Rates Are Proportional to Concentrations

We start with a simple kinetic process, the interconversion between tw o states, as in Equation (13.1), 

The forward rate coefficient kj giv es the probability that an A molecule will convert to a B, per unit time. The number of A molecules that convert to B per unit time is the product of (the number of A molecules) x (the probability that an A molecule converts to a B per unit time). A rate coefficient kr describes the probability per unit time of a reverse conversion, from B to A. The overall rates of change of fA(t)] and [B(t)] are the creation rates minus the disappearance rates, 

Equations (19.2) and (19.3) are coupled both the quantities [A(t)] and [B(t) appear in both equations. Such equations can be solved by standard matrix methods. Texts on chemical kinetics give the details 11,2]. 

If the backward rate is much smaller than the forward rate, k,- s k/. Equation (19.2) simplifies to 

To express the time dependence explicitly, rearrange and integrate Equation (19.4) from time t = 0 to t to get 

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In chemical kinetics, the reaction rates are proportional to concentrations or to some power of the concentrations. Phenomenological equations, however, require that the reaction velocities are proportional to the thermodynamic force or affinity. Affinity, in turn, is proportional to the logarithms of concentrations. Consider a monomolecular... 

These investigations have provided valuable support for the applicability of the Debye-Hiickei limiting law, and for the validity of the assumption that rates are proportional to concentrations of activated complexes. 

The processes of transport of matter by diffusion and by convection are both, in a first approximation, first order processes. This means that transport rates are proportional to concentrations or concentration differences. Therefore it is a reasonable assumption that mass transfer is also a first order process. 

Equilibrium occurs when the rates of forward and reverse reactions are equal. These rates are proportional to concentrations of reactants and products respectively. Let k/ and kr be designated as forward and reverse-rate coefficients. Then at equilibrium. 

Because partial pressures are proportional to concentrations, rate laws for gas-phase reactions can also be expressed in terms of partial pressures, for instance, as... 

Studies of the polarized IBTILE provide a fundamental knowledge that makes it possible to explain phenomena occurring at the membranes of ion-selective electrodes. In addition, the rates of ion transfer and assisted ion transfer reactions are proportional to concentrations, which is a basis of an amperometric ion-selective (sensitive) electrode. 

DSC can be used effectively in the isothermal mode as well. In this case, the container with the sample is inserted into the DSC preheated to the desired test temperature. This type of experiment should be performed to examine systems for induction periods that occur with autocatalytic reactions and with inhibitor depletion reactions. (Reactions with induction periods can give misleading results in the DSC operated with increasing temperature scans.) Autocatalytic reactions are those whose rates are proportional to the concentration of one or more of the reaction products. Some hydroperoxides and peroxy esters exhibit autocatalytic decomposition. Inhibitor depletion can be a serious problem with certain vinyl monomers, such as styrene and acrylic acid, that can initiate polymerization at ambient temperatures and then selfheat into runaways. Isothermal DSC tests can be used to determine a time to runaway that is related to the inhibitor concentration. 

Assume that the rates are proportional to the concentrations of the acid catalysts, and that the stoichiometry represents the mechanism, thus second order. 

The initiation and propagation reactions typically show fractional orders of dependence of rate on alkyllithium. The situation is quite complex. The fractional orders for initiation and propagation are seldom the same and often vary depending on the monomer, solvent, and initiator and their absolute as well as relative concentrations. For styrene polymerization by n-butyllithium in aromatic solvents, the initiation and propagation rates are proportional to only the and -powers of n-butyllithium concentration, respectively. These results have been interpreted in terms of the following association equilibria... 

Catalyst-Inhibitor Conversion. The system 2,6,10,14-tetramethyl-pentadecane-bis(N-butylsalicylaldimino)cobalt(II) at 50°C. illustrates well the observed catalyst-inhibitor conversion (Figure 2). At low concentrations up to M/20,000 the metal chelate is a conventional catalyst no induction period is observed, and the reproducible initial autoxidation rates are proportional to the square root of catalyst concentration. From the curves shown in Figure 2 catalyst deactivation becomes apparent at... 

Individual rates usually are determined under conditions in which the rate is zero order in alkene. In the competitive reactions, however, the relative rates are proportional to the relative concentrations of the competing unsaturated compounds, i.e. first order. Presumably, the mechanisms are unchanged by the competitor, and accordingly, the relative individual rates and the relative competitive rates are determined by the difference in energy of the same transition states but different ground states in the former the alkene is bound to the catalyst, in the latter it is free. 

Because partial pressures are proportional to concentrations, rate laws for gas-phase reactions can also be expressed in terms of partial pressures, for instance, as rate = kPx for a first-order reaction of a gas X. What are the units for the rate constants when partial pressures are expressed in Torr and time is expressed in seconds for (a) zero-order reactions ... 

In the areas b (pH > 7) and u (pH 5) the contributions of the base-catalyzed and uncatalyzed reactions dominate, respectively. The slopes in region b are +1 (the rates are proportional to base concentration c0He with an apparent rate coefficient = k K /Kyj for base catalysis), but base... 

Little is known about the kinetics of the bioprocesses. A reasonable assumption is that the reaction rates are proportional to the amount of micro-organisms catalyzing the reactions. The influence of the other reactants is more complex, for example a nutrient in high concentration often has an inhibiting effect. Moreover, factors such as pH, salt concentrations, temperature, can have effects that are difficult to quantify. For this reason, we assume first-order kinetics and include all the other factors influencing the process rate in two Damkohler numbers. The following dimensionless reactor model is obtained ... 

The paraffins dehydrogenation on platinum-alumina catalysts proceeds with constant rate up to some time-on-stream after which a slow deactivation of the catalysts takes place Since relative changes of the catalyst activity ( characterized by reaction rate) are proportional to relative amounts of the deposited coke it can suppose that coke formation is the main reason of deactivation. Deactivation can be related with an attainment of a threshold in coke concentration (Co) on catalysts. The threshold amounts are 1.8 wt.% for A-I, 6,8% and 2.2% for A-II and A-IXI catalysts respectively. The isobutane dehydrogenation in non-stationary region (C > Co) is described by the following kinetic equation ... 

For the reaction A + A — P, we can write —dAJdt = k(A)(A) = k A2. In this case, the rate involves a higher power of the concentration (n = 1 + 1 = 2) and the reaction is second-order. For the reaction A -I- B —> P, the rate is proportional to the first power of each reactant and —d(A)(B)/dt = k(A)(B). The reaction is first-order with respect to A or B, but the overall reaction is second-order, because the right-hand side of the equation contains the product of two concentrations. The value of tV2 will depend on the initial reactant concentration(s), and the second-order rate constants have the dimensions of reciprocal concentration times time. Reactions of higher order, such as third order, are relatively rare, and their rates are proportional to the product of three concentration terms. 

According to eqns 5.16, all rates are proportional to the concentration of A and thus, since A reacts to completion, to the distance from the state at complete conversion. Moreover, this being so, the decay or formation rate of each participant is proportional to the fractional distance, x, from the state at complete conversion ... 

Zeroth-order rates are proportional to or k, independent of the concentrations of reactants or products. The rates of solution of quartz and many other silicate minerals are zeroth-order (see Sections 2.7.8 and 2.7.9). 

First-order rates are proportional to the concentrations of a reactant or product. An important example is radioactive decay (Section 2.7.2), which generally is irreversible and has only a forward rate and rate constant, for which... 

K inches of surface chemical reactions are governed by the elementary reactions llial constitute the reaction mechanism. In accordance with the principle of mass in lion, rates are proportional to surface concentrations (or surface densities) of piulicipating species surface sites, specifically adsorbed solutes, and non-npiTilically adsorbed solutes. The site stoichiometry model presented above is nih il used to express rate equations for surface chemical reactions. 

Transport processes are concerned with the flow of mass, momentum, and energy in fluids in nonuniform states. For normal liquids near equilibrium, the transport rates are proportional to the gradients of concentration, mass velocity, and temperature and the coefficients of diffusion, viscosity, and thermal conductivity are the respective proportionality constants. Various cross coefficients such as those of binary and thermal diffusion arise in Reciprocal processes expressing the effects of combined gradients of concentration and temperature. 

Manipulate the data to produce data triplets (of rate, concentration reactant 1, concentration reactant 2). If the raw data are proportional to concentration, this means differentiating the raw data, and integrating the raw data if they are proportional to rate. 

For other classes of normal paraffins, such as ra-pentane and ra-hep-tane, and for unsaturated hydrocarbons (C2H4, C3H6) the reaction rates are proportional to the first power of hydrocarbon concentrations. 

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