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Lecture 31 At the beginning of the semester I stated that we can describe the realms of kinetics and thermodynamics by stating that thermodynamics tells us what can happen, while kinetics tells us how fast it can happen. For example, thermodynamics tells us that the reaction H 2 (g) + Cl 2 (g) → 2HCl(g) has an equilibrium constant greater than 10^37 , yet we can place a mixture of H 2 and Cl 2 in a darkened container for upwards of a hundred years without a noticeable reaction occurring. Does this mean that thermodynamics is wrong? Of course not! What it does point out is that while thermodynamics tells us a great deal about what processes are favored, it tells us nothing of the rate at which they occur. Of course this is a crude simplification of thermodynamics. We've already seen from our treatment of thermodynamics that in addition to showing what can happen, we can predict a wide variety of equilibrium properties. Perhaps this last phrase with its emphasis on the word equilibrium helps bring about another level of understanding of the different realms of thermodynamics and kinetics. Thermodynamics is limited to systems in equilibrium, while kinetics can treat systems in disequilibrium. Kinetics has a wide array of applications. By studying the concentration dependence of reactions we can determine the optimum concentration conditions for running a reaction, the rate-determining step of the mechanism, and the mechanism of the reaction itself. The temperature dependence of reactions tells us the optimum temperature for running reactions, the energy of activation for the reaction and information about microscopic mechanisms. Most of the efforts of modern chemical kinetics have been focused on elucidation of
microscopic mechanisms. To help understand the distinction between microscopic and macroscopic mechanisms consider the reaction H 2 + 2ICl → I 2 + 2 HCl When we study the concentration dependence of the reaction rates we discover that it is consistent with the reaction being broken down into two steps, both of which involve reactions between no more than two molecules. H 2 + ICl → HI + HCl HI + ICl → I 2 + HCl These reactions are called elementary reactions because they proceed as written in a single step. This breakdown of a complex reaction into a series of elementary reactions is what we mean when we talk about a macroscopic mechanism. Macroscopic mechanisms of reactions are what we'll be focusing on in Chem 309. When we discuss microscopic mechanisms , we're considering an elementary reaction like the first step H 2 + ICl → HI + HCl and asking questions like
- How do transitional, vibrational, and rotational energies affect the efficiency of the reaction?
- How do the bond lengths change as the reaction proceeds?
- What, if any, are the intermediates and how stable are they?
- How do reaction geometries affect reaction rates?
- How does energy flow between the various degrees of freedom of the molecule? In other words, a microscopic mechanism is a detailed description of all of the processes that can affect an elementary reaction, and of the processes that occur during an elementary reaction.
In contrast the reaction H 2 + Br 2 → 2HBr, while having a very complex mechanism, results in little or no buildup of reactive intermediates so that the stoichiometry of the overall reaction is the reaction stoichiometry at all times. This type of stoichiometry is called time independent stoichiometry. Suppose we have a time independent reaction 2A + 3B → Y + 2Z and at time T, the amounts of our reactants are given by nA, nB, nY and nZ, while initially at time 0, the amounts are nA0, nB0, nY0, and nZ0. We can relate the changes in the amounts of these various reactants and products by n - n -2 =
n - n -3 =
n - n 1 =
n - n 2 A A 0 B B 0 Y Y 0 Z Z 0
In general the change in amount of the ith^ substance in the reaction mixture is given by
n - ni i 0 = ν i ζ
where ζ is our extent of reaction from chemical thermodynamics. Note again that the extent of reaction is a positive number when our reaction proceeds from reactants to products, and is the same for all species in the reaction. We are interested, of course, in rates of reactions , so we need to quickly define our terms. For our reaction 2A + 3B → Y + 2Z, the rate of consumption of A will be referred to as vA. If we assume that the volume of the solution is constant, the rate of consumption for a reactant is defined as the negative of the
change in concentration of A per unit time,
vA = -d^ dt [^ A ] ,
while for products, we speak of the rate of formation, given by
vY = d Ydt [^^ ].
Note that if we plot [A] vs t, in general the slope changes with time so that the rate of change of concentration at time t 1 is not equal to the rate of change of concentration at some other time t 2. If we take the slope of the change of concentration with time at t = 0, we call the rate the initial rate of consumption for reactants, or the initial rate of formation for products. The rates of change of the various species in the system are related by their stoichiometric coefficients by vA = 2/3vB = -2vY = -vZ. Clearly it is confusing to have a different rate describing the kinetics of each of the species in the reaction mixture. We can avoid this by defining the rate of reaction as
v ≡ dtd ⎛⎜⎝^ V^ ζ⎞⎟⎠.
With the simplifying assumption that the volume remains constant during the reaction this
constant pressure, where volume changes can be substantial, the more complicated treatment can be necessary. All of you should be familiar with rate equations from general chemistry. I'd like to review them quickly. In general, rate equations are equations that relate the rate of a chemical reaction to the concentrations of the species present in the reaction mixture, i.e.
v = f ( [ A ] (^) , (^) [ B ] (^) , (^) [ C ]...).
Note that this equation can be something very complicated such as
[ ] [ ] [ ] [ ]
1 3/ 2 2 1 v k^ A^ C = (^) k A + k (^) − B
or something as simple as
v = k 1 (^) [ A ]
In these example equations, the substances A, B, C etc, can be reactants, products, intermediates or catalysts. For some reactions, the rate of reaction can be expressed by an particularly simple equation of the form v = k[A]α[B]β… where the concentrations of reactants and catalysts can be included in the equation and where k, α, and β are independent of the concentration and of time. It is important to emphasize that this simpler rate equation is not going to work for all reactions. It is also important to recognize that these rate equations are strictly empirical. The rate of consumption of a given species will be represented by the equation vA = kA[A]α^ [B]β
where α and β are the same exponents as for the overall reaction, and kA is related to k by kA = k|νA| So for our reaction 2A + 3B → Y + 2Z, k = kA/2 = kB/3. The exponent α is called the order of the reaction with respect to A. Similarly the exponent β
will be called the order of the reaction with respect to B. α and β are called partial orders. The sum of the partial orders, α + β + ... = n,
is called the order of the reaction. Once again I wish to emphasize that α and β are empirical quantities. They are not necessarily integral, and not necessarily related to the reaction stoichiometry. The simplest case of a rate law is when the rate is linear with the concentration of a single reactant, i.e. v = k[A]. This is called a first order reaction. If the rate depends on the square of the concentration of a single reactant, or the product of two reactant concentrations, i.e., v = k[A]^2 or v = k[A][B], the reaction is said to be second order. The constant k, once again is called the rate constant. The rate constant depends on the temperature and the chemical identity of the reactants. Its dimensions depend on the order of the reaction. For example, for any reaction we have
The molecularity of an elementary reaction is the number of reactant particles (atoms, molecules, free radicals or ions) that are involved. If one reactant particle is involved then the reaction is said to be unimolecular. If two are involved the reaction is bimolecular, three trimolecular, etc. Note that in general, the molecularity is distinct from the order. In general, the molecularity tells us mechanistic details of a reaction, while the order simply tells us the dependence of the reaction rate on the concentration of a reactant. Composite reactions are those that involve more than one elementary reaction. The term molecularity has no meaning for a composite reaction. It is convenient to number these reactions in such a way that reverse reactions are identified easily. Thus the reverse of reaction 1 is referred to as reaction -1, and the rates and rate constants are denoted by corresponding subscripts, for example v 1 , v-1, k 1 and k-1. A composite reaction mechanism sometimes includes a cycle of reactions such that certain reaction intermediates consumed in one step are generated in another. If such a cycle is repeated more than once, the reaction is known as a chain reaction. For example, if hydrogen and bromine react and the product HBr is removed as fast as it is formed the reaction is believed to proceed by the following steps:
(1) Br 2 (^) ⎯ →⎯ k^1 2 Br Initiation (2) Br + H (^) 2 ⎯ →⎯ k^2 HBr + H Chain Propagation (3) H + Br 2 (^) ⎯ →⎯ k^3 HBr + Br Ditto (-1) Br + Br ⎯⎯ k −^1 → Br 2 Termination Reactions (2) and (3) constitute a cycle and are known as chain-propagating steps; in reaction (2) a bromine atom is produced which is consumed in reaction (3) while in reaction (3)
a hydrogen atom is produced which is consumed in reaction (2). Under normal conditions this cycle occurs a number of times, and the reaction is therefore a chain reaction. Reaction (1), which produces active intermediates, is known as an initiation reaction, and its reverse, reaction (-1) is called a termination step or chain ending step. Chain reactions always involve initiation and termination steps and two or more chain-propagating steps.
concentrations were changed. The alternative method is to follow the concentration of one sample over time, and to determine the rate at various points in the course of the reaction. Plots of log v vs log [ ] then yield the order. Orders determined this way are labeled nt. It is important to note that nt is not necessarily equal to nc. The reason is that when the reaction is followed over time, intermediates may build up which change the rate of reaction, and therefore the apparent order of the reaction. We see from this that nc and nt each provide useful information that the other does not. nc shows the concentration dependence under initial conditions. nt shows us the order based on conditions averaged over the whole course of the reaction. In addition, comparison of the two shows whether additional species need to be considered in the rate law. Once nc or nt is determined for one of the reactants, the procedure is repeated to determine the overall rate law. The alternative method for determining the order of reaction for each substance is called the method of integration. Basically this method is to guess an order for the reaction, see what the concentration dependence should be and compare the prediction with the experimental results. We will solve the concentration equation for three cases - first order, second order in one substance, and second order overall with two reactants having partial orders of one. The simplest example of a first order reaction would be an elementary reaction of the form A → Z, but more complex reactions like
2A → Z
or A + B → Z could be first order reactions as well. Remember, there is not necessarily any relationship between the stoichiometry of a reaction and its order. For the first of these reactions, at t = 0, we have [A] = a 0 , and at some time t later we have [A] = a 0 – x, where x is the amount of A consumed at time t. Therefore the rate of consumption of A is given by
vA = (^) - d(a - x)dt^0 = dxdt.
Since the reaction is first order we can write dx dt = k (a - x)A^^0 Separation of variables and integration yields
- ln (a 0 -x ) = kAt + c Finally evaluation of the constant of integration yields our integrated rate equation
ln ( (^) a - x 0 a^0 ) = k tA
Therefore if our reaction is first order, a plot of ln ( (^) a - x 0 a^0 ) vs t will be linear with a slope of kA.
Note that this result is exactly equivalent to
[ ] ln [^ A^0^ ] k tA A
⎜⎝ ⎟⎠^.
There are two possibilities for second order reactions. The first is that the reaction is second order in a single reactant, i.e.,
dx dt = k^ A^ (a - x)(b - x)^0 This is a fairly difficult integral to solve, but can be solved in one of three ways. The oldest and least recommended is to use the method of partial integration. It works fine but it’s just too much work. The second is to look it up in a table of integrals. Third, you could use a symbolic manipulation program like Mathematica or Maples to do the integration. No matter what method you use, when you carry out the integral and evaluate the constants of integration, the rate law is 1 a - b [
b (a - x) 0 0 a (b - x) ] = kt
0 0 ln 0 0
There are tables available where you can find the integrated rate laws for a wide variety of reaction orders and a wide variety of initial conditions. In each case you determine the order of your reaction by plotting your data according to the appropriate integrated rate equation and testing for linearity. A reaction of general order n in one reactant would have the rate expression v d^^ [^ A ]^ k [ ] n = − (^) dt = A.
A simple integration of this expression yields the result
( ) (^) [ ] 1 [ ] 01
n (^1) A n^^ −^ An^ − kt
− ⎜⎝^ ⎟⎠^ ,
where [A] is the concentration of A at time t, and [A] 0 is the initial concentration of A. When the reaction rate depends on the concentration of only one reactant, it is relatively easy to determine the rate law. It is more work for a reaction in which the rate depends on several species, whether they are reactants, products, or catalysts. If the rate of consumption is
relatively slow, then we can simply use the method of initial rates, varying the initial concentration of each species in turn. However, this method presupposes that the concentrations of the species which we are not varying are either constant or are changing slowly enough that their changes do not affect the apparent order of the reaction. As an example, consider the bromination of acetone. The reaction here is CH 3 COCH 3 + Br 2 → CH 3 COCH 2 Br + HBr. The rate of reaction is a function of the concentrations of bromine, acetone, and a catalyst, H+, so our rate equation is v = k [H+]α[CH 3 COCH 3 ]β[Br 2 ]γ,
where α, β and γ are the orders we wish to determine. Now suppose that while we are examining the effect of changing the bromine concentration on the reaction rate, the acetone concentration is also changing rapidly. This would distort the results of the bromine study. One solution to this is the method of isolation in which all of the species in the reaction except one are present in large excess. In effect what this does is to keep the concentrations of all species but one approximately constant. This means that the observed order of the reaction will be the partial order of the species under consideration. Thus if our rate law is v = k [A]α[B]β[C]γ,
and we isolate A, the order of the reaction will be α. If we now repeat the experiment isolating B and C in turn, we can determine the overall rate law for the reaction. The fact that we can isolate a single component, and therefore simplify the rate law means that the method of integration can be used for relatively complex systems, since the integrated rate laws involving only a single substance are all simple.
A + B → Z,
where a 0 ≠ b 0. Since the initial amounts of the substances are not equal, the half-lives cannot be equal. Clearly in this case it is not meaningful to speak of reaction half-lives. In general, it is meaningful to speak of reaction half-lives only when the reactants are present in their stoichiometric ratios. For example, in the case we just discussed, if a 0 = b 0 , then we can talk about reaction half-lives. Similarly for a reaction with stoichiometry A + 3B → Z, we can only talk about reaction half lives if b 0 = 3a 0. Otherwise it is only meaningful to speak of reactant half-lives. We can write an equation for the half life of a general reaction of order n>1 for a single reactant, based on our integrated equation for a reaction with order n. It is
( )[ ]
1 1/ 2 01
n t (^) k n An
− −
We can see from this equation that a plot of the log of t1/2 vs the log of [A] 0 can be used to determine the reaction order. To see this we take the log of the equation above to get
( ) (^ )^ [^ ]
1 log (^) 1/ 2 log 2 11 1 log 0
n t (^) k n n
= − − − A.
Thus a plot of log t1/2 vs log [A] 0 will have a slope of n-1, where n is the order. Half-lives have the useful property that they can be used to compare how fast two reactions with different orders will proceed. This is one case where the half life is more transparent than the rate constant, since the rate constants for reactions with different orders will have different units and will be difficult to compare. We can also use half-lives for a relatively
quick determination of whether a reaction is first order. If the reaction is first order we can measure the half-life for two or more different initial concentrations, and it will not change, while for higher order reactions the half life will be dependent on the initial concentration. How do these methods for determining rate laws compare? If you know very little about the reaction the differential method is best for the following reasons. First, if the log- log plot is linear, then the reaction has an order. Second, by distinguishing between nt and nc it helps us detect the presence of intermediates in the reaction. Third, it allows us to find non- integral orders if they are present. The main advantage of the integral method is that it may provide more accurate rate constants than the differential method. A disadvantage is that a reaction may have a non-integral order, yet fit one of our integrated rate laws fairly well. This means that it is not too hard to get at least slightly incorrect rate constants from the integral method. A second disadvantage is that the integral method gives us only nt, since n is obtained by following the concentration over time.
