The Molecular View of Equilibrium
In other sections, we show that equilibrium constants exist, how they can be determined, and how they can be used, but we have not explained why they are sometimes large and sometimes small. In order to answer this and related questions we must switch our point of view from the macroscopic to the microscopic level.
The cisDescribes the relationship between two atoms or groups of atoms, each attached to one of two doubly bonded carbon atoms and located on the same side of the double bond. Also refers to groups located adjacent to each other in an octahedral or square planar coordination complex.-transDescribes the relationship between two atoms or groups of atoms, each attached to one of two doubly bonded carbon atoms and located on the opposite side of the double bond. Also refers to groups located opposite to each other in an octahedral or square planar coordination complex. isomerism of difluoroethene provides a simple example for examination on the molecular level. As we saw earlier, if 1 mol cis isomer of this compoundA substance made up of two or more elements and having those elements present in definite proportions; a compound can be decomposed into two or more different substances. is heated to 623 K, it is gradually transformed into the equilibrium mixtureA combination of two or more substances in which the substances retain their chemical identity. of ⅔ mol cis isomer and ⅓ mol trans isomer. From a macroscopic point of view, we might conclude that the reaction stops once equilibrium is attained, but on the microscopic level, such a statement is obviously untrue. As cis molecules move around a container at 623 K, they occasionally suffer a collision with sufficient energyA system's capacity to do work. to flip them over to the trans conformationIn an ion or molecule, one of many spatial arrangements of atoms that differ by rotation about single bonds, but that do not differ in which atoms are attached to each other.. This conversion process does not suddenly stop when equilibrium has been attained. There are plenty of cis molecules present in the equilibrium mixture, and they continue to collide with their fellow molecules and hence to flip over from the cis to the trans conformation. Thus the forward reaction
cis-C2H2F2 → trans-C2H2F2
must continue to occur in the equilibrium mixture.
If this were the only reaction taking place in the flask, the eventual result would be that all the cis isomer would be converted to the trans isomer and no equilibrium mixture would result. This does not happen because the reverse reaction
trans-C2H2F2 → cis-C2H2F2
is also occurring at the same time. The trans molecules in the equilibrium mixture are also continually being bombarded by their fellow molecules. Many of these collisions have sufficient energy to flip the trans molecule back into the cis form. Thus cis molecules are not only being consumed but also being produced in the flask. The concentrationA measure of the ratio of the quantity of a substance to the quantity of solvent, solution, or ore. Also, the process of making something more concentrated. of the cis molecules remains constant because the rate at which they are being consumed is exactly balanced by the rate at which they are being produced. In other words, for every cis molecule which flips to the trans conformation in one part of the container, there will be on the average a molecule of trans isomer flipping in the reverse direction in another part of the container. This constant reshuffle of molecules between the cis and the trans forms can then continue indefinitely without any net change in the concentration of either species.
This molecular interpretation of equilibrium is not confined to the particular example of cis-trans isomerism or even just to chemical reactions. In the section on phase transitions we describe vaporThe gaseous state of a substance that typically exists as a liquid or solid; a gas at a temperature near or below the boiling point of the corresponding liquid.-liquidA state of matter in which the atomic-scale particles remain close together but are able to change their positions so that the matter takes the shape of its container equilibria in which the condensationThe process in which a liquid forms from gas or vapor of the same substance. A chemical reaction in which two molecules combine to form a very small molecule and a larger molecule than either of the two reactants. of vapor just balanced the evaporation of liquid. Any state of equilibrium corresponds to a situation in which the rate at which the forward reaction is occurring is exactly equal to the rate at which the reverse reaction is occurring. Since these rates are equal, there is no net change in the concentration of any of the reactants or products with time. Because an equilibrium corresponds to a balance between the forward and the reverse reaction in this way, we use double arrows () in the equation.
As another example of chemical equilibrium, let us take a 0.001 M solutionA mixture of one or more substances dissolved in a solvent to give a homogeneous mixture. of acetic acidIn Arrhenius theory, a substance that produces hydrogen ions (hydronium ions) in aqueous solution. In Bronsted-Lowry theory, a hydrogen-ion (proton) donor. In Lewis theory, a species that accepts a pair of electrons to form a covalent bond. at 25°C. As seen in the section on weak acids, slightly more than 10 percent of the acetic acid molecules in this solution have ionized according to the equation
CH3COOH + H2O CH3COO– + H3O+
Although the concentrations of all species in this solution remain constant with time, this does not mean that acetic acid molecules have stopped transferring protons to water molecules. The concentrations remain constant because the reverse reaction (transfer of a proton from a hydronium ion to an acetate ion) is occurring at the same rate as the forward reaction. For every hydronium ion produced by a proton transfer somewhere in the solution, another hydronium ion is losing its proton somewhere else. The net result is that the concentration of the hydronium ion remains constant, and with it the concentrations of the other species involved.
The microscopic view of equilibrium as a constant reshuffle of chemical species is often given a special name and referred to as a dynamic equilibriumA self-establishing state in which the concentrations of reactant and product species remains constant forever; called dynamic because it is reached when opposing processes occur at the same rate; if a change in conditions causes a system not to be at equilibrium, the system will return to equilibrium in a way that partially counteracts the change in conditions.. It is important to realize that once a dynamic equilibrium has been set up, a particular atom will sometimes turn up as part of one of the product molecules and sometimes as part of one of the reactant molecules. In the ionizationA process in which an atom, molecule, or negative ion loses an electron; a process in which a covalent molecule reacts with a solvent to form positive and negative ions; for example, a weak acid reacting with water to form its conjugate base (an anion) and a hydrogen (hydronium) ion. equilibrium of acetic acid just considered, suppose for the sake of argument that we could identify a particular oxygen atom in some way. If we could now follow the history of this atom, we would find that it would sometimes form part of an acetic acid molecule, CH3CO*OH and sometimes part of an acetate ion, CH3CO*O–. Since the acid is about 10 percent ionized, we would find on the average that our labeled oxygen would spend only 10 percent of its time as part of an acetate ion and the remaining 90 percent as part of an acetic acid molecule.
In the laboratory the heavy isotopeOne of two or more samples of an element whose atoms differ in the number of neutrons found in the nucleus. of oxygen, 188O, could be used.
Figure 1 gives a computer simulation of this dynamic view of equilibrium for the cis-trans isomerization equilibrium of difluoroethene at 623 K. Suppose that it were possible to color one of the molecules in the equilibrium mixture and to photograph it at regular intervals thereafter under a high-power microscope. The figure shows the kind of results we could expect. In each “photograph” the molecule appears at a different orientation― between photos it has been bounced around quite a bit by collisions with other molecules. A careful inspection reveals a more important fact. Three of the nine photographs show the screened molecule in the trans conformation, while in the other six the molecule has the cis conformation. If we went on taking a large number of photographs of this molecule, we would find on the average that this ratio would be preserved. On the average one-third of the photographs would show the trans conformation, while the other two-thirds would show the cis form. The situation is much as though we had a conventional six-sided cubic die with the word trans printed on two sides and the word cis printed on the other four. If we roll this die a sufficient number of times, it will come up cis twice as often as trans.
When we roll dice, each of them singly is called a die.
We can now begin to see why it is that the equilibrium constant for this reaction is 0.5 at 623 K. Not only the screened molecule, but every other molecule in the equilibrium mixture as well, spends one-third of its time in the trans conformation and two-thirds in the cis conformation. At any given time, therefore, we will find a mixture in which one-third of the molecules are trans while the other two-thirds are cis. The ratio of trans to cis will be 1:2, or 0.5, as found experimentally.
Such an explanation is still not the whole story. We still need to explain why the dice are loaded against the trans conformation of a molecule of difluoroethene. Why is it that this isomer is less likely to occur in the constant reshuffle between the two conformations caused by the continual random collisions of molecules with each other in the equilibrium mixture? The reason is that the trans isomer is slightly higher in energy than the cis form. The enthalpyA thermodynamic state function, symbol H, that equals internal energy plus pressure x volume; the change in enthalpy corresponds to the energy transferred as a result of a temperature difference (heat transfer) when a reaction occurs at constant pressure. change ΔHm for the reaction
cis-C2H2F2 → trans-C2H2F2
has the small, but nevertheless positive, value of +3.88 kJ mol–1.
In any collection of molecules, energy is constantly being transferred from one molecule to another as they collide. In this continual interchange of energy, inevitably some molecules acquire more kinetic or potential energy than their fellow molecules. However, when considering the distribution of kinetic energy, the probability of a molecule’s acquiring a given energy depends on the magnitude of that energy. The higher the energy, the less likely it is to occur. In the constant reshuffle of energies, molecules which are even slightly higher in energy than their fellows occur slightly less often. In the case under consideration, since a molecule of trans-difluoroethene is 3.88 kJ mol–1 higher in energy than the cis form, it will occur less often than the cis form in an equilibrium mixture in which each molecule is constantly flipping back and forth between the two forms. In other words, the die is loaded against the trans configurationThe 3-D arrangement of atoms about a chiral center in a molecule. because it is higher in energy.
We can now also understand why the position of equilibrium for this reaction varies with temperatureA physical property that indicates whether one object can transfer thermal energy to another object.. At a very low temperature it will be a rare occurrence for a molecule to acquire the extra 3.88 kJ mol–1 needed if it is to have the trans configuration. Accordingly, we expect the equilibrium mixture to contain many fewer trans molecules than cis molecules and the equilibrium constant
to be quite small. Indeed, if it were possible to take the system down to absolute zeroThe minimum possible temperature: 0 K, -273.15 °C, -459.67 °F., no trans molecules would be possible and the value of Kc would be zero. As the temperature is increased, the probability of a molecule acquiring enough energy to assume the trans configuration increases, and increases along with it. Kc cannot increase indefinitely, though. If the temperature is high enough, the energy difference between the cis and trans configurations becomes insignificant compared with the average kinetic energy of a molecule. Both forms will then be almost equally probable, and the value of Kc will be very slightly less than 1.
Very few chemical equilibria are as simple as the cis-trans isomerism of difluoroethene. In almost all other cases the position of equilibrium is governed not only by the energy difference between reactants and products but also by a probability factor which is independent of the energy. A simple example of such a probability factor is provided by the equilibrium between the ring and chain forms of 1,4-butanediol. The projection formula for this alcoholAn organic compound containing the functional group -OH. is
When a dilute solution (about 0.04 M) of this compound is prepared in carbon tetrachloride, the two ends of the molecule hydrogen-bond together to form a ring containing seven atomsThe smallest particle of an element that can be involved in chemical combination with another element; an atom consists of protons and neutrons in a tiny, very dense nucleus, surrounded by electrons, which occupy most of its volume., and an equilibrium between the ring form and the chain form is set up:
At 70°C (343 K) the equilibrium constant for this reaction is found by experiment to be
So that there are about 22 percent more chain molecules than ring molecules in the equilibrium mixture at this temperature. This is contrary to what one would expect if energy were the only factor involved in determining the position of equilibrium. The chain form of the molecule is obviously higher in energy than the ring form, since energy must be supplied in order to break the hydrogen bondAn attractive force, either intramolecular or intermolecular, between an electronegative atom and a hydrogen atom attached to another electronegative atom.. Indeed ΔHm for Eq. (1) is found to be +13.5 kJ mol–1. On the basis of our previous argument, if the chain form is higher in energy, it should be less probable than the ring form and Kc should be less than 1. The experimental fact that Kc is greater than 1 means that there is some other factor making the chain form more probable than would be the case if energy alone were responsible.
In order to see what this other factor is, let us suppose that we can somehow investigate our system at a temperature which is so high that the energy difference between the ring and the chain form is insignificant. (In practice, the carbon tetrachloride solventThe substance to which a solute is added to make a solution. would boil off long before this temperature was obtained.) At such a temperature, if the energy difference between the two forms were the only factor determining their relative probability, we would expect the ring and chain forms to be equally probable. It is easy to see, though, why this would not actually be the case.
The chain form of 1,4-butanediol is a very flexible molecule because of the free rotation which is possible around the three carbon-carbon bonds and the two carbon-oxygen bonds. As it collides with its fellow molecules, it can adopt a very large number of different conformations. Figure 13.5 illustrates some of these conformations, selected at random by a computer. Again we can take these as representing a series of photographs of the same molecule taken at regular intervals, say every 5 min. For the sake of clarity, though, we have kept the position of two of the carbon atoms the same for each photograph. In actual fact the molecule would be rotating as well as flexing, and the orientation of the bond joining these two carbon atoms would be continually changing. We should also remember that the molecule has undergone very many collisions between successive photographs.
The important thing to notice about this random selection of nine conformations shown in Fig. 2 is that in none of them are the two oxygen atoms close enough to allow the formation of a hydrogen bond. At this high temperature, because of the large number of ways in which a butanediol molecule can arrange itself in space, it will only occasionally adopt a ring conformation like that shown in Fig. 3 in which the two oxygen atoms are close enough to hydrogen bond to each other. In the equilibrium mixture at high temperature, therefore, there will be many more chain molecules than ring molecules and Kc will be much larger than 1.
This situation begins to change as the temperature is reduced. With decreasing temperature, a smaller and smaller fraction of the molecules will have enough energy to break open a hydrogen bond. In consequence the ring form will become more probable. Once having formed, a ring will be able to survive quite a few collisions with other molecules before it is broken open again. Indeed, if we could reduce the temperature of our system close to absolute zero without the solvent freezingThe process of forming a solid from a liquid., then we would find that virtually all the butanediol molecules were in the ring form. Once formed, a ring would almost never experience a collision which was sufficiently energetic to break it open. At such a low temperature the value of Kc would be very close to zero.
The two equilibria we have just described have both been very simple examples, especially since they each involved only one reactant and one product. Nevertheless the principles we have discovered in discussing them apply to all chemical equilibria. We can always interpret the equilibrium constant of a chemical reaction as the product of two factors, each of which has a meaning on the microscopic level:
K = energy factor × probability factor (2)
The energy factor takes account of the fact that either the products or the reactants are higher in energy and hence less probable in the constant reshuffling of energy that goes on between molecules. The probability factor takes account of the fact that even if there were no energy difference between products and reactants, one or the other would still be more probable. This greater probability arises because of the larger number of ways in which the products (or reactants) can be realized on the molecular level.
In general the energy factor has the greatest effect on the value of the equilibrium constant at low temperatures. The lower the temperature, the very much lower the probability of a high-energy species occurring by chance in the constant energy reshuffle. At higher temperatures the effect of the energy factor becomes less pronounced and the probability factor becomes more important. If the temperature can be raised high enough, the probability factor eventually predominates.