Change of Concentration with Time

A rate law tells you how the rate of a reaction depends on reactant concentrations at a particular moment. But often you would like to have a mathematical relationship showing how a reactant concentration changes ovct a period of time. Such an equation would be directly comparable to the experimental data, which are usually obtained as concentrations at various times. In addition to summarizing the experimental data, this equation would predict concentrations for all times. Using it, you could answer questions such as How long does it take for this reaction to be 50% complete to be 90% complete  

Using calculus, we can transform a rate law into a mathematical relationship between concentration and time called an int rated rate law. Because we will work only with the final equations, we need not go into the derivations here. We will look in some depth at first-order reactions and briefly at second-order and zero-order 

The general derivation using calculus is as follows. Substituting [A] for [NjOj], the rate law becomes 

Integrated Rate Laws (Concentration-Time Equations) 

First-Order Rate Law Let us first look at first-ordo- rate laws. The decomposition 

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In the previous section, we detailed diffusion equations and generalized mass balance equations. We now turn to their practical uses in the pharmaceutical sciences. Mass transport problems can be classified as steady or unsteady. In steady mass transport there is no change of concentration with time [3], characterized mathematically by... 

Differential rate equation The equation relating the rate of change of concentration with time of a reactant or product. 

The rate of a reaction is usually measured in terms of the change of concentration, with time, of one of the reactants or products, - d [reactant]/clt or +r/ [products]/r/t, and is usually expressed as moles per liter per second, or M s . We have already seen how this information might be used to derive the rate law and mechanism of the reaction. Now we are concerned, as kineticists, with measuring experimentally the concentration change as a function of the time that has elapsed since the initiation of the reaction. In principle, any property of the reactants or products that is related to its concentration can be used. A large number of properties have been tried. 

This expresses the rate of change of concentration with time at given coordinates (t, x, y, z) in terms of second space derivatives and three different diffusion coefficients. It is theoretically possible for D to be direction-dependent (in anisotropic media) but for a solute in solution, it is equal in all directions and usually the same everywhere, so (2.3) simplifies to... 

Estimate the change of concentrations with time for the following elementary reaction X + Y = 2Z... 

Figure 9.5. Spatial integral averages for the catalytic oxidation of CH30H to CH20 (a) change of concentration with time, (b) change of temperature with time, (c) change of temperature with concentration when the time varies between 0 and 1. The parameters used...

The central topic of the book is the rates of chemical reactions or elementary steps. A rate rx states the number of moles of species i formed (if positive) or consumed (if negative) by a chemical reaction or reactions per unit volume and unit time. It is a process rate as distinct from a rate of change. The distinction is important. A rate of change is an observable phenomenon of nature, e.g., a change of concentration with time, and is the combined result of all contributing process rates. ... 

Zero-order reaction can be expressed by the equation, dc/dt = -K, where dc/dt is the rate of change of concentration with time and k is the rate constant. In a zero-order reaction, a plot of concentration against time will produce a straight line whose slope is equal to -K and the half-life of the reactant is equal to % Co/K where Co is the initial concentration of the reactant. 

Note dCldt = rate of change of concentration with time (t), ko = zero-order rate constant, kt = first-order rate constant AH = the enthalpy change C0 = concentration available to react at time zero and C, = concentration available to react at time t. 

The rate of a chemical reaction is defined as the rate of change of the concentration of one of its components, either a reactant or a product. The experimental investigation of reaction rates therefore depends on being able to monitor the change of concentration with time. Classical procedures for reactions that take place in hours or minutes make use of a variety of techniques for determining concentration, such as spectroscopy and electrochemistry. Very fest reactions are studied spectroscopically. Spectroscopic procedures are available for monitoring reactions that are initiated by a rapid pulse of electromagnetic radiation and are over in a few femtoseconds (1 fe = 10 s). 

This paper is aimed at clarification of the change of concentration with time in the liquid phase before crystallization starts. To find optimum conditions for the commercial production of pure zeolites of the types A and faujasite, the reaction of fine-particle amorphous silica with sodium alu-minate solution was studied at 20°, 40°, and 75°C. The liquid phase separated by filtration nucleates the zeolite types Ay sodalite, phillipsite, and faujasite, depending on stirring time before liquid-solid separation. Quite similar conditions are observed in precipitated sodium aluminosilicate gels and mother liquor. 

Figure 2. Change of concentration with time and temperature in the 2 phases, using different Si02 sources...

The concentration-versus-time profiles for each of the reactants and products in Figure 3.1 are curved. This means that the rate of change of concentration with time for each of these species is not constant in each case it will vary continuously as the reaction progresses. 

It becomes cumbersome to keep using the qualifying term, instantaneous . From now on when we discuss, or calculate, a rate of change of concentration with time we shall always understand it to mean an instantaneous rate at a specific time. 

Jf is the number density (the number of atoms per unit volume). Eiquation (7.85) refers to diffusion in the z direction. The minus sign indicates that flux increases in the direction of negative concentration gradient. The time dependence of diffusive behaviour (which applies if a distribution is established at some time and is then allowed to evolve) is governed by Fick s second law, which gives the rate of change of concentration with time ... 

If only the overall rate equation of a reaction is known, no information on the change of concentrations with time can be obtained. Therefore it is necessary to determine the mechanism of a reaction - that means the sequence of all the elementary partial steps. Then the rate law can be derived unambiguously in the following way ... 

Special devices - chromatographs - are installed at the exit to record the change of concentration with time as the mixture is passing by. The typical form of recordings (chromatograms) is depicted in Fig. 6.11. 

The diffusion coefficient of dissolved ions or neutral species such as oxygen in aqueous solution is on the order of 1 X 10 cm s h Under nonsteady state conditions diffusion is described by Pick s second law, which states that the change of concentration with time is equal to the difference of the diffusive fluxes in and out of a given volume element. 

In most solid-state systems, it is more convenient to investigate diffusion by determining the change of concentration with time (dc/dt). Tick s second law, derived from Equation 10.10, states that... 

THE CHANGE OF CONCENTRATION WITH TIME We learn that rate equations can be written to express how concentrations change with time and look at several examples of rate equations zero-order, first-order, and second-order reactions. 

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