Rate equation
Its value may depend on conditions such as temperature, ionic strength, surface area of an adsorbent, or light irradiation.
The rate equation of a reaction with an assumed multi-step mechanism can often be derived theoretically using quasi-steady state assumptions from the underlying elementary reactions, and compared with the experimental rate equation as a test of the assumed mechanism.
The equation may involve a fractional order, and may depend on the concentration of an intermediate species.
is defined as where νi is the stoichiometric coefficient for chemical Xi, with a negative sign for a reactant.
This predicts that the rate depends only on the concentrations of the reactants, raised to the powers of their stoichiometric coefficients.
[8] The differential rate equation for an elementary reaction using mathematical product notation is: Where: The natural logarithm of the power-law rate equation is This can be used to estimate the order of reaction of each reactant.
[11][12] The partial order with respect to a given reactant can be evaluated by the method of flooding (or of isolation) of Ostwald.
in excess) is determined by a series of similar experiments with a range of initial concentration
[14] This may occur when there is a bottleneck which limits the number of reactant molecules that can react at the same time, for example if the reaction requires contact with an enzyme or a catalytic surface.
[16] Similarly, reactions with heterogeneous catalysis can be zero order if the catalytic surface is saturated.
For example, ring-opening metathesis polymerization using third-generation Grubbs catalyst exhibits zero order behavior in catalyst due to the reversible inhibition that occurs between pyridine and the ruthenium center.
[14] Although not affecting the above math, the majority of first order reactions proceed via intermolecular collisions.
For example, in the reaction of aryldiazonium ions with nucleophiles in aqueous solution, ArN+2 + X− → ArX + N2, the rate equation is
The second type includes nucleophilic addition-elimination reactions, such as the alkaline hydrolysis of ethyl acetate:[22] This reaction is first-order in each reactant and second-order overall: If the same hydrolysis reaction is catalyzed by imidazole, the rate equation becomes[22] The rate is first-order in one reactant (ethyl acetate), and also first-order in imidazole, which as a catalyst does not appear in the overall chemical equation.
Another well-known class of second-order reactions are the SN2 (bimolecular nucleophilic substitution) reactions, such as the reaction of n-butyl bromide with sodium iodide in acetone: This same compound can be made to undergo a bimolecular (E2) elimination reaction, another common type of second-order reaction, if the sodium iodide and acetone are replaced with sodium tert-butoxide as the salt and tert-butanol as the solvent: If the concentration of a reactant remains constant (because it is a catalyst, or because it is in great excess with respect to the other reactants), its concentration can be included in the rate constant, leading to a pseudo–first-order (or occasionally pseudo–second-order) rate equation.
One way to obtain a pseudo-first order reaction is to use a large excess of one reactant (say, [B]≫[A]) so that, as the reaction progresses, only a small fraction of the reactant in excess (B) is consumed, and its concentration can be considered to stay constant.
For example, the hydrolysis of esters by dilute mineral acids follows pseudo-first order kinetics, where the concentration of water is constant because it is present in large excess: The hydrolysis of sucrose (C12H22O11) in acid solution is often cited as a first-order reaction with rate
[27] The order of a chain reaction can be rationalized using the steady state approximation for the concentration of reactive intermediates such as free radicals.
Another type of mixed-order rate law has a denominator of two or more terms, often because the identity of the rate-determining step depends on the values of the concentrations.
This is zero-order with respect to hexacyanoferrate (III) at the onset of the reaction (when its concentration is high and the ruthenium catalyst is quickly regenerated), but changes to first-order when its concentration decreases and the regeneration of catalyst becomes rate-determining.
A pair of forward and reverse reactions may occur simultaneously with comparable speeds.
this is simply The derivative is negative because this is the rate of the reaction going from A to P, and therefore the concentration of A is decreasing.
Then: Since: the reaction rate becomes: which results in: A plot of the negative natural logarithm of the concentration of A in time minus the concentration at equilibrium versus time t gives a straight line with slope k1 + k−1.
[32] If the concentration at the time t = 0 is different from above, the simplifications above are invalid, and a system of differential equations must be solved.
However, this system can also be solved exactly to yield the following generalized expressions: When the equilibrium constant is close to unity and the reaction rates very fast for instance in conformational analysis of molecules, other methods are required for the determination of rate constants for instance by complete lineshape analysis in NMR spectroscopy.
When a substance reacts simultaneously to give two different products, a parallel or competitive reaction is said to take place.
Therefore, previous equation for [C] can only be used for low concentrations of [C] compared to [A]0 The most general description of a chemical reaction network considers a number
-th reaction can then be written in the generic form which is often written in the equivalent form Here The rate of such a reaction can be inferred by the law of mass action which denotes the flux of molecules per unit time and unit volume.
One can define the stoichiometric matrix denoting the net extent of molecules of
In this case, the rate of the forward and backward reactions are equal, a principle called detailed balance.
