Collision theory of gaseous reactions

In this section we consider the factors that determine the magnitude of the rate coefficient for a gaseous bimolecular reaction. 

If the effective circular cross section of the molecule for collisions has diameter p, two molecules will collide if their centers come within a distance p of each other. If we now imagine that all of the molecules except one (which we will call X) shrink to points, X will still collide with the other molecules when they come within a distance p of each other provided that we artificially expand the diameter of X to 2p. Now, in unit time the expanded molecule X, which has an artificial radius p, will sweep out a volume vrp c, where c is the average speed of a molecule. Therefore, if there are n point molecules per unit volume, and we assume that all of these molecules are stationary, the number of molecules with which X will collide in unit time will be iTfrcn  

If we now consider molecules A (in concentration colliding with molecules B (in concentration /ig), the number of collisions per second that one A molecule makes with the B molecules is TTp cn. This expression gives the maximum chemical reaction rate, assuming that each collision between A and B molecules results in a reaction. 

Exercise 3.5. Calculate the approximate maximum rate for a gaseous 

Solution. We must first determine the number of molecules per unit volume ( t,) of each gas at I atm and O C. This can be found using the ideal gas equation in the form of Eq. (1.8g) 

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The Collision Theory of Bimolecular Gaseous Reactions. This is the earliest theory of reaction rates. Since reaction between two species takes place only when they are in contact, it is reasonable to suppose that the reactant species must collide before they react. Since our knowledge of molecular collisions is more complete for the gaseous phase than for the liquid phase (in the latter case we speak of encounters rather than collisions), we will restrict our discussion to bimolecular reactions in the gaseous phase. 

It is easy to understand a bimolecular reaction on the basis of collision theory. Thus, when two molecules A and B collide their relative kinetic energy exceeds the threshold energy, the collision may result in the breaking of bonds and the formation of new bonds. But how can one account for a unimolecular reaction If we assume that in such a reaction (A — P) the molecule A acquires the necessary activation energy by colliding with another molecule, then the reaction should obey second-order kinetics and not the first-order kinetics which is actually observed in several unimolecular gaseous reactions. A satisfactory theory of these reactions was proposed by F.A. Lindemann in 1922. According to him, a unimolecular reaction... 

The atomic processes that are occurring (under conditions of equilibrium or non equilibrium) may be described by statistical mechanics. Since we are assuming gaseous- or liquid-phase reactions, collision theory applies. In other words, the molecules must collide for a reaction to occur. Hence, the rate of a reaction is proportional to the number of collisions per second. This number, in turn, is proportional to the concentrations of the species combining. Normally, chemical equations, like the one given above, are stoichiometric statements. The coefficients in the equation give the number of moles of reactants and products. However, if (and only if) the chemical equation is also valid in terms of what the molecules are doing, the reaction is said to be an elementary reaction. In this case we can write the rate laws for the forward and reverse reactions as Vf = kf[A]"[B]6 and vr = kr[C]c, respectively, where kj and kr are rate constants and the exponents are equal to the coefficients in the balanced chemical equation. The net reaction rate, r, for an elementary reaction represented by Eq. 2.32 is thus... 

Summary.—The mechanism of the activation process in gaseous systems has been investigated from the point of view of (1) activation by radiation (2) activation by collision. An increase in the radiation density of possible activating frequencies has resulted in no increased reaction velocity. The study of the bimolecular decomposition of nitrous oxide at low pressures has led to the conclusion that the reaction is entirely heterogeneous at these pressures. A study of the unimolecular decomposition of nitrogen pentoxide between pressures of 7io mm. Hg and 2 X 10 3 mm. Hg shows no alteration in the rate of reaction such as was found by Hirst and Rideal but follows exactly the rate determined by Daniels and Johnson at high pressures. No diminution of the reaction velocity as might be ex-expected from Lindemann s theory was observed. 

The activated-complex theory provides a plausible explanation of the first-order rate of unimolecular gaseous reactions. In such a reaction the reacting molecules gain the energy of activation by colhsion with other molecules. This might be thought of as a second-order process, since the number of collisions is proportional to the square of the concentration. However, Lindemann showed in 1922 that activation by collision could result in first-order rates. If A is an activated molecule of reactant, the equilibrium between A and A and reaction to products B can be represented as... 

As the fundamental concepts of chemical kinetics developed, there was a strong interest in studying chemical reactions in the gas phase. At low pressures the reacting molecules in a gaseous solution are far from one another, and the theoretical description of equilibrium thermodynamic properties was well developed. Thus, the kinetic theory of gases and collision processes was applied first to construct a model for chemical reaction kinetics. This was followed by transition state theory and a more detailed understanding of elementary reactions on the basis of quantum mechanics. Eventually, these concepts were applied to reactions in liquid solutions with consideration of the role of the non-reacting medium, that is, the solvent. 

The most simple chemical reactions are the homogeneous gaseous reactions, and when they involve simple enough molecular species, they should be described fairly well J>y some simple treatments familiar to all graduate students in physics and chemistry, namely the collision theory and the activated state method. This attitude is shared among most people who are more interested in the practical results of these methods than in the physical pictures which are involved. However, a critical inspection of the grounds of these methods shows some obscure points which could invalidate to some extent the treatments themselves. 

In many cases, there must be energy transfer between the reacting molecules. For reactions that take place in the gas phase, molecular collisions constitute the vehicle for energy transfer, and our description of gas phase reactions begins with a kinetic theory approach to collisions of gaseous molecules. In simplest terms, the two requirements that must be met for a reaction to occur are (1) a collision must occur and (2) the molecules must possess sufficient energy to cause a reaction to occur. It will be shown that this treatment is not sufficient to explain reactions in the gas phase, but it is the starting point for the theory. 

One of the major mechanistic steps during heterogeneous chemical reactions is the collision between the gaseous species and the catalytically active solid adsorbent. According to kinetic theory the rate at which these collisions occur is given by... 

The fact that it is the volume concentrations which determine the rate of a simple gaseous reaction (and not, for example, the mole fractions) receives a simple interpretation as soon as it is assumed that reaction takes place at the collision of the reacting molecules. According to kinetic theory the number of collisions, per unit time and volume, of the molecules A and B is proportional to the product [A] [5] of their concentrations. These considerations are very familiar and need not be elaborated. For the present it is sufficient to emphasize that the customary use of volume concentrations in kinetics, including the kinetics of liquids, has its origin (a) in the experimental results obtained from gas reactions and (6) in the support obtained from the kinetic theory of gases. However, this question will be referred to again in 15 5. 

The effect of a catalyst on the rate equation is to increase the value of the rate constant. Table 16.11 summarizes the changes that affect the rate of reaction and the rate constant. Rate constants are unaffected by changes in concentration and are only affected by temperature (as described by the Arrhenius equation) or the presence of a catalyst, which provides a new pathway or reaction mechanism. Rates increase with concentration and pressure (if gaseous reactants are involved), which can be accounted for by simple collision theory (Chapter 6). 

Two theories have been developed for the prediction of rates of elementary reactions. One is the collision theory [15] based on the classical kinetic theory of gases and applicable to gaseous reactions only. The other is the transition state theory [27] making use of statistical mechanics and quantum mechanics. As an example, the collision theory predicts the rate of the elementary reaction (1.1.14) as... 

Collision theory. It is "estimated that reaction occurred roughly at every collision between sodium, Na, and methyl iodide, CH3I What statement can you make about the energy of activation for the gaseous reaction Na-I-CH, Na -I-CH, ... 

There are two types of molecular theories on reaction rates collision theory and transitimr state theory. According to the coUision theory, the upper limit of a bimolecular rate constant of a gaseous reaction is the molecular collision frequency obtained by gas kinetics. The molecular coUtsimr frequency Zab between molecules, A and B, is given by... 

A reaction between reacting molecules or ions can lake place only when they come sufficiently close tc ether. The rate of reaction will thus depend on the frequency with which the reacting particles collide (Collision Theory). The increase in concentration of one or more reactants by admitting their additional amounts (or by increasing the pressure in case of gaseous reactants) will increase the frequency of collision and is therefore expected to increase the rate of the reaction. 

Transport of the gas to the surface and the initial interaction. The first step in heterogeneous reactions involving the uptake and reaction of gases into the liquid phase is diffusion of the gas to the interface. At the interface, the gas molecule either bounces off or is taken up at the surface. These steps involve, then, gaseous diffusion, which is determined by the gas-phase diffusion coefficient (Dg) and the gas-surface collision frequency given by kinetic molecular theory. 

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