First-order reactions rate constant

In the case of enzymes, even slight changes in the steric arrangement and mobility of the amino acid residues which participate in catalysis can lead to loss of activity. Nevertheless, a relatively high pressure is often required to inhibit enzymes. But the pressure required can be reduced by increasing the temperature, as shown in Fig. 2.41 for a-amylase. While a pressure of 550 MPa is required at 25 °C to inactivate the enzyme with a rate constant (first order reaction) of A = 0.01 min , a pressure of only 340 MPa is required at 50 °C. 

STRATEGY We need to plot the natural logarithm of the reactant concentration as a function of t. If we get a straight line, the reaction is first order and the slope of the graph is —k. We could use a spreadsheet program or the Living Graph Determination of Rate Constant (first-order rate law) on the Weh site for this book to make the plot. 

For first-order reactions, the plots were straight lines of slope -kQ /2.303, where is the first-order rate constant. Mixed order reactions gave curves on both the zero and first-order plots. 

A reaction carried out in the laboratory employed spherical catalyst pellets of approximate total volume = 1.5 cm. The observed rate was 5 X 10 mol/cm cat-s. It is known from other data that the intrinsic rate constant (first order) is 2.75 x 10 s . What would be the dimension of a spherical particle that would yield the same observed rate D f = 0.02cm /s. 

Figure 35. The distribution curve of enzymes with respect to the reaction rate constant (first-order) in enzymatic catalysis.

Cotton fabric filler. ite constants for melamine-formaldehyde resins at pH 7.7 Reaction Temp. (°Q Second order rate constant First order rate constant of ... 

Kinetic measurements were performed employii UV-vis spectroscopy (Perkin Elmer "K2, X5 or 12 spectrophotometer) using quartz cuvettes of 1 cm pathlength at 25 0.1 C. Second-order rate constants of the reaction of methyl vinyl ketone (4.8) with cyclopentadiene (4.6) were determined from the pseudo-first-order rate constants obtained by followirg the absorption of 4.6 at 253-260 nm in the presence of an excess of 4.8. Typical concentrations were [4.8] = 18 mM and [4.6] = 0.1 mM. In order to ensure rapid dissolution of 4.6, this compound was added from a stock solution of 5.0 )j1 in 2.00 g of 1-propanol. In order to prevent evaporation of the extremely volatile 4.6, the cuvettes were filled almost completely and sealed carefully. The water used for the experiments with MeReOj was degassed by purging with argon for 0.5 hours prior to the measurements. All rate constants were reproducible to within 3%. 

When the dienophile does not bind to the micelle, reaction will take place exclusively in the aqueous phase so that the second-order rate constant of the reaction in the this phase (k,) is directly related to the ratio of the observed pseudo-first-order rate constant and the concentration of diene that is left in this phase. 

For weU-defined reaction zones and irreversible, first-order reactions, the relative reaction and transport rates are expressed as the Hatta number, Ha (16). Ha equals (k- / l ) where k- = reaction rate constant, = molecular diffusivity of reactant, and k- = mass-transfer coefficient. Reaction... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Figure 3-10. Rate constants for two competing eiementary first order reactions.

AHp/a) is the heat of reaction, which is a function of temperature, but it is assumed constant. The rate expression for a first order reaction is (-1- ) = koC e-E T... 

A = rate constant (pre-exponential factor from Arrhenius equation k = A exp (-E /RT), sec (i.e., for a first order reaction) B = reduced activation energy, K C = liquid heat capacity of the product (J/kg K)... 

For a first-order reaction, therefore, a plot of In Ca (or log Ca) vs. / is linear, and the first-order rate constant can be obtained from the slope. A first-order rate constant has the dimension time , the usual unit being second. ... 

A third method, or phenomenon, capable of generating a pseudo reaction order is exemplified by a first-order solution reaction of a substance in the presence of its solid phase. Then if the dissolution rate of the solid is greater than the reaction rate of the dissolved solute, the solute concentration is maintained constant by the solubility equilibrium and the first-order reaction becomes a pseudo-zero-order reaction. 

A parameter such as a rate constant is usually obtained as a consequence of various arithmetic manipulations, and in order to estimate the uncertainly (error) in the parameter we must know how this error is related to the uncertainties in the quantities that contribute to the parameter. For example, Eq. (2-33) for a pseudo-first-order reaction defines k, which can be determined by a semilogarithmic plot according to Eq. (2-6). By a method to be described later in this section the uncertainty in itobs (expressed as its variance associated with cb. Thus, we need to know how the errors in fcobs and cb are propagated into the rate constant k. 

Suppose that Cy = 0, Cz = 0, as is often the case. Then the final product concentrations are found by setting f = < in Eqs. (3-12) and (3-13) we obtain Cy = ( ki/k and c = c /Ji lk. The half-life for the production of Y is then given by Eq. (3-12), setting Cy = Cyl2 when t = t i. We find ha = In Hk, and the same result is obtained for product Z. Thus, the products are generated in first-order reactions with the same half-life, even though they have different rate constants. 

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

Thus, if Ca and Cb can both be measured as functions of time, a plot of v/ca vs. Cb allows the rate constants to be estimated. (If it is known that B is also consumed in the first-order reaction, mass balance allows cb to be easily expressed in terms of Ca-) The rate v(Ca) is the tangent to the curve Ca = f(t) at concentration Ca-This can be determined graphically, analytically, or with computer processing of the concentration-time data. Mata-Perez and Perez-Benito show an example of this treatment for parallel uncatalyzed and autocatalyzed reactions. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

If k is much larger than k", Eq. (6-64) takes the form of Eq. (6-61) for the fraction Fhs thus we may expect the experimental rate constant to be a sigmoid function of pH. If k" is larger than k, the / -pH plot should resemble the Fs-pH plot. Equation (6-64) is a very important relationship for the description of pH effects on reaction rates. Most sigmoid pH-rate profiles can be quantitatively accounted for with its use. Relatively minor modifications [such as the addition of rate terms first-order in H or OH to Eq. (6-63)] can often extend the description over the entire pH range. 

From this expression, it is obvious that the rate is proportional to the concentration of A, and k is the proportionality constant, or rate constant, k has the units of (time) usually sec is a function of [A] to the first power, or, in the terminology of kinetics, v is first-order with respect to A. For an elementary reaction, the order for any reactant is given by its exponent in the rate equation. The number of molecules that must simultaneously interact is defined as the molecularity of the reaction. Thus, the simple elementary reaction of A P is a first-order reaction. Figure 14.4 portrays the course of a first-order reaction as a function of time. The rate of decay of a radioactive isotope, like or is a first-order reaction, as is an intramolecular rearrangement, such as A P. Both are unimolecular reactions (the molecularity equals 1). 

A first-order reaction plot I ho rate constant for a first-order reaction can be determined from the slope of a plot of ln[A] versus time. The reaction illustrated, at 67°C. 

Notice that for a first-order reaction the rate constant has the units of reciprocal time, for example, min-1. This suggests a simple physical interpretation of k (at least where k is small) it is the fraction of reactant decomposing in unit time. For a first-order reaction in which... 

The quotient Mi/k-i is the experimentally observed rate constant for the reaction, which is found to be second-order in NO and first-order in Cl2, as predicted by this mechanism. 

---