Gavin Howe
The rate law of the first step is a fundamental concept in chemistry, specifically in the field of chemical kinetics. It refers to the rate-determining step, which is the slowest step in a chemical reaction that dictates the overall speed at which the reaction occurs. This concept is akin to the neck of a funnel, where the flow of water is constrained by the width of the neck rather than the rate at which water is poured. The rate law is expressed as an equation that correlates the reaction rate with the concentration of reactants, typically increasing as reactant concentration rises. The rate-determining step is crucial for comprehending and optimizing processes like catalysis and combustion. It aids in predicting the rate equation for comparison with the experimental rate law. The identification of the rate-limiting step involves determining the reactants and their reaction orders in the chemical equation, followed by substituting known values into the rate law equation.
| Characteristics | Values |
|---|---|
| Rate-determining step | The slowest step of a chemical reaction that determines the speed (rate) at which the overall reaction proceeds |
| Rate equation | Derived by the slowest step in the reaction |
| Reaction order | The degree of the reaction rate |
| Rate-limiting reaction | The slowest reaction in the overall reaction mechanism of a multi-step/complex reaction |
| Reaction rate | Depends on the slowest step |
| Rate law | An equation that relates the reaction rate to the concentration of reactants |
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What You'll Learn
The rate-determining step is the slowest step
In chemical kinetics, the rate-determining step or rate-limiting step is the slowest step in a chemical reaction. It is the step that determines the speed or rate at which the overall reaction proceeds. This slowest step can be compared to the neck of a funnel, which limits the rate at which water flows through it, rather than the rate at which water is poured into the funnel.
The rate-determining step is important for the optimization and understanding of many chemical processes such as catalysis and combustion. It is identified by predicting the rate law for each possible step and comparing the predictions with the experimental law. For example, in the reaction between NO2 and CO, if the reaction occurred in a single step, the reaction rate would be proportional to the rate of collisions between NO2 and CO. However, the observed reaction rate is second-order in NO2 and zero-order in CO, indicating that the rate is determined by a step involving two NO2 molecules, with the CO molecule entering in a faster subsequent step.
The rate equation is derived from the rate-determining step, with the rate being equal to the rate constant of the slowest step multiplied by the concentrations of the reactants raised to their reaction order. For instance, in the reaction between NO2 and F2, the first step is the slowest, with the rate equation being: rate = k1[NO2][F2].
The rate-determining step is not always the first step of a reaction, but when it is, the overall rate of the reaction is simply the rate of the first step. For example, in the reaction between NO2 and F2, the first step is the rate-determining step, with the rate equation being: rate = k1[NO2][F2]. This means that the overall rate law is also rate = k1[NO2][F2].
In some cases, the rate-determining step may not be the first step, but rather a subsequent step. For example, in the reaction between NO3 and CO, the first step is a slow step in the reverse direction, but the overall rate of reaction is determined by the rate of formation of the final product (CO2), which is the rate of the second step.
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The rate equation is derived from the slowest step
In chemical kinetics, the overall rate of a reaction is often determined by the slowest step, known as the rate-determining step (RDS) or rate-limiting step. The rate equation is derived from this slowest step in the reaction. The rate equation is set up by writing the rate as equal to the rate constant of the slowest step multiplied by the concentrations of the reactant or reactants raised to their reaction order.
For example, consider the reaction of carbon monoxide and nitrogen dioxide to form carbon dioxide and nitrogen oxide. The rate equation would usually be r=k[NO2][CO] as the rate depends on the reaction of the nitrogen dioxide and carbon monoxide in a one-step reaction. However, if this reaction occurred in multiple steps and was dependent on the collision of two nitrogen dioxide molecules, the rate equation would instead be r = k[NO2]2.
Another example is the unimolecular nucleophilic substitution (SN1) reaction in organic chemistry, where the first, rate-determining step is unimolecular. The reaction of basic hydrolysis of tert-butyl bromide (t-C4H9Br) by aqueous sodium hydroxide proceeds in two steps. The first step is the formation of a carbocation, which is slow and determines the rate, while the second step with OH− is much faster. Therefore, the overall rate is independent of the concentration of OH−.
The rate-determining step can be compared to the neck of a funnel. The rate at which water flows through a funnel is limited by the width of the neck and not by the rate at which water is poured in. Similarly, the slow step of a reaction determines the rate of the overall reaction. Not all reactions have a rate-determining step and it only exists if one step is significantly slower than the other steps.
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Reaction orders in the rate-limiting chemical equation
The rate-determining step, also known as the rate-limiting step, is the slowest step of a chemical reaction, which determines the rate at which the overall reaction proceeds. In other words, the slowest step of a chemical reaction can be determined by setting up reaction mechanisms.
The rate law for a first-order reaction depends on the concentration of only one reactant, although other reactants may be present. The rate equation is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters. The rate equation for a zero-order reaction is independent of the concentration of a reactant, so changing its concentration has no effect on the rate of the reaction.
The reaction order is the relationship between the concentrations of species and the rate of a reaction. The order of a rate law is the sum of the exponents of its concentration terms. The reaction order is the exponent to which the concentration of a particular reactant is raised, and it indicates to what extent the concentration of a species affects the rate of a reaction.
In the context of a rate-limiting reaction, the first step is the slowest reaction in the overall reaction mechanism of a multi-step or complex reaction. If the first step in a reaction mechanism is rate-limiting, the overall rate of the complex reaction is determined by the rate of the rate-limiting reaction.
For example, consider the reaction:
$2 NO_{(g)} + 2 H_{2(g)} \rightarrow N_{2(g)} + 2 H_{2}O_{(g)}$
The reaction mechanism is:
$2 NO_{(g)} + H_{2(g)} \rightarrow N_{2(g)} + H_{2}O_{2(g)} (slow)$
$H_{2}O_{2(g)} + H_{2(g)} \rightarrow 2 H_{2}O_{(g)} (fast)$
The reactants in the rate-limiting step are $NO$ and $H_2$. Since the overall reaction order is second order, it can be assumed that $m + n = 2$, so $m$ and $n$ are both $1$. The rate law for the rate-limiting reaction is:
$$rate=k\left [NO \right ]^{1} \left [H_{2} \right ]^{1}$$
This also means that the overall rate law is:
$$rate=k\left [NO \right ] \left [H_{2} \right]$$
Similar examples can be found in Wikipedia's article on the rate-determining step and Study.com's article on identifying the rate law for a reaction with a rate-limiting first step.
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Rate law for the overall reaction
Rate laws provide a mathematical description of how changes in the amount of a substance affect the rate of a chemical reaction. They are determined experimentally and cannot be predicted by reaction stoichiometry. The order of reaction describes how much a change in the amount of each substance affects the overall rate, and the overall order of a reaction is the sum of the orders for each substance present in the reaction.
The rate law or rate equation is a mathematical expression that describes the relationship between the rate of a chemical reaction and the concentration of its reactants. In general, a rate law takes the form:
[A]^m*[B]^n*[C]^p* = *k*
Where [A], [B], and [C] represent the molar concentrations of reactants, and k is the rate constant, which is specific for a particular reaction at a particular temperature. The exponents m, n, and p are usually positive integers, but they can also be fractions or negative numbers. The rate constant k and the exponents must be determined experimentally by observing how the rate of reaction changes as the concentrations of the reactants are changed.
For example, if m = 1 and n = 2, the reaction is first order in A and second order in B. The overall reaction order is the sum of the orders for each reactant, so in this case, the reaction is third order overall (1 + 2 = 3).
The rate of a first-order reaction depends on the concentration of only one reactant (a unimolecular reaction). Other reactants can be present, but their concentration has no effect on the rate. The unit of k for a first-order reaction is s^-1. The rate of the overall reaction depends on the slowest step, so the overall reaction will be first order when the reaction of the energized reactant is slower than the collision step.
The rate equation for a zero-order reaction has the unit of k as mol dm^-3 s^-1. Many enzyme-catalyzed reactions are zero order, provided that the reactant concentration is much greater than the enzyme concentration, which controls the rate, so the enzyme is saturated.
Differential rate laws are used to express the rate of a reaction in terms of the change in the concentration of reactants (d [R]) over a small interval of time (dt). They can be used to calculate the instantaneous rate of a reaction, which is the reaction rate under a very small time interval.
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Rate laws for elementary reactions
The rate law is a mathematical expression that illustrates how the rate of product formation varies with the concentrations of reactants. It defines the rate of a reaction as the change in product concentration over time. This rate is proportional to a rate constant, usually denoted as 'k', and the concentrations of reactants. The relationship may not always be linear, and the exponent 'x' indicates the rate's variability with reactant concentrations.
An elementary step is a proposed expression of the reaction rate for an elementary reaction. The rate of an elementary step is always written according to the proposed equation. Elementary processes are written to show how a chemical reaction progresses, leading to an overall reaction, and the order for an elementary process is the order of the reaction. The slowest elementary step is the rate-determining step.
There are two types of elementary steps: unimolecular and bimolecular. A unimolecular step involves a single molecule or ion decomposing by itself and is always a first-order reaction. The rate of a unimolecular step depends on the concentration of the intermediate and the rate constant of the process. A bimolecular step involves two reacting molecules or ions, and the rates for these steps are second-order. The rate of a bimolecular step depends on the concentration of both species and the rate constant for the reaction.
The rate law for an elementary reaction can be determined by first looking at the coefficient for each reactant. The coefficient of each reactant is the power to which you raise the reactant concentration when writing the rate law. Rate constants (k) are determined experimentally and are multiplied by the reactant concentrations.
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Frequently asked questions
It is the slowest step of a chemical reaction that determines the speed (rate) at which the overall reaction proceeds.
The rate-determining step can be identified by predicting the rate law for each possible choice and comparing the different predictions with the experimental law.
The rate law equation relates the reaction rate to the concentration of reactants. The rate equation is derived by the slowest step in the reaction.
The rate law for a first-order reaction depends on the concentration of only one reactant. The rate increases when the concentration of the reactant increases.
