The Relation Between Reactant Concentration and Time

Comment Note that the reaction is first order in H2, whereas the stoichiometric coefficient for H2 in the balanced equation is 2. The order of a reactant is not related to the stoichiometric coefficient of the reactant in the overall balanced equation. 

Practice Exercise The reaction of peroxydisulfate ion (S2OI ) with iodide ion (I ) is 

From the following data collected at a certain temperature, determine the rate law and calculate the rate constant. 

Rate law expressions enable us to calculate the rate of a reaction from the rate constant and reactant concentrations. The rate laws can also be used to determine the concentrations of reactants at any time during the course of a reaction. We will illustrate this application by first considering two of the most common rate laws—those applying to reactions that are first order overall and those applying to reactions that are second order overall. 

A first-order reaction is a reaction whose rate depends on the reactant concentration raised to the first power. In a first-order reaction of the type 

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THE RELATION BETWEEN REACTANT CONCENTRATION AND TIME Review Questions... 

In general, the gasoline in China contains H2S, S02 and other organic sulphides. There is a minimum of 30-40 ppm S02 and other sulphides in the exhaust. It is well known that the sulphides possess strong toxicity for basic metal oxide catalysts (ref. 14). Therefore, a study on catalytic resistance to S02 poisoning is important. The relation between catalytic activity and reaction time when the reactant contains S02 (50 ppm) was determined. It was found that the conversion of CO remained 81.6% for catalyst A after 112 hours and 76.1% for catalyst B after 64 hours in micro-reactor conditions. However S02 poison was reversible for catalyst A. When S02 concentration decreased and temperature increased, the activity of the poisoned catalyst could be restored. The data are shown in Table 8 and... 

We can now give precise definitions of some words and phrases that are often used loosely. A reaction rate is a rate of change of the concentration of some chemical species at a particular moment it is derived from a set of observations, in which the course of a reaction is monitored over a period of time. Such observations are the basic experimental data. By analysing the relation between these rates and the corresponding concentrations, one obtains a rate law that fits both the data and (often) some standard form (e.g., first-order, in which the rate is proportional to the concentration of one of the reactants). These rate laws, along with known structural data, may be given some interpretation in terms of reaction kinetics, one would describe a scheme of molecular motions that would explain the rate law. Often there are several alternative schemes that would be consistent with a particular set of data further experimentation would then be needed in order to choose between them. Such schemes in terms of chemical kinetics describe events on the microscopic scale, involving atoms and molecules, as distinct from rate laws which are expressed in terms of macromolecular quantities (time and concentration). The schemes may in turn be interpreted in terms of a reaction mechanism, which relates them to chemical dynamics, i.e., to theories of how molecules behave, in terms either of some particular model with limited scope (such as collison theory, or transition-state theory) or of the more fundamental body of theory based upon quantum mechanics. 

The relation between E and t is S-shaped (curve 2 in Fig. 12.10). In the initial part we see the nonfaradaic charging current. The faradaic process starts when certain values of potential are attained, and a typical potential arrest arises in the curve. When zero reactant concentration is approached, the potential again moves strongly in the negative direction (toward potentials where a new electrode reaction will start, e.g., cathodic hydrogen evolution). It thus becomes possible to determine the transition time fiinj precisely. Knowing this time, we can use Eq. (11.9) to find the reactant s bulk concentration or, when the concentration is known, its diffusion coefficient. 

This is the integrated rate equation for a first-order reaction. When dealing with first-order reactions it is customary to use not only the rate constant, k for the reaction but also the related quantity half-life of the reaction. The half-life of a reaction refers to the time required for the concentration of the reactant to decrease to half of its initial value. For the first-order reaction under consideration, the relation between the rate constant k and the half life t0 5 can be obtained as follows ... 

A reaction has the rate equation, ra = kCaCb. The reactor initially contains only reactant A. B is a gas of limited solubility. It is charged at a rate sufficient to keep the solution saturated at a concentration Cb0. Find the relation between the time and the variable feed rate of B, Vb, sufficient to keep the solution saturated as the reaction goes on. 

Various properties of crystals can be used to inspect c,( ,r), provided that appropriate detectors for the intensity of input and output signals are available. If the monitor response is sufficiently fast, one may determine the time dependence of solid state reactions. The monitoring of reactants and/or reaction products can serve this purpose, but the relation between signal intensity (property) and concentration Cj) must always be established first. Since functions of state are related to one another in a unique way, any equilibrium property can, in principle, be used to determine However, the necessary assumption of local equilibrium must still be... 

Between the two discontinuous transitions, the system may exist in either a high-iodide steady state or a low-iodide steady state. Of course the concentrations of all the other species in the system reflect the differences in iodide concentration in particular, the iodate concentration is given by the conservation relation above. Which state the system ochibits depends on its past history—whether it has been approached from hi or low flow rates. Thus, for any particular flow rate in the bistability range, the extent of reaction may be very high or very low, even though the reactant concentrations, residence time, and so forth are identical. 

If one employs reactants in precisely stoichiometric proportions, the class II and class III rate expressions will reduce to the mathematical form of the class I rate function. Because the mathematical principles employed in deriving the relations between the extent of reaction per unit volume (or the concentrations of the various species) and time are similar to those used in Sections 3.1.1.1 and 3.1.1.2, we list only the most interesting results for third-order reactions. 

A solution of the transport and kinetic equations resulted in the relation between the ratio (ttr.kin/ ftr.d)" the kinetic parameter X =ksTtr. This relation could be applied on QevE, EQat and ECi E reaction mechanisms. The transition time Ttr,kin established in the system with chemical perturbation is compared with the transition time observed under the diffusion control, Ttr,di at the given conditions (constant exponent, growing drop-surface and the same bulk concentration of reactants c t). The value of can be calculated as... 

If the density of the reaction mixture remains constant during the conversion, the concentrations of the reactants and the degree of conversion can be calculated as a function of time, as will be shown in the next sections. However, there are also reactions where the volume of the reaction mixture changes during reaction. For these cases we need to know the relation between the density of the reaction mixture and the degree of conversion. In general such relations are not easy to find. However, a simple relation may be used in the case of additive molar volumes of the components of the reaction mixture. In that case there is a simple relationship between the density of the mixture, the degree of conversion, and the relative volume increase on complete conversion (o ... 

If the relation between the density of the reaction mixture and the reactant concentration is known, the latter can be calculated as a function of time. However, if we describe the progress of the reaction in terms of the degree of conversion, see eq. (3.8), we d the same solution as eq. (3.15), which indicates that volume changes during reaction do not influence the rate of conversion of first order reactions. 

This is a set of coupled differential equations. They can be solved with the help of an additional mass balance that describes the relation between the reactant concentrations c, and c at the time t ... 

To arrive at a model describing this sort of process, we proceeed in three steps. First the mass transfer and the reaction of reactant A (present in the gas phase) is described in terms of a quasi steady state. From this we find an effective rate constant. In the second step a mass balance is made for the gas flow, describing the relation between the concentrations of reactant A in the in- en outgoing streams. In the third step a mass balance is made for reactant B in the liquid phase over a certain peri( of time. The conc tration of B changes with time,with a rate coefficient found in the first step. 

It is often experimentally convenient to use an analytical method that provides an instrumental signal that is proportional to concentration, rather than providing an absolute concentration, and such methods readily yield the ratio clc°. Solution absorbance, fluorescence intensity, and conductance are examples of this type of instrument response. The requirements are that the reactants and products both give a signal that is directly proportional to their concentrations and that there be an experimentally usable change in the observed property as the reactants are transformed into the products. We take absorption spectroscopy as an example, so that Beer s law is the functional relationship between absorbance and concentration. Let A be the reactant and Z the product. We then require that Ea ez, where e signifies a molar absorptivity. As initial conditions (t = 0) we set Ca = ca and cz = 0. The mass balance relationship Eq. (2-47) relates Ca and cz, where c is the product concentration at infinity time, that is, when the reaction is essentially complete. 

Reactant A of concentration Ca0 is charged then reactant B is pumped in at a rate Vb relation between the time and the amount of apply it to the numerical case following. 

In chemical equilibria, the energy relations between the reactants and the products are governed by thermodynamics without concerning the intermediate states or time. In chemical kinetics, the time variable is introduced and rate of change of concentration of reactants or products with respect to time is followed. The chemical kinetics is thus, concerned with the quantitative determination of rate of chemical reactions and of the factors upon which the rates depend. With the knowledge of effect of various factors, such as concentration, pressure, temperature, medium, effect of catalyst etc., on reaction rate, one can consider an interpretation of the empirical laws in terms of reaction mechanism. Let us first define the terms such as rate, rate constant, order, molecularity etc. before going into detail. 

Kc does not depend on concentrations but depends on temperature. At the given temperature, if the equilibrium concentrations of C and D are higher than those of A and B and indicates high value of Kc, then A and B have reacted to a considerable extent. On the other hand, if Kc is small, there will be little of C and D at equilibrium. Thus, the extent of chemical reaction is determined by equilibrium constant and is not related in any simple way to the rate or velocity of reaction with which the chemical change takes place. The reaction between two reactants may occur to almost completion, but the time for even very small fraction of the molecules to react may be extremely long. 

The theory of equilibrium is treated on the basis of thermodynamics considering only the initial and final states. Time or intermediate states have no concern. However, there is a close relationship between the theory of rates and the theory of equilibria, in spite of there being no general relation between equilibrium and rate of reaction. A good approximation of equilibrium can be regarded between the reactants and activated state and the concentration of activated complex can, therefore, be calculated by ordinary equilibrium theory and probability of decomposition of activated complex and hence the rate of reaction can be known. 

The concentration of the radioactive nuclide (reactant, such as Sm) decreases exponentially, which is referred to as radioactive decay. The concentration of the daughter nuclides (products, including Nd and He) grows, which is referred to as radiogenic growth. Note the difference between Equations l-47b and l-47c. In the former equation, the concentration of Nd at time t is expressed as a function of the initial Sm concentration. Hence, from the initial state, one can calculate how the Nd concentration would evolve. In the latter equation, the concentration of Nd at time t is expressed as a function of the Sm concentration also at time t. Let s now define time t as the present time. Then [ Nd] is related to the present amount of Sm, the age (time since Sm and Nd were fractionated), and the initial amount of Nd. Therefore, Equation l-47b represents forward calculation, and Equation l-47c represents an inverse problem to obtain either the age, or the initial concentration, or both. Equation l-47d assumes that there are no other ot-decay nuclides. However, U and Th are usually present in a rock or mineral, and their contribution to " He usually dominates and must be added to Equation l-47d. 

The reaction between olefins and ozone produces light that can be measured and related to the concentration of the reactants. One of the preferred methods for measuring ambient ozone concentrations utilizes the chemiluminescence generated in the ozone-ethylene reaction for detection. Recently, Hills and Zimmerman (16) described the use of this detection principle for determining hydrocarbon concentrations. They utilized the chemiluminescence created when ozone reacts with isoprene for development of a continuous, fast-response isoprene analyzer. This real-time isoprene system is reported to be linear over three orders of magnitude and to have a detection limit of about 1 ppbv. Because the system doesn t include a preseparation of hydrocarbons, interferences from other olefins (ethylene, propylene, and so forth) could occur. Thus far the chemiluminescent detector has been used to monitor isoprene emissions under conditions in which the concentrations of olefins that could interfere are negligible compared to those of the biogenic hydrocarbon. 

FIGURE 21-1 Left, variation with time of the concentrations of reactant R and product P. Right, relation between half-time and fraction of reactant... 

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