms of entropy of activation i.e. ΔS*., 7. This theory tells nothing about the mechanism 7. The formation of the activated complex is very, much correlated with the actual mechanism going, involved., on., , Copyright © Mandeep Dalal

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158, , A Textbook of Physical Chemistry – Volume I, ,  Problems, Q 1. Discuss the fundamental concept of the effect of temperature on the reaction-rate with special reference, to Maxwell-Boltzmann distribution of energies., Q 2. Derive Arrhenius equation for the rate of reactions. How it can be used to determine the activation energy?, Q 3. Deduce the rate expression for the first order opposed by first-order reactions. How it can be used to yield, the value of backward reaction too?, Q 4. Derive and discuss the rate law for second-order opposed by second order., Q 5. What are the consecutive reactions? Discuss the condition required to apply the steady-state, approximation to a typical consecutive reaction., Q 6. Define concurrent or parallel reactions. Derive the rate law for the first-order case., Q 7. Discuss the collision of the theory bimolecular reaction when the colliding particles are dissimilar., Q 8. What are unimolecular reactions? How they are treated in the collision framework?, Q 9. Discuss the limitations of collision theory and the incorporation of the steric factor?, Q 10. Derive the rate law in the activated complex framework. How it can be used to determine the entropy of, activation?, Q 11. What is the single-sphere model of ionic reactions? How is it different from the double sphere model?, Q 12. Discuss the influence of ionic strength on the rate of ionic reactions of the third order., Q 13. Give five points of difference between the collision and activated complex theory., , Copyright © Mandeep Dalal

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LEGAL NOTICE, This document is an excerpt from the book entitled “A, Textbook of Physical Chemistry – Volume 1 by, Mandeep Dalal”, and is the intellectual property of the, Author/Publisher. The content of this document is, protected by international copyright law and is valid, only for the personal preview of the user who has, originally downloaded it from the publisher’s website, (www.dalalinstitute.com). Any act of copying (including, plagiarizing its language) or sharing this document will, result in severe civil and criminal prosecution to the, maximum extent possible under law., , This is a low resolution version only for preview purpose. If you, want to read the full book, please consider buying., , Buy the complete book with TOC navigation, high resolution, images and no watermark.

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Table of Contents, CHAPTER 1 ................................................................................................................................................ 11, Quantum Mechanics – I ........................................................................................................................ 11, , , Postulates of Quantum Mechanics .................................................................................................. 11, , , , Derivation of Schrodinger Wave Equation...................................................................................... 16, , , , Max-Born Interpretation of Wave Functions .................................................................................. 21, , , , The Heisenberg’s Uncertainty Principle.......................................................................................... 24, , , , Quantum Mechanical Operators and Their Commutation Relations............................................... 29, , , , Hermitian Operators – Elementary Ideas, Quantum Mechanical Operator for Linear Momentum,, Angular Momentum and Energy as Hermitian Operator ................................................................. 52, , , , The Average Value of the Square of Hermitian Operators ............................................................. 62, , , , Commuting Operators and Uncertainty Principle (x & p; E & t) .................................................... 63, , , , Schrodinger Wave Equation for a Particle in One Dimensional Box.............................................. 65, , , , Evaluation of Average Position, Average Momentum and Determination of Uncertainty in Position, and Momentum and Hence Heisenberg’s Uncertainty Principle..................................................... 70, , , , Pictorial Representation of the Wave Equation of a Particle in One Dimensional Box and Its, Influence on the Kinetic Energy of the Particle in Each Successive Quantum Level ..................... 75, , , , Lowest Energy of the Particle ......................................................................................................... 80, , , , Problems .......................................................................................................................................... 82, , , , Bibliography .................................................................................................................................... 83, , CHAPTER 2 ................................................................................................................................................ 84, Thermodynamics – I .............................................................................................................................. 84, , , Brief Resume of First and Second Law of Thermodynamics .......................................................... 84, , , , Entropy Changes in Reversible and Irreversible Processes ............................................................. 87, , , , Variation of Entropy with Temperature, Pressure and Volume ...................................................... 92, , , , Entropy Concept as a Measure of Unavailable Energy and Criteria for the Spontaneity of Reaction, ...........................................................................................................................................................94, , , , Free Energy, Enthalpy Functions and Their Significance, Criteria for Spontaneity of a Process ... 98, , , , Partial Molar Quantities (Free Energy, Volume, Heat Concept) ................................................... 104, , , , Gibb’s-Duhem Equation ................................................................................................................ 108, , , , Problems ........................................................................................................................................ 111, , , , Bibliography .................................................................................................................................. 112

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CHAPTER 3 .............................................................................................................................................. 113, Chemical Dynamics – I ........................................................................................................................ 113, , , Effect of Temperature on Reaction Rates ...................................................................................... 113, , , , Rate Law for Opposing Reactions of Ist Order and IInd Order..................................................... 119, , , , Rate Law for Consecutive & Parallel Reactions of Ist Order Reactions ....................................... 127, , , , Collision Theory of Reaction Rates and Its Limitations ............................................................... 135, , , , Steric Factor................................................................................................................................... 141, , , , Activated Complex Theory ........................................................................................................... 143, , , , Ionic Reactions: Single and Double Sphere Models ..................................................................... 147, , , , Influence of Solvent and Ionic Strength ........................................................................................ 152, , , , The Comparison of Collision and Activated Complex Theory ..................................................... 157, , , , Problems ........................................................................................................................................ 158, , , , Bibliography .................................................................................................................................. 159, , CHAPTER 4 .............................................................................................................................................. 160, Electrochemistry – I: Ion-Ion Interactions ..................................................................................... 160, , , The Debye-Huckel Theory of Ion-Ion Interactions ....................................................................... 160, , , , Potential and Excess Charge Density as a Function of Distance from the Central Ion ................. 168, , , , Debye-Huckel Reciprocal Length ................................................................................................. 173, , , , Ionic Cloud and Its Contribution to the Total Potential ................................................................ 176, , , , Debye-Huckel Limiting Law of Activity Coefficients and Its Limitations ................................... 178, , , , Ion-Size Effect on Potential ........................................................................................................... 185, , , , Ion-Size Parameter and the Theoretical Mean - Activity Coefficient in the Case of Ionic Clouds with, Finite-Sized Ions ............................................................................................................................ 187, , , , Debye-Huckel-Onsager Treatment for Aqueous Solutions and Its Limitations ............................ 190, , , , Debye-Huckel-Onsager Theory for Non-Aqueous Solutions........................................................ 195, , , , The Solvent Effect on the Mobility at Infinite Dilution ................................................................ 196, , , , Equivalent Conductivity (Λ) vs Concentration C1/2 as a Function of the Solvent ......................... 198, , , , Effect of Ion Association Upon Conductivity (Debye-Huckel-Bjerrum Equation) ...................... 200, , , , Problems ........................................................................................................................................ 209, , , , Bibliography .................................................................................................................................. 210, , CHAPTER 5 .............................................................................................................................................. 211, Quantum Mechanics – II .................................................................................................................... 211, , , Schrodinger Wave Equation for a Particle in a Three Dimensional Box ...................................... 211

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, , The Concept of Degeneracy Among Energy Levels for a Particle in Three Dimensional Box .... 215, , , , Schrodinger Wave Equation for a Linear Harmonic Oscillator & Its Solution by Polynomial Method, .........................................................................................................................................................217, , , , Zero Point Energy of a Particle Possessing Harmonic Motion and Its Consequence ................... 229, , , , Schrodinger Wave Equation for Three Dimensional Rigid Rotator .............................................. 231, , , , Energy of Rigid Rotator ................................................................................................................ 241, , , , Space Quantization ........................................................................................................................ 243, , , , Schrodinger Wave Equation for Hydrogen Atom: Separation of Variable in Polar Spherical, Coordinates and Its Solution ......................................................................................................... 247, , , , Principal, Azimuthal and Magnetic Quantum Numbers and the Magnitude of Their Values ....... 268, , , , Probability Distribution Function .................................................................................................. 276, , , , Radial Distribution Function ......................................................................................................... 278, , , , Shape of Atomic Orbitals (s, p & d) .............................................................................................. 281, , , , Problems ........................................................................................................................................ 287, , , , Bibliography .................................................................................................................................. 288, , CHAPTER 6 .............................................................................................................................................. 289, Thermodynamics – II ........................................................................................................................... 289, , , Clausius-Clapeyron Equation ........................................................................................................ 289, , , , Law of Mass Action and Its Thermodynamic Derivation ............................................................. 293, , , , Third Law of Thermodynamics (Nernst Heat Theorem, Determination of Absolute Entropy,, Unattainability of Absolute Zero) And Its Limitation ................................................................... 296, , , , Phase Diagram for Two Completely Miscible Components Systems ........................................... 304, , , , Eutectic Systems (Calculation of Eutectic Point) .......................................................................... 311, , , , Systems Forming Solid Compounds AxBy with Congruent and Incongruent Melting Points ....... 321, , , , Phase Diagram and Thermodynamic Treatment of Solid Solutions.............................................. 332, , , , Problems ........................................................................................................................................ 342, , , , Bibliography .................................................................................................................................. 343, , CHAPTER 7 .............................................................................................................................................. 344, Chemical Dynamics – II ...................................................................................................................... 344, , , Chain Reactions: Hydrogen-Bromine Reaction, Pyrolysis of Acetaldehyde, Decomposition of, Ethane ............................................................................................................................................ 344, , , , Photochemical Reactions (Hydrogen-Bromine & Hydrogen-Chlorine Reactions) ....................... 352, , , , General Treatment of Chain Reactions (Ortho-Para Hydrogen Conversion and Hydrogen-Bromine, Reactions) ....................................................................................................................................... 358

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, , Apparent Activation Energy of Chain Reactions .......................................................................... 362, , , , Chain Length ................................................................................................................................. 364, , , , Rice-Herzfeld Mechanism of Organic Molecules Decomposition (Acetaldehyde) ...................... 366, , , , Branching Chain Reactions and Explosions (H2-O2 Reaction) ..................................................... 368, , , , Kinetics of (One Intermediate) Enzymatic Reaction: Michaelis-Menten Treatment .................... 371, , , , Evaluation of Michaelis's Constant for Enzyme-Substrate Binding by Lineweaver-Burk Plot and, Eadie-Hofstee Methods ................................................................................................................. 375, , , , Competitive and Non-Competitive Inhibition ............................................................................... 378, , , , Problems ........................................................................................................................................ 388, , , , Bibliography .................................................................................................................................. 389, , CHAPTER 8 .............................................................................................................................................. 390, Electrochemistry – II: Ion Transport in Solutions ....................................................................... 390, , , Ionic Movement Under the Influence of an Electric Field ............................................................ 390, , , , Mobility of Ions ............................................................................................................................. 393, , , , Ionic Drift Velocity and Its Relation with Current Density .......................................................... 394, , , , Einstein Relation Between the Absolute Mobility and Diffusion Coefficient .............................. 398, , , , The Stokes-Einstein Relation ........................................................................................................ 401, , , , The Nernst-Einstein Equation ....................................................................................................... 403, , , , Walden’s Rule ............................................................................................................................... 404, , , , The Rate-Process Approach to Ionic Migration ............................................................................ 406, , , , The Rate-Process Equation for Equivalent Conductivity .............................................................. 410, , , , Total Driving Force for Ionic Transport: Nernst-Planck Flux Equation ....................................... 412, , , , Ionic Drift and Diffusion Potential ................................................................................................ 416, , , , The Onsager Phenomenological Equations ................................................................................... 418, , , , The Basic Equation for the Diffusion ............................................................................................ 419, , , , Planck-Henderson Equation for the Diffusion Potential ............................................................... 422, , , , Problems ........................................................................................................................................ 425, , , , Bibliography .................................................................................................................................. 426, , INDEX ......................................................................................................................................................... 427