Rate constants

The rate constant (kr) of a reaction is a concentration-independent measure of the velocity of a reaction. For a first-order reaction, kr has units of (time) for a second-order reaction, kr has units of (concentration) (time) . In general, the rate constant of an nth-order reaction has units of (concentration) ( (time) .  

At equilibrium the net rate is by definition zero. If we compare the rate equation for an elementary step, e.g the rate equation 

The constant k which appears in the rate equation (9.1) is known as the rate constant, the rate coefficient, or the specific rate the first expression wili be used in this book. The units of the rate constant vary with the order of the reaction. Suppose, for example, that a reaction is of the first order, i.e, 

The units corresponding to other orders can easily be worked out. 

Note that since the rale in general depends on the reactant or product with which we are concerned, the rate constant also reflects this dependence. 

Some results for a reaction between two substances A and B are shown below  

Assuming that the reaction has a simple order, i.e., that the rate equation is of the form 

In the rate laws mentioned so far, there has always been a proportionality constant, the rate constant (k), which is a fundamental property of any given chemical reaction. A rate must have the units of concentration per unit time, often moles per liter per second. For a first-order process such as the SnI reaction, the rate constant k must have the units of reciprocal seconds (s ) (if we use the second as our time unit. Fig. 8.11). 

Rate = concentration/time = (rate constant) (concentration) rearranging we have j x (s) x = k 

FIGURE8.il For a first-order reaction, the rate constant (k) has the units of time = s.  

For a second-order reaction, k must have units of reciprocal moles per liter per second [(mol/L) s ] (Fig. 8.12). 

With the rate, reactant concentrations, and reaction orders known, the sole remaining unknown in the rate law is the rate constant, k. The rate constant is specific for a particular reaction at a particular temperature. The experiments with the reaction of O2 and NO were run at the same temperature, so we can use data from any to solve for k. From experiment I in Table 16.2, for instance, we obtain 

CHAPTER 16 Kinetics Rates and Mechanisms of Chemical Reactions 

Units of the Rate Constant k for Several Overall Reaction Orders 

Always check that the values of fc for a series are constant within experimental error. To three significant figures, the average value of k for the five experiments in Table 16.2 is 1.72x10- L /mol -s. 

Note the units for the rate constant. With concentrations in mol/L and the reaction rate in units of mol/L-time, the units for k depend on the order of the reaction and, of course, the time unit. The units for k in our example, L /mol -s, are required to give a rate with units of mol/L-s  

The rate constant will always have these units for an overall third-order reaction with the time unit in seconds. Table 16.3 shows the units of k for some common overall reaction orders, but you can always determine the units mathematically. 

An intravenous infusion involves a continuous flow of drug into a patient at a rate defined by the infusion rate constant, Rini, with units of mass/time. Discussions of infusion normally present the infusion rate constant as inf, which may be confused with a true reaction rate constant. Therefore, this presentation of infusion uses a less ambiguous variable, Rmi, for the infusion rate constant. 

The mathematical relationship between Cp and time for an IV infusion is shown in Equation 7.17.1 

This equation assumes a one-compartment model and constant, first-order elimination. Based on Equation 7.17, the e kt term approaches 0 as t increases, and Cp approaches Rmi/kx]Vd. The value of 7 inf/A eiVd corresponds to Cpss (Equation 7.18). 

In a strict sense, Cp never achieves Cpss. In practice, infusion of a drug for a period of four /1/2 values achieves between 90% and 95% of Cpss. 

PROBLEM For a 70-kg patient, the Vd of morphine is 230 L, and CLy is 1,680 mL/min. Calculate the infusion rate (Winl ) of morphine required to sustain a Cpss of 15 ng/mL in a 70-kg patient 

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Hammen equation A correlation between the structure and reactivity in the side chain derivatives of aromatic compounds. Its derivation follows from many comparisons between rate constants for various reactions and the equilibrium constants for other reactions, or other functions of molecules which can be measured (e g. the i.r. carbonyl group stretching frequency). For example the dissociation constants of a series of para substituted (O2N —, MeO —, Cl —, etc.) benzoic acids correlate with the rate constant k for the alkaline hydrolysis of para substituted benzyl chlorides. If log Kq is plotted against log k, the data fall on a straight line. Similar results are obtained for meta substituted derivatives but not for orthosubstituted derivatives. 

Derive Eq. XI-IS, assuming a Langmuir adsorption process described in Eq. XI-2, where ka and kd are the adsorption and desorption rate constants. Treat the solution... 

The rate constants k and ki may be related to the concepts of the preceding section as follows. First, k is simply the reciprocal of the adsorption time, that is. 

Mention was made in Section XVIII-2E of programmed desorption this technique gives specific information about both the adsorption and the desorption of specific molecular states, at least when applied to single-crystal surfaces. The kinetic theory involved is essentially that used in Section XVI-3A. It will be recalled that the adsorption rate was there taken to be simply the rate at which molecules from the gas phase would strike a site area times the fraction of unoccupied sites. If the adsorption is activated, the fraction of molecules hitting and sticking that can proceed to a chemisorbed state is given by exp(-E /RT). The adsorption rate constant of Eq. XVII-13 becomes... 

Just as the surface and apparent kinetics are related through the adsorption isotherm, the surface or true activation energy and the apparent activation energy are related through the heat of adsorption. The apparent rate constant k in these equations contains two temperature-dependent quantities, the true rate constant k and the parameter b. Thus... 

It was pointed out that a bimolecular reaction can be accelerated by a catalyst just from a concentration effect. As an illustrative calculation, assume that A and B react in the gas phase with 1 1 stoichiometry and according to a bimolecular rate law, with the second-order rate constant k equal to 10 1 mol" see" at 0°C. Now, assuming that an equimolar mixture of the gases is condensed to a liquid film on a catalyst surface and the rate constant in the condensed liquid solution is taken to be the same as for the gas phase reaction, calculate the ratio of half times for reaction in the gas phase and on the catalyst surface at 0°C. Assume further that the density of the liquid phase is 1000 times that of the gas phase. 

We now make two coimections with topics discussed earlier. First, at the begiiming of this section we defined 1/Jj as the rate constant for population decay and 1/J2 as the rate constant for coherence decay. Equation (A1.6.63) shows that for spontaneous emission MT = y, while 1/J2 = y/2 comparing with equation (A1.6.60) we see that for spontaneous emission, 1/J2 = 0- Second, note that y is the rate constant for population transfer due to spontaneous emission it is identical to the Einstein A coefficient which we defined in equation (Al.6.3). 

With the help of U, an expression for the rate constant for the reaction... 

Table A3.4.2 Rate laws, reaction order, and rate constants.

If certain species are present in large excess, their concentration stays approximately constant during the course of a reaction. In this case the dependence of the reaction rate on the concentration of these species can be included in an effective rate constant The dependence on the concentrations of the remaining species then defines the apparent order of the reaction. Take for example equation (A3,4.10) with e. The... 

The rate constant in this case is of the order of 10 s depending on the rovibronic level considered. 

Experimentally, one finds the same first-order rate law as for monomolecular reactions, but with an effective rate constant /rthat now depends on [M],... 

The correct treatment of the mechanism (equation (A3.4.25), equation (A3.4.26) and equation (A3.4.27), which goes back to Lindemann [18] and Hinshelwood [19], also describes the pressure dependence of the effective rate constant in the low-pressure limit ([M] < [CHoNC], see section A3.4.8.2). 

The second-order rate law for bimolecular reactions is empirically well confinned. Figure A3.4.3 shows the example of methyl radical recombination (equation (A3.4.36)) in a graphical representation following equation (A3.4.38) [22, 23 and 24]. For this example the bimolecular rate constant is... 

If the dominant contributions /r,[M.] are approximately constant, this leads to pseudo second-order kinetics with an effective rate constant... 

A bimoleciilar reaction can be regarded as a reactive collision with a reaction cross section a that depends on the relative translational energy of the reactant molecules A and B (masses and m ). The specific rate constant k(E ) can thus fonnally be written in tenns of an effective reaction cross section o, multiplied by the relative centre of mass velocity... 

Simple collision theories neglect the internal quantum state dependence of a. The rate constant as a function of temperature T results as a thennal average over the Maxwell-Boltzmaim velocity distribution p Ef. 

We use the symbol for Boltzmaim s constant to distingiush it from tire rate constant k. Equation (A3.4.85) defines the thennal average reaction cross section (a). 

Combining equation (A3,4,95). equation (A3,4,96) and equation (A3.4.97) one obtains the first Eyring equation for iinimolecular rate constants ... 

The quasi-equilibrium assumption in the above canonical fonn of the transition state theory usually gives an upper bound to the real rate constant. This is sometimes corrected for by multiplying (A3.4.98) and (A3.4.99) with a transmission coefifiwient 0 < k < 1. 

A specific unimolecular rate constant for the decay of a highly excited molecule at energy E and angular momentum J takes the fomr... 

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

The collision partners may be any molecule present in the reaction mixture, i.e., inert bath gas molecules, but also reactant or product species. The activation k and deactivation krate constants in equation (A3.4.125) therefore represent the effective average rate constants. 

This yields die qiiasi-stationaty reaction rate with an effective unimolecular rate constant... 

The effective rate law correctly describes the pressure dependence of unimolecular reaction rates at least qualitatively. This is illustrated in figure A3,4,9. In the lunit of high pressures, i.e. large [M], becomes independent of [M] yielding the high-pressure rate constant of an effective first-order rate law. At very low pressures, product fonnation becomes much faster than deactivation. A j now depends linearly on [M]. This corresponds to an effective second-order rate law with the pseudo first-order rate constant Aq ... 

Figure A3.4.9. Pressure dependence of the effective unimolecular rate constant. Schematic fall-off curve for the Lindemaim-FIinshelwood mechanism. A is the (constant) high-pressure limit of the effective rate constant...

This leads to the quasi-stationary rate constant of equation (A3,4,133) if 4k + k +k f, which is... 

The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... 

The resulting rate law agrees with the fonn found experimentally. Of course the postidated mechanism can only be proven by measuring the rate constants of the individual elementary steps separately and comparing calculated rates of equation (A3.4.148) with observed rates of HBr fomiation. 

Table A3.4.3. Rate constants for the reaction of with O, [73], The rate constants are given in temis of the...

Quack M and Troe J 1974 Specific rate constants of unimoiecuiar processes ii. Adiabatic channei modei Ber. Bunsenges. Phys. Chem. 78 240-52... 

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