Chemical Equilibrium
Rates of Reaction & Equilibrium Constants
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| Unit 29
Chemical Equilibrium Rates of Reaction & Equilibrium Constants |
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Overview
Physical chemistry includes the study of chemical kinetics or reaction kinetics, which emphasize the role of reaction rates in a chemical reactions. Analyzing the influence of different reaction conditions on the reaction rate gives information about the reaction mechanism. By analyzing the reaction rate, and comparing these data with various proposed mechanisms, we can identify the mechanism indirectly as the one which best fits the data. This is valuable, as the metastable transition state of a chemical reaction is typically short-lived and exceedingly difficult to isolate Thus, though we have no formal proof that the intermediary molecular conformation exists, we have a reaction rate constant, k, which convinces us that it does indeed exist.
Rates of Reaction
In 1864, Peter Waage pioneered the development of chemical kinetics by formulating the law of mass action, which states that the speed of a chemical reaction is proportional to the quantity (or concentration) of the reacting substances. To put it simply, chemical reactions occur more rapidly if there are more initial reactants present.
Chemical equilibrium is the state in which the concentrations of the reactants and products have no net change over time. Usually, this state results when the forward reactions proceed at the same rate as their reverse reactions. The rates of the forward and reverse reactions are generally not zero but, being equal, there are no net changes in any of the reactant or product concentrations. This process is known as dynamic equilibrium.
Consider the use of square brackets to represent the concentration of any given species, in a basic chemical reaction where (in the forward direction) reactants A and B yield products C and D. Let a, b, c, d be the stoichiometric coefficients for chemical species A, B, C, D respectively.
a [A] + b [b] = c [c] + d [D]
The rate of reaction is the change in the molar concentration of a product per unit time. This is equivalent to either the rate of formation of product or the rate of disappearance of reactants, given in units of moles per second.
Note the use of the "=" sign rather than using an arrow. The purpose of this is to indicate that this is a reversible reaction which can proceed in either the forward or the reverse directions. In actuality, most chemical reactions are reversible. [ Irreversible reactions can be viewed as an extreme, "special case" of reversible reactions ].
General Rate of Reaction
Consider the rate of the reaction in which 2 moles of reactant A produce 3 moles of reactant by by the following reaction scheme:
2 A = 2 B + C
We can base the reaction rate either on the 1) disappearance of reactant A, or 2) formation of products B or C.
But if we use reactant A and product C, the rates will not be the same. The reason is that it takes 2 moles of reactant A to produce 1 mole of product C. I.E. reactant A disappears twice as fast as C forms. A better alternative is to describe a general rate of reaction that has the same value, regardless of of which reactant or product we study. Thus, to say that reactant A disappears twice as fast as C forms is equivalent to saying that product C forms only half as fast as reactant A disappears.
The general (or normalized) rate of reaction is given by dividing the rate of change in concentration of a reactant or product by the its stoichiometric coefficient. If we apply this methodology to the general reaction:
a [A] + b [b] = c [c] + d [D]
Then the general rate of reaction is given by:
It may be helpful to note this "normalization" process is not species specific. That is to say that we could choose any of reactants or products to be the species upon which our reaction rate is based. This simply requires that we scale the coefficients of the other products and reactants appropriately. In any reaction rate study, it is usually a particular reactant or product that is being monitored. We can then express the rate of reaction using a stoichiometric coefficient of 1 for that particular species, and scaling the others appropriately. (This procedure reminds us that it is not the absolute values of the stoichiometric coefficients which is most important. Rather it is the relative ratios between them that defines the relative molar quantities).
The Rate Law
The rate law for a chemical reaction relates the rate of reaction to the concentration of reactants. Consider again a general reaction which proceeds as follows:
a [A] + b [B] = c [C] + d [D]
If we assume that the initial time is at t = 0, then we can express the general rate of reaction in terms of each individual component as follows:
The rate law for the forward reaction is of the form:
The exponents m and n are generally small whole numbers -- but they have been found in some instances to be non-integral. It is important to note here that the exponents can only be determined by experiment. It should not be assumed that are in direct coincidence with the stoichiometric coefficients - though in many instances they are identical. The primary function of the exponents is their relationship to the mechanism of reaction. It is often by way of these exponents that we are able to deduce the details of the reaction mechanism.
The values the exponents in a rate law establish the order of a reaction. If m = 1, the reaction is first order in A. If n = 2, the reaction is second order in B. The overall order of f reaction is given by the sun of the exponents. The constant of proportionality is the rate constant, k, which is determined strictly by experiment. With the rate law and a value of k, we can calculate the initial rates for any initial concentration of reactants.
Note: The units of the rate constant, k (e.g. moles per second for zero order reaction) will vary depending upon the values of the exponents. It can easily be shown that the units are given by:
One way of determining the rate law exponents involves a series of experiments in which the initial concentrations of some reactants are held constant while others are varied in convenient multiples.
Zero - Order Reactions
Some (though relatively few) reactions are independent of the concentration of reactants. In these cases, other factors than reactant concentrations typically control how fast the reactants can enter into a reaction. For example, photo-initiated reactions (e.g. photosynthesis) may require the absorption of light, in which case the intensity of light determines the reaction rate.
In the rate law for a reaction that is zero order overall, the sum of the exponents is zero, and the rate law is given by:
In such a case, a plot of concentration vs. time (for various initial concentrations) is a straight line with negative slope. The rate of reaction is constant, and is equal to the reaction constant k and to the negative of the slope.
First - Order Reactions
First-order reactions are those in which a single reactant yields products, as follows:
A --> Products
The rate law for such a reaction is simply given by:
In this case, we can use the integrated rate law which describes the concentration of a reactant as a function of time to slve for the reaction constant k as follows.
= Concentration of reactant A after an
elapsed time t
=
Initial concentration of reactant A
Again, this is the equation of a straight line. This becomes more apparent if we write it as:
Then if we plot ln [A] vs. time t, the slope of the line is again the negative of the reaction constant k. If the plot is not a straight line, then the reaction is not first-order.
Second - Order Reactions
In a second-order reaction, the sum of the exponents is equal to 2. For example:
This reaction is first order in NO, first order in O3, and second order overall.
A simpler possibility for a second-order reaction is a reaction with a single reactant:
A --> Products
One example of this type of reaction is the decomposition of hydrogen iodide into hydrogen and iodine (all gaseous states).
2 HI = H2 + I2
The integrated rate law that expresses [ A ] as a function of time for these second-order reaction has the following rm:
From this equation, we see that a graph of 1/[ A ] vs. time is a straight line. The slope of the line is equal to the reaction constant k, and the y-intercept (t = 0) is equal to the initial concentration of reactant A.
Reaction Mechanisms
A few simple reactions consist can be understood completely using only the single step suggested by the balanced equation for the reaction. But many reactions involve the simultaneous contact (or collision) of several different species. Consider the reaction between nitrogen monoxide and oxygen in the formation of urban smog (all vapors).
2 NO + O2 = 2 NO2
It is highly unlikely that 2 NO molecules and one O2 molecule would collide simultaneously, just as it is highly unlikely that 3 basketballs might simultaneously collide in midair in a warm-up or practice session. Instead, the overall reaction is accomplished in several individual simpler steps. Most elementary steps such as these involve only 2 species.
Thus, a reaction mechanism is typically characterized by a series or sequence of simpler steps that ultimately lead from the initial reactants the the final products of a reaction. Most importantly:
1) Any proposed scenarios for such a mechanism must inevitably account for the experimentally determined rate law.
2) The mechanism must also be consistent with the stoichiometry of the overall or net reaction.
Elementary Reactions
Consider the following reaction:
CO + NO2 = CO2 + NO
It has been shown experimentally that the rate law for the forward reaction is given by:
R = k [NO2] 2
One possible mechanism by which this reaction takes place is:
The net reaction would then be given as initially stated:
CO + NO2 --> CO2 + NO
Each step is called an elementary step, and each has its own rate law and molecularity. All of the elementary steps must add up to the original reaction. The molecularity of an elementary reaction refers to the number of free particles (atoms, ions or molecules) which enter into the reaction. According to collision theory, the successful collisions will be the ones having sufficient energy (activation energy) at the moment of impact to break the existing bonds and form new bonds, resulting in the formation of the products of the reaction.
1) A unimolecular reaction is one in which a single molecule dissociates.
2) A bimolecular reaction is one in which the two molecules collide effectively to produce a proper reaction intermediate, transition state or activated complex.
3) A termolecular reaction, requiring the simultaneous collision of three free particles, is much less likely to occur than unimolecular or bimolecular reactions.
** Note: The exponents in the rate law for an elementary reaction are identical to the stoichiometric coefficients in the chemical equation for the reaction. (Recall that this is generally not the case with the rate law for the overall reaction).
Elementary reactions are reversible -- that is, the forward and reverse reactions occur simultaneously. Some reach a state of equilibrium in which the rates of forward and reverse reactions are equal. More importantly, one elementary reaction may be much slower than all the others. In most cases, this will be the rate-determining step -- the crucial step in establishing the rate of the overall reaction.
When determining the overall rate law for a reaction, the slow step is the step that determines the reaction rate. In the reaction above between carbon monoxide and nitrogen dioxide, because it involves the collision of 2 NO2 molecules, it is a bimolecular reaction with a rate law of R = k [NO2] 2.
( slow ) 2 NO2 --> NO3 + NO
Note: In this particular case, the rate of reaction R depends solely on the concentration of a single reactant, and is independent of the concentration of the second reactant. The exponents determine the order of reaction and depend on the reaction mechanism. Thus, we call this a second-order reaction, as it is only dependent on the concentration of a single second-order reactant. . Another type of second-order reaction would be dependent on two first-order reactants, such as R = k [A] [B] where A and B are reactants.
Note the coincidence between the exponent (2) on the NO2 in the rate-determining (slow) elementary reaction step and the order (2) of the overall reaction.
Let us now reconsider the above reaction involving the formation of urban smog.
2 NO + O2 = 2 NO2
Many reactions occur by a mechanism that has a fast reversible step followed by a slow step. The first step is an elementary reaction in which the reactants form an intermediate (rate constant k1). But the metastable intermediate readily decomposes back into the initial reactants in a reverse reaction (rate constant k1). Here we assume that the rates of the forward and reverse reactions rapidly equilibrate (or reach a state of rapid equilibrium).
However, a small amount of the metastable reaction intermediate is removed by reacting in a slow second step, the rate-determining step, to form the final products (rate constant k2).
The rate law for the net (smog-producing) reaction is investigated experimentally and found to be given by the following expression:
Rate = k [NO2] 2 [O2]
A simple mechanism with only one step would account for the observed rate law. But this would be a highly unlikely termolecular reaction involving the simultaneous collision of 3 vapor phase molecules. A more plausible mechanism is outlined as follows:
(fast) 2 NO = N2O2 k1
(slow) N2O2 + O2 = 2 NO2 k2
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(net) 2 NO + O2 = 2 NO2 k
If we base the rate law on the slow rate-determining step alone, we get:
Rate = k2 [ N2O2 ] [ O2 ]
Recall, however, that we can state the experimentally determined rate law only in terms of substances or species in the net equation. I.E. Short-lived reaction intermediates which are impossible to isolate cannot be used to describe the rate law for a given reaction.
Thus, we must eliminate the [ N2O2 ] from the expression for the rate of reaction as follows:
1) Start with the equation for the first elementary reaction, and assume that equilibrium is rapidly established.
2) Describe those rates in terms of the formation and disappearance of N2O2 and set the rate laws equal.
3) Solve for [ N2O2 ].
2 NO = N2O2
Rate of formation of N2O2 = Rate of disappearance of N2O2
Substituting into the expression for the rate of reaction:
Since the slow reaction (reaction constant k2) will be the rate-determining reaction, then it will ultimately determine the net (or overall) reaction constant k. Thus k = k2. Substitution into the above expression, the rate of the net reaction will be given by:
as determined experimentally. Thus the proposed mechanism is consistent with the observed rate law. Note that in this second-order reaction, the order (2) is identical to the sum of the stoichiometric coefficients in the elementary rate-determining step (2), but not equal to the sum of the exponents in the overall expression for the rate law (3).
Organic reactions can be organized into several basic types. In general, there are four types of elementary bimolecular steps:
1) Addition, 2) Elimination, 3) Substitution, 4) Rearrangement.
Some reactions fit into more than one category. For example, some substitution reactions follow an addition-elimination pathway. This overview isn't intended to include every single organic reaction. Rather, it is intended to cover the basic reactions.
Chemical Equilibrium
In a state of chemical equilibrium, rate of reaction for the forward reaction is the same as that of the reverse reaction. Another way of saying this is that the concentrations of reactants and products have reached a steady state and remain fixed or constant. I.E. The amount of each reactant and product being consumed is the same as that being produced. Thus there is no net change in the concentrations of any of the reactants or products over time.
Thus, if we bail water out of a boat at the same rate that it manages to seep in, then the depth of the pool of water at the bottom of the boat will never change.
The equilibrium constant K is one way of expressing these fixed concentrations in relation to each other. If they are indeed fixed, then there should be one number (K) which expresses their collective amounts under these conditions. K should have a constant value regardless of the initial concentrations of reactants and products. Thus, for the original hypothetical reaction:
a [A] + b [b] = c [c] + d [D]
The equilibrium constant K is given by the following expression:
In terms of the forward and reverse reactions, it is important to note here that at equilibrium, the rates of these respective reactions are equal. Thus we have an alternative way of expressing K in terms of k1 and k2 as follows.
Rearranging this expression yields the following result:
The ratio of reaction rate constants for the forward and reverse reactions is equivalent to the equilibrium constant for the net or overall reaction.
In general, reactions which liberate heat are characterized by products which have a more stable chemical configuration, or lower energy. Since nature favors lower energy configurations, exothermic processes are generally spontaneous and favor reaction products. Such reactions are typically characterized by a larger equilibrium constant as follows:
Products favored: K >> 1
Alternatively, endothermic processes absorb heat energy during the reaction. Thus, they are typically characterized by products which have a less stable chemical configuration, or higher energy. Thus, endothermic processes generally favor reactants, and are extremely sluggish in the absence of a catalyst. Such reactions are typically characterized by a smaller equilibrium constant as follows:
Reactants favored: K << 1
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