Activation Energy and Temperature Dependence of Rate Constants

TABLE 14.2 Summary of the Kinetics of Zero-Order, First-Order, and Second-Order Reactions  

Order Rate Law Time Equation Half-Life 

Matty of the known zero-order reactions take place on a metal surface. An example is the decomposition of nitrous oxide (N2O) to nitrogen and oxygen in the presence of platinum (Pt)  

When all the binding sites on Pt are occupied, the rate becomes constant regardless of the amount of N2O present in the gas phase. As we will see in Section 14.6, another well-studied zero-order reaction occurs in enzyme catalysis. 

Third-order and higher order reactions are quite complex they are not presented in this book. Table 14.2 summarizes the kinetics of zero-order, first-order, and second-order reactions. 

Activation Energy and Temperature Dependence of Rate Constants 

The kinetic molecular theory of gases (p. 201) postulates that gas molecules frequently collide with one another. Therefore, it seans logical to assume— and it is generally true—that chemical reactions occur as a result of collisions between reacting molecules. In terms of the collision theory of chemical kinetics, then, we expect the rate of a reaction to be directly proportional to the number of molecular collisions per second, or to the frequency of molecular collisions  

This simple relationship explains the dependence of reaction rate on concentration. 

The reaction is first order in both A and B and obeys second-order kinetics. 

The coUision theory is intuitively appealing, but the relationship between rate and molecular coUisions is more complicated than you might expect. The implication of the coUision theory is that a reaction always occurs when an A and a B molecule coUide. However, not aU coUisions lead to reactions. Calculations based on the kinetic molecular theory show that, at ordinary pressures (say, 1 attn) and temperatures (say, 298 K), there are about 1 X 10 binary coUisions (coUisions between two molecules) in 1 mL of volume every second in the gas phase. Even more coUisions per second occur in Uquids. If every binary coUision led to a product, then most reactions would be complete almost instantaneously. In practice, we find that the rates of reactions differ greatly. This means that, in many cases, coUisions alone do not guarantee that a reaction will take place. 

---

ACTIVATION ENERGY AND TEMPERATURE DEPENDENCE OF RATE CONSTANTS... 

Since ktw is expected to have a small or zero activation energy, the temperature dependence of Pi is determined by /kb (the reciprocal of the branching rate constant), and Pi will thus decrease with an increase in temperature, as is observed. 

Calculate Arrhenius factors and activation energies from measnrements of the temperature dependence of rate constants (Section 18.5, Problems 35-40). 

The diagram opposite which distinguishes between a transition state and an intermediate also shows the Gibbs energy of activation, AG, for each step in the two-step reaction path. Enthalpies and entropies of activation, AH and obtained from temperature dependence of rate constants, can shed light on mechanisms. Equation 25.6 gives the relationship between the rate constant, temperature and activation parameters. 

Temperature-dependence of rate constants activation energy and entropy... 

Unlike water and aqueous solutions, this intermediate s decay is slower (see Figure 4) and follows a second-order reaction law (Figure 5) with a rate constant of (1.1 0.2). 1010M 1 sec."1. Analysis of temperature dependence of rate constant gives the activation energy of observed reaction as 2 0.5 kcal./mole. 

Temperature Dependence of Rate Constants To react, molecules must possess energy equal to or greater than the activation energy. The rate constant generally increases with increasing temperature. The Arrhenius eqiration relates the rate constant to activation energy and temperature. 

Meanwhile, the pre-exponential factor A in the Arrhenius Eq. (2.39) is the temperature independent factor related to reaction frequency. Comparing the Eq. (2.33) for the collision theory and Eq. (2.38) with the transition state theory, the pre-exponential factors in these theories contain temperature dependences of T and T respectively. Experimentally, for most of reactions for which the activation energy is not close to zero, the temperature dependence of the reaction rate constants are known to be determined almost solely by exponential factor, and the Arrhenius expression holds as a good approximation. Only for the reaction with near-zero activation energy, the temperature dependence of the pre-exponential factor appears explicitly, and the deviation from the Arrhenius expression can be validated. In this case, an approximated equation modifying the Arrhenius expression can be used. 

The temperature dependence of rate constants for both gaseous and liquid-state reactions is usually well described by the Arrhenius formula, Eq. (12.3-2). For activation-limited reactions, the activation energies are roughly equal to those for gas-phase reactions. This is as expected, since the collisional activation is very similar to that of gaseous reactions. In the case of diffusion-limited reactions, the temperature dependence of the rate constant is governed by the diffusion coefficients. Diffusion coefficients in liquids commonly have a temperature dependence given by Eq. (10.4-5), which is also of the same form as the Arrhenius formula ... 

This representation is completely parallel to the temperature dependence of rate constants for reactions of small molecules, isj, Ep, and are, therefore, the activation energies for initiation, propagation, and termination reactions, respectively. The values are tabulated extensively in the Polymer Handbook [4]. The temperature dependence of and can be easily found by substituting Eq. (5.7.1) in Eqs. (5.3.14) and (5.4.4) to get... 

Use the Arrhenius equation and temperature dependence of a rate constant to determine an activation energy (Example 13.8). 

An apparent compensation effect can result from errors in the experimental data used for an Arrhenium plot. Besides trivial errors, there may also occur errors in the calculation of rate constants, for instance when a homogeneous and a heterogeneous reaction occur simultaneously or when a heterogeneous reaction undergoes a change from a certain reaction order to another order. A temperature dependence of the activation energy, and the variability of the effective surface of the catalyst with temperature, especially caused by diffusion processes, may also account for apparent compensation effects. 

Racemization (1,2-oxygen migration) occurred between 2a and 2b (97T12203). The racemization took place slowly in solution at room temperature and obeyed reversible first-order kinetics krac = 4.3 x 10 6 s-1 (25°C, CH2C12). Tlie AH and AS, calculated from rate constants at temperatures between 20° and 40°C, were 24.3 kcal/mol and -2.0 eu, respectively. The rate of the racemization was hardly dependent on either the concentration of the solution or the polarity of the solvent examined. The low activation energy and the analogy of the mechanism to that for the dispro-... 

The expressions above depend on light intensity I, reactant concentration A, activation energy, and temperature by way of the rate constant... 

The activation energy for the second stage k ) and enthalpy for the first equilibrium K ) have been determined from the temperature-dependent decomposition rate constants obtained in the presence and absence of boric acid. 

All SFC processes operate at above the critical temperature (Tc) of supercritical fluids. Temperature is a critical controlling variable of the SFC process based on both thermodynamic and kinetic considerations. First, solubility is a function of temperature, and this will determine the supersaturation ratio or the driving force for the crystallization of individual polymorphs. Second, the kinetics of polymorphic transformation is governed by the Arrhenius law and is also temperature dependent. The rate constant of the conversion is related to the activation energy and the mass transfer process involved (i.e., diffusion, evaporation, or mixing in supercritical fluids). 

The Arrhenius equation expressing the dependence of the rate constant on activation energy and temperature. 

---