Units first-order rate constant

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. ... 

We can reach two useful conclusions from the forms of these equations First, the plots of these integrated equations can be made with data on concentration ratios rather than absolute concentrations second, a first-order (or pseudo-first-order) rate constant can be evaluated without knowing any absolute concentration, whereas zero-order and second-order rate constants require for their evaluation knowledge of an absolute concentration at some point in the data treatment process. This second conclusion is obviously related to the units of the rate constants of the several orders. 

The Arrhenius equation relates the rate constant k of an elementary reaction to the absolute temperature T R is the gas constant. The parameter is the activation energy, with dimensions of energy per mole, and A is the preexponential factor, which has the units of k. If A is a first-order rate constant, A has the units seconds, so it is sometimes called the frequency factor. 

A first-order rate constant has the dimension time, but all other rate constants include a concentration unit. It follows that a change of concentration scale results in a change in the magnitude of such a rate constant. From the equilibrium assumption of transition state theory we developed these equations in Chapter 5 ... 

The first-order rate constants have units of s 1 and the second-order rate constants have units of L mol 1 s"1. 

The activation parameters from transition state theory are thermodynamic functions of state. To emphasize that, they are sometimes designated A H (or AH%) and A. 3 4 These values are the standard changes in enthalpy or entropy accompanying the transformation of one mole of the reactants, each at a concentration of 1 M, to one mole of the transition state, also at 1 M. A reference state of 1 mole per liter pertains because the rate constants are expressed with concentrations on the molar scale. Were some other unit of concentration used, say the millimolar scale, values of AS would be different for other than a first-order rate constant. 

One other point should be noted. The dimensions of the right-hand side of Eq. (7-57) are time-. That is, only first-order rate constants appear to be permitted, when in fact, the derivation assumed a bimolecular mechanism. The problem is entirely artificial, arising from the different ways in which concentration units are ordinarily dealt. 

Since the value of the first-order rate constant is not given, X and cannot be calculated directly. The reaction rate per unit volume of catalyst 9tt. = rikC i (equation 10.217),... 

Where kf is the rate constant for the exchange of an unspecified one of the four coordinated water molecules recalculated to second-order units [ is the first-order rate constant (s 1) for the exchange of a particular coordinated solvent molecule]. 

Note that because kon is a second-order rate constant, and koff is a first-order rate constant, the units of Ka will be reciprocal molarity and the units of Kd will be molarity. 

The rates of each of the environmentally important chemical processes are influenced by numerous parameters, but most processes are described mathematically by only one or two variables. For example, the rate of biodegradation varies for each chemical with time, microbial population characteristics, temperature, pH, and other reactants. In modeling efforts, however, this rate can be approximated by a first-order rate constant (in units of time). 

First-order rate constants are used to describe reactions of the type A — B. In the simple mechanism for enzyme catalysis, the reactions leading away from ES in both directions are of this type. The velocity of ES disappearance by any single pathway (such as the ones labeled k2 and k3) depends on the fraction of ES molecules that have sufficient energy to get across the specific activation barrier (hump) and decompose along a specific route. ES gets this energy from collision with solvent and from thermal motions in ES itself. The velocity of a first-order reaction depends linearly on the amount of ES left at any time. Since velocity has units of molar per minute (M/min) and ES has units of molar (M), the little k (first-order rate constant) must have units of reciprocal minutes (1/min, or min ). Since only one molecule of ES is involved in the reaction, this case is called first-order kinetics. The velocity depends on the substrate concentration raised to the first power (v = /c[A]). 

The turnover number, or kCM (pronounced kay kat ), is another way of expressing Vmax. It s Vmax divided by the total concentration of enzyme (Vmax/E,). The kcat is a specific activity in which the amount of enzyme is expressed in micromoles rather than milligrams. The actual units of kcat are micromoles of product per minute per micromole of enzyme. Frequently, the micromoles cancel (even though they re not exactly the same), to give you units of reciprocal minutes (min-1). Notice that this has the same units as a first-order rate constant (see later, or see Chap. 24). The kcat is the first-order rate constant for conversion of the enzyme-substrate complex to product. For a very simple mechanism, such as the one shown earlier, kcat would be equal to k3. For more complex... 

The first-order rate constant k will have the units of s 1. 

The rate constant k is obtained from such a first-order rate graph as (—1 x gradient) if the time axis is given with units of seconds. Accordingly, the units of the first-order rate constant are s-1. 

In this equation, k is the first-order rate constant. It has units of s . ... 

The standard enthalpy difference between reactant(s) of a reaction and the activated complex in the transition state at the same temperature and pressure. It is symbolized by AH and is equal to (E - RT), where E is the energy of activation, R is the molar gas constant, and T is the absolute temperature (provided that all non-first-order rate constants are expressed in temperature-independent concentration units, such as molarity, and are measured at a fixed temperature and pressure). Formally, this quantity is the enthalpy of activation at constant pressure. See Transition-State Theory (Thermodynamics) Transition-State Theory Gibbs Free Energy of Activation Entropy of Activation Volume of Activation... 

C) Rate constants describe the rate of a reaction as a function of starting concentration(s). A first-order rate constant describes a reaction whose rate depends on the concentration of one component only. A first-order rate constant has units of inverse time (usually s" ). 

Experimental. In order to study the nucleophilic properties of 13 it was necessary to add excess I " to the solutions to prevent precipitation of I2. The rate of formation of CoCCN I-3 was followed spectrophotometrically after the I3 " in aliquots of the solution taken at suitable time intervals was reduced to I by arsenite ion. A typical set of experiments was carried out at 40°C. and unit ionic strength, with all solutions containing 0.5/1/ 1 and variable I3 " at a maximum concentration of 0.28M, the approximate upper limit imposed by solubility restrictions. The results are presented in Figure 3 as a plot of k the symbol used for the pseudo first-order rate constant for this system, vs. l/(lf). It is apparent that 13 is a remarkably efficient nucleophile, with a reaction rate considerably greater than that found for I at comparable concentrations. The points in Figure 3 also show detectable deviation from linearity, despite the limited range of 13 " concentration which was available. 

In acidic solution the aquation of Co(CN)5N3 3 was found to be acid-catalyzed In the absence of anions other than CIO, the only reaction products were Co(CN)5OH2 "2 and HN3. Typical results obtained at unit ionic strength and 40°C. are presented in column 2 of Table V as pseudo first-order rate constants. 

We can use this to eliminate b from eqn (1.43). If we also divide throughout by the reactor volume V and call q/V the flow rate kf (it has units of (time)-1 and so is like a first-order rate constant)... 

Our aim is to determine the concentration of A in the reactor as a function of time and in terms of the experimental conditions (inflow concentrations, pumping rates, etc.). We need to obtain the equation which governs the rate at which the concentration of A is changing within the reactor. This mass-balance equation will have contributions from the reaction kinetics (the rate equation) and from the inflow and outflow terms. In the simplest case the reactor is fed by a stream of liquid with a volume flow rate of q dm3 s 1 in which the concentration of A is a0. If the volume of the reactor is V dm3, then the average time spent by a molecule in the reactor is V/q s. This is called the mean residence time, tres. The inverse of fres has units of s-1 we will call this the flow rate kf, and see that it plays the role of a pseudo-first-order rate constant. We denote the concentration of A in the reactor itself by a. 

In this reaction, the rate depends only on the concentration of S. This is called a first-order reaction. The factor A is a proportionality constant that reflects the probability of reaction under a given set of conditions (pH, temperature, and so forth). Here, A is a first-order rate constant and has units of reciprocal time, such as s "1. If a first-order reaction has a rate constant k of 0.03 s-1,... 

The constant kcat is a first-order rate constant and hence has units of reciprocal time. It is also called the... 

Relation Between Single-Bubble Collision Efficiency and the First Order Rate Constant for Droplet Collection First a parameter which describes the frequency at which a continuous-stirred vessel volume is swept by air bubbles must be found. The total volume swept by rising bubbles per unit time is equal to the volume swept by a single bubble as it rises to the liquid surface ( multiplied by the number of bubbles generated per unit time (A). ... 

The units of k2 are M 1 s 1. If [B] is present at unit activity, the rate is /c2[A], a quantity with units of s We can see that the bimolecular, or second-order, rate constant for reaction of A with B may be compared with first-order constants when the second reactant B is present at unit activity. In many real situations, reactant B is present in large excess and in a virtually constant concentration. The reaction is pseudo-first order and the experimentally observed rate constant /c2[B] is an apparent first-order rate constant. The bimolecular rate constant k2 can be obtained by dividing the apparent constant by [B]. 

The variations of velocity with temperature over the range 5°-35° were measured for several substrates (9). Linear Arrhenius plots of these data were obtained which allowed calculation of the empirical activation energy and AHt. AF values at 25° were calculated from the first-order rate constants k0 expressed in units of sec-1. These values and the entropies of activation are given in Table III. The Km values for glutamine have also been determined over this temperature range and found not to vary from the value at 25° (4.32 mM) by an experimentally significant amount. 

Taking therefore unit area of the core surface as the basis for the reaction rate, and writing the first order rate constant as ks, then the rate at which moles of A are consumed in the reaction is given by ... 

First-order rate constant with respect to hydrogen (here expressed as A", the rate constant per unit mass of catalyst, which is often the practice for heterogeneous reactions catalysed by solids, as in Chapter 3) ... 

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