Rate Equation
The dependence of the rate of a reaction on the concentrations of the reagents is expressed by a rate equation.
Studies of the rate of reactions usually give information on the decrease in the amount of one of the reactants or on the increase in the amount of a product that occurs in some time interval. If the reaction system is one of constant or near constant volume, the change in the amount of reagent will correspond to a change in the concentration of the reagent. For liquid systems the rate of the molar concentrations reaction is often expressed in terms of the rate of change of the molar concentrations of a reagent. For constant volume gaseous systems it is generally more convenient to deal with the partial pressure is proportional to the concentration n/V.
The rate of a reaction can be defined as the derivative with respect to time of the extent of the reaction ξ. Consider a generalised reaction
aA + bB cC +dD
the number of moles of a product, C for example, is given as the reaction proceeds by nc = cξ. The number of moles of a reactant, A for example, is given by nA = a (1 - ξ). In view of such relations, the rate of the reaction can be expressed as
Rate of reaction = dξ/dt = -1/a dn_{A}/dt = -1/b dn_{B}/dt = 1/c dn_{C}/dt = 1/d dn_{D}/dt
This definition is often modified, but the name of reaction is retained when changes in convenience rather than changes in numbers of moles are used.
Rate of reaction = 1/c d[C]/dt = 1/d d [D]/dt = -1/a d [A]/dt = -1/b d [B]/dt
In practice, the rate is generally determined by following the rate with which the concentration of one or more of the reactants is used up. In spectral cases, however, the presence and amount of a product might be the best indication of the extent of the reaction.
A large number of reactions have rates that, at a given temperature, are proportional to the concentration of one or two of the reactants, with each reactant raised to a small integral power. If reactions are considered in which A and B represent possible reactants, the rate equations for reactions with such concentration dependence would be one of them
Rate of reaction = k [A] first order
Rate of reaction = k [A]^{2} or k[A] [B] second order
Reactions that proceed according to such simple rate equations are said to be reactions of the first or second order, as indicated. As we shall see, not all reactions have such simple rate laws. Some involve concentrations raised to nonintegral powers; others consist of more elaborate algebraic expressions. There are, however, enough reactions that are simple first or second order at least under certain conditions to make the ideal of the order at least under certain conditions to make the idea of the order of a reaction useful.
The rate of a reaction, as the above equations illustrate, generally depends on the concentrations of one or more of the reactants. Thus the rate changes as the reaction proceeds. This complication can be avoided by studying initial rates. These are the rates of the reaction during the initial stages when the concentrations of the reactants have not changed appreciably.
Example: a commonly used kinetic study in the chemistry laboratory course makes use of the oxidation of the oxide ion I- by peroxydisulfate, S_{2}O_{8}^{2-} . The reaction can be followed by observing the increase in absorption at 414 nm, this absorption being due to the colored I_{3}^{-}- ion that is formed. The initial rate is obtained from the slope of a plot of the remaining I^{-} concentrations versus time for results collected over periods upto about 10 min. some student values for initial rates, for the reaction at 20˚C, are as follows:
[I^{-}] |
0.0108 |
0.0216 |
0.0324 |
0.0090 |
0.0090 |
0.0090 |
S_{2}O_{8}^{2-} |
0.0060 |
0.0060 |
0.0060 |
0.0060 |
0.0120 |
0.0240 |
Rate × 10^{5}, (mol L^{-1})^{-1} min |
2.4 |
4.4 |
7.0 |
2.0 |
3.7 |
6.1 |
From these data obtain an expression that describes how the rate depends on the concentrations of the reactants.
Solution: for some reactions the rate equations are complicated mathematical expressions. Often, however, rate equations are simple expressions involving the products of the reactants raised to integral powers. Here we try an expression of the form rate = k[I^{-}]m S_{2}O_{8}^{2-}n.
The values of m and n can be obtained by first taking logarithms of the expected rate equation to obtain log (rate) = log k + m log [I^{-}] + n log S_{2}O_{8}^{2-}. The first three data, for which the peroxysulfate concentration is constant, can be used to obtain the value of m. when the iodine concentrations and the initial rates of the first two data are substituted and the resulting equations are subtracted, we have log (4.4/2.4) = m log (0.0216/0.0108), giving m = 0.61/0.69 = 0.88. Similar use of the second and third data sets gives m = 0.16/0.16 = 1.00. If m is an integer, we assign to it the value 1.
A similar treatment with data pairs from the final three sets of data gives n values of 0.90 and 1.23. These suggest that n = 1.
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