The relationship between concentration and time can be derived from the rate law and calculus Integration of the rate laws gives the integrated rate laws Integrate laws give concentration as a function of time Integrated laws can get very complicated, so only a few simple forms will be considered

First order reactions Rate law is: rate = k [A] The integrate rate law can be expressed as: [A]0 is [A] at t (time) = 0 [A]t is [A] at t = t e = base of natural logarithms = 2.71828…

Graphical methods can be used to determine the first-order rate constant, note

A plot of ln[A]t versus t gives a straight line with a slope of -k The decomposition of N2O5. (a) A graph of concentration versus time for the decomposition at 45oC. (b) A straight line is obtained from a logarithm versus time plot. The slope is negative the rate constant.

The simplest second-order rate law has the form The integrated form of this equation is

Graphical methods can also be applied to second-order reactions A plot of 1/[B]t versus t gives a straight line with a slope of k Second-order kinetics. A plot of 1/[HI] versus time (using the data in Table 15.1).

The amount of time required for half of a reactant to disappear is called the half-life, t1/2 The half-life of a first-order reaction is not affected by the initial concentration

First-order radioactive decay of iodine-131 First-order radioactive decay of iodine-131. The initial concentration is represented by [I]0.

The half-life of a second-order reactions does depend on the initial concentration

One of the simplest models to explain reactions rates is collision theory According to collision theory, the rate of reaction is proportional to the effective number of collisions per second among the reacting molecules An effective collision is one that actually gives product molecules The number of all types of collisions increase with concentration, including effective collisions

There are a number of reasons why only a small fraction of all the collisions leads to the formation of product: Only a small fraction of the collisions are energetic enough to lead to products Molecular orientation is important because a collision on the “wrong side” of a reacting species cannot produce any product This becomes more important as the complexity of the reactants increases

The key step in the decomposition of NO2Cl to NO2 and Cl2 is the collision of a Cl atom with a NO2Cl molecules. (a) A poorly orientated collision. (b) An effectively orientated collision.

The minimum energy kinetic energy the colliding particles must have is called the activation energy, Ea In a successful collision, the activation energy changes to potential energy as the bonds rearrange to for products Activation energies can be large, so only a small fraction of the well-orientated, colliding molecules have it Temperature increases increase the average kinetic energy of the reacting particles

Kinetic energy distribution for a reaction at two different temperatures. At the higher temperature, a larger fraction of the collisions have sufficient energy for reaction to occur. The shaded area under the curves represent the reacting fraction of the collisions.

Transition state theory explains what happens when reactant particles come together Potential-energy diagrams are used to help visualize the relationship between the activation energy and the development of total potential energy The potential energy is plotted against reaction coordinate or reaction progress

The potential-energy diagram for an exothermic reaction The potential-energy diagram for an exothermic reaction. The extent of reaction is represented as the reaction coordinate.

A successful (a) and unsuccessful (b) collision for an exothermic reaction.

Activation energies and heats of reactions can be determined from potential-energy diagrams Potential-energy diagram for an endothermic reaction. The heat of reaction and activation energy are labeled.

Reactions generally have different activation energies in the forward and reverse direction Activation energy barrier for the forward and reverse reactions.

The brief moment during a successful collision that the reactant bonds are partially broken and the product bonds are partially formed is called the transition state The potential energy of the transition state is a maximum of the potential-energy diagram The unstable chemical species that “exists” momentarily is called the activated complex

Formation of the activated complex in the reaction between NO2Cl and Cl. NO2Cl+ClNO2+Cl2

The activation energy is related to the rate constant by the Arrhenius equation k = rate constant Ea = activation energy e = base of the natural logarithm R = gas constant = 8.314 J mol-1 K-1 T = Kelvin temperature A = frequency factor or pre-exponential factor

The Arrhenius equation can be put in standard slope-intercept form by taking the natural logarithm A plot of ln k versus (1/T) gives a straight line with slope = -Ea/RT

The activation energy can be related to the rate constant at two temperatures The reaction’s mechanism is the series of simple reactions called elementary processes The rate law of an elementary process can be written from its chemical equation

The overall rate law determined for the mechanism must agree with the observed rate law The exponents in the rate law for an elementary process are equal to the coefficients of the reactants in chemical equation

Multistep reactions are common The sum of the element processes must give the overall reaction The slow set in a multistep reaction limits how fast the final products can form and is called the rate-determining or rate-limiting step Simultaneous collisions between three or more particles is extremely rate

A reaction that depended a three-body collision would be extremely slow Thus, reaction mechanism seldom include elementary process that involve more than two-body or bimolecular collisions Consider the reaction The mechanism is thought to be

The second step is the rate-limiting step, which gives N2O2 is a reactive intermediate, and can be eliminated from the expression

The first step is a fast equilibrium At equilibrium, the rate of the forward and reverse reaction are equal

Substituting, the rate law becomes Which is consistent with the experimental rate law

Thus a larger fraction of the collisions are effective A catalyst is a substance that changes the rate of a chemical reaction without itself being used up Positive catalysts speed up reactions Negative catalysts or inhibitors slow reactions (Positive) catalysts speed reactions by allowing the rate-limiting step to proceed with a lower activation energy Thus a larger fraction of the collisions are effective

(a) The catalyst provides an alternate, low-energy path from the reactants to the products. (b) A larger fraction of molecules have sufficient energy to react when the catalyzed path is available.

Catalysts can be divided into two groups Homogeneous catalysts exist in the same phase as the reactants Heterogeneous catalysts exist in a separate phase NO2 is a homogeneous catalyst for the production of sulfuric acid in the lead chamber process The mechanism is:

The second step is slow, but is catalyzed by NO2:

Heterogeneous catalysts are typically solids Consider the synthesis of ammonia from hydrogen and nitrogen by the Haber process The reaction takes place on the surface of an iron catalyst that contains traces of aluminum and potassium oxides The hydrogen and nitrogen binds to the catalyst lowering the activation energy

The Haber process. Catalytic formation of ammonia molecules from hydrogen and nitrogen on the surface of a catalyst.