Kinetics

Defining the Rate Law

The rate law is a fundamental equation in chemical kinetics that describes the rate of a chemical reaction as a function of the concentrations of its reactants. It quantifies the relationship between the reaction rate, a rate constant (k), and the molar concentrations of the reactants. For a generic reaction represented as aA + bB → cC + dD, the rate law is expressed as Rate = k[A]^x[B]^y. It is crucial to understand that the exponents 'x' and 'y' in the rate law are not necessarily equal to the stoichiometric coefficients 'a' and 'b' from the balanced chemical equation; instead, they are experimentally determined values that define the order of the reaction with respect to each reactant. The sum of these exponents, x + y, gives the overall reaction order.

Illustrative Example of Reaction Orders

Consider the reaction 2H₂(g) + 2NO(g) → 2H₂O(g) + N₂(g). If experimental data reveals that this reaction is second order with respect to NO and first order with respect to H₂, then the rate law would be written as Rate = k[NO]²[H₂]¹. In this case, the overall reaction order is the sum of the individual orders, 2 + 1 = 3, classifying it as a third-order reaction.

Experimental Determination of Rate Laws

Rate laws are determined experimentally, typically by measuring the initial reaction rates under varying reactant concentrations. By systematically changing the concentration of one reactant while keeping others constant, and then measuring the corresponding initial reaction rates, the relationship between reactant concentrations and the reaction rate can be deduced. This process allows for the determination of the reaction order with respect to each reactant.

Practical Application: Determining Rate Laws

To determine a rate law from experimental data, one must identify experiments where the concentration of only one reactant changes while the others remain constant. By comparing the initial rates from these experiments, the reaction order for that specific reactant can be found. * If changing the concentration of a reactant (A) has no effect on the reaction rate, the reaction is zero-order with respect to A. * If a directly proportional change in rate occurs when the concentration of A is changed, the reaction is first-order with respect to A. * If the rate changes by the square of the change in concentration of A, the reaction is second-order with respect to A. Let's apply this method to the reaction 2H₂(g) + 2NO(g) → 2H₂O(g) + N₂(g) using the following experimental data:

Experiment	Initial [NO]/mol dm-3	Initial [H2]/mol dm-3	Initial rate/ mol(N2) dm-3 s-1
1	0.100	0.100	2.53 x 10-6
2	0.100	0.200	5.05 x 10-6
3	0.200	0.100	1.01 x 10-5
4	0.300	0.100	2.28 x 10-5

Calculating Reaction Orders and the Rate Constant

Using the provided experimental data, we can determine the reaction orders. To find the order with respect to NO, we compare experiments 3 and 4, where [H₂] is constant. The ratio of [NO] is 0.300/0.200 = 1.5, and the ratio of the rates is (2.28 x 10^-5)/(1.01 x 10^-5) ≈ 2.25. Since (1.5)² = 2.25, the reaction is second-order with respect to NO. To find the order with respect to H₂, we compare experiments 1 and 2, where [NO] is constant. The ratio of [H₂] is 0.200/0.100 = 2, and the ratio of the rates is (5.05 x 10^-6)/(2.53 x 10^-6) ≈ 2. Since (2)¹ = 2, the reaction is first-order with respect to H₂. Therefore, the rate law for this reaction is Rate = k[NO]²[H₂]. The overall reaction order is 2 + 1 = 3, making it a third-order reaction. To determine the value of the rate constant (k), data from any of the experimental trials can be substituted into the derived rate law.

Characteristics and Units of the Rate Constant (k)

The rate constant (k) is a proportionality constant in the rate law that is unique for each specific reaction. Its value is determined experimentally and is dependent on temperature, increasing with higher temperatures. The units of the rate constant vary depending on the overall order of the reaction, ensuring that the units of the rate law consistently yield units of concentration per unit time (e.g., mol dm^-3 s^-1).

Zero Order	First Order	Second Order	Third Order
Rate = k	Rate = k[A]	e.g. rate = k[A]2	e.g. rate = k[A]3
k = units of rate = mol dm-3 s-1	k = units of rate/units of concentration = mol dm-3 s-1/mol dm-3 = s-1	k = units of rate/(units of concentration)2 = mol dm-3 s-1/(mol dm-3)2 = mol-1 dm3 s-1	k = units of rate/(units of concentration)3 = mol dm-3 s-1/(mol dm-3)3 = mol-2 dm6 s-1

Zero-Order Reactions

For a zero-order reaction, the rate law is expressed as Rate = k[A]⁰, which simplifies to Rate = k. This indicates that the concentration of reactant A has no influence on the reaction rate; the rate remains constant regardless of changes in [A].

First-Order Reactions

In a first-order reaction, the rate law is given by Rate = k[A]. This signifies that the reaction rate is directly proportional to the concentration of reactant A. Doubling the concentration of A will double the reaction rate.

Second-Order Reactions

For a second-order reaction, the rate law is Rate = k[A]². This means the reaction rate is proportional to the square of the concentration of reactant A. If the concentration of A is doubled, the reaction rate will increase by a factor of four.

Overview of Reaction Orders

The concept of reaction orders is crucial for understanding how reactant concentrations affect the speed of a chemical reaction. The graphical representations below summarize the relationship between reactant concentration and reaction rate for different orders.

Understanding Reaction Mechanisms

Reaction mechanisms represent theoretical sequences of events that describe how a chemical reaction proceeds at a molecular level. While these mechanisms are formulated based on experimental evidence, it is crucial to understand that they cannot be directly observed or definitively proven. Instead, they are the most plausible explanations for the observed kinetics and products of a reaction. A reaction mechanism is composed of a series of individual steps, each of which is referred to as an elementary step.

The Role of Intermediates in Reaction Pathways

Intermediates are chemical species that are produced in one elementary step of a reaction mechanism and subsequently consumed in a later elementary step. They do not appear in the
overall balanced chemical equation because they cancel out
when all the individual elementary steps are summed. The
fundamental principle governing reaction mechanisms is that the sum of all elementary steps must precisely equal the overall balanced chemical equation for the reaction.

Molecularity of Elementary Steps

Molecularity describes the number of reactant particles involved in a single elementary step of a reaction. This concept helps classify the nature of the collision or transformation occurring in that specific step.

• Unimolecular: An elementary step is unimolecular if it involves only a single reactant particle undergoing a transformation, such as a decomposition or rearrangement.
• Bimolecular: A bimolecular elementary step involves the collision and reaction of two reactant particles.
• Termolecular: Termolecular elementary steps involve the simultaneous collision of three reactant particles. These are considerably rarer than unimolecular or bimolecular steps due to the low probability of three particles colliding effectively at the same instant.

Identifying the Rate-Determining Step

The rate-determining step, also known as the rate-limiting step, is the slowest elementary step in a reaction mechanism. This step dictates the overall rate of the entire reaction because products can only form as quickly as they are produced in this slowest step. Consequently, the activation energy for the overall reaction is effectively determined by the activation energy of this rate-determining step. It is also important to note that intermediate species may be present after the first step of a multi-step reaction, and their formation and consumption are often linked to the rate-determining step.

Deriving Rate Laws from Reaction Mechanisms

Rate laws can be formulated for each individual elementary step within a reaction mechanism. For an elementary step, the rate law is directly determined by the stoichiometry of that step: the concentration of each reactant in the elementary step is included in the rate law, raised to the power of its stoichiometric coefficient. Crucially, the overall rate law for the
                                                                     entire reaction is
                                                                     derived from the rate-
                                                                     determining step. If a 
reaction consists of only a single elementary step, then the rate law for that step is identical to the overall rate law for the reaction.

Example of a Reaction Mechanism

Temperature's Influence on the Rate Constant

Increasing the temperature of a reaction system significantly impacts the reaction rate, even though it does not alter the concentrations of the reactants. According to the rate law, this observed increase in reaction rate must be attributed to a change in the value of the rate constant, *k*. Therefore, the rate constant is inherently dependent on temperature. It is important to note that the magnitude of this effect is also influenced by the activation energy (*E_a*). If the activation energy is initially low, a temperature increase will have a less pronounced effect on the reaction rate. Conversely, for reactions with a high activation energy, even a modest increase in temperature can lead to a much more significant acceleration of the reaction.

The Arrhenius Equation: Quantifying Temperature Dependence

The Arrhenius equation provides a fundamental mathematical relationship that links temperature, the rate constant (*k*), and the activation energy (*E_a*). This powerful equation allows chemists to quantify how the rate constant changes with temperature and, crucially, to determine the activation energy of a reaction from experimental data. The Arrhenius equation is a key formula provided in Section 1 of the IB Chemistry data booklet.

Determining Activation Energy with an Arrhenius Plot

Once the rate law for a reaction has been established, the rate constant (*k*) can be experimentally determined at various temperatures. This data can then be used to construct an Arrhenius plot, which is a graphical method for determining the activation energy (*E_a*). This plot is generated by taking the natural logarithm of both sides of the Arrhenius equation, transforming it into a linear form. When ln *k* is plotted against 1/*T* (where *T* is the absolute temperature in Kelvin), a straight line is obtained. The slope of this line is equal to -*E_a*/R, where R is the ideal gas constant (8.31 J K^-1 mol^-1). Therefore, the activation energy can be calculated by multiplying the negative of the slope by the gas constant: *E_a* = -slope × R.

Example of Arrhenius Plot Calculation

Calculating Activation Energy from Two Temperatures

 The activation energy can also be calculated without constructing a full Arrhenius plot, provided that the rate constants are known at two different temperatures. This method utilizes a derived form of the Arrhenius equation. By subtracting the natural logarithm of the Arrhenius equation at one temperature from that at another temperature, a simplified equation is obtained that directly relates the two rate constants, the two temperatures, and the activation energy. This allows for a direct calculation of *E_a*.

Example of Two-Point Activation Energy Calculation