Let's return to the two conditions we've considered in which the compartments are at the same height (the beans on the left and right have the same energy).  It is important to realize that this picture of jumping beans is not very realistic when applied to molecules.  The vast majority of molecules would appear to be quivering in this picture, and a very small percentage of the molecules would have enough energy to be represented by a jump.  However, in even the tiniest bit of matter there are an enormous number of molecules, so a small percentage might still represent a substantial amount.  On average, the beans are actually a little bit higher than the bottom of the box.

Both of these sets of beans have reached a state where the average number of beans (averaged over a period of time) in each compartment is staying constant - an equilibrium condition.  This situation arises because the number of jumps from left to right over a period of time is exactly matched by the number of jumps from right to left, so that there is no net change.

Actually, if we consider the bottom of the compartments to be divided into spaces which can be occupied by a single bean, we can see that each of these spaces has the same chance of being occupied whether it is on the right or left side of the barrier. Of course, there is a narrow strip (about the width of a bean) along the barrier which the center of a bean cannot enter.  When the compartments are at the same height, the ratio of the numbers of beans on the left and right sides is determined by the ratio of the areas of the left and right sides.  This effect of chance or probability affects the thermodynamic equilibrium as an entropic effect, which is related to the ratio of the number of beans to the area of the compartment.

Here we have two different equilibrium conditions which come about when the jumps across the barrier from left-to-right and jumps from right-to-left are occurring at the same rate.  All of the crossovers above are occurring at approximately the same rate, even though the average numbers of beans in the compartments are different.  This occurs because these average numbers of beans in each compartment are related to the area in which they are contained - the average group of four beans is in the largest area and the average group of two beans is in the smallest area.  The number of crossover jumps in a given period of time are determined by the ratio of the number of beans to the area of the compartment they are jumping from.  This ratio is similar to concentration, though we normally think of concentration as the number of moles or molecules divided by the volume in which they are contained.

Now we can see that these two equilibrium conditions are different only because of the location of the barrier, which affects the sizes of the compartments.  If we think of an equilibrium constant (K) as the ratio of the concentration of beans on the right to the concentration of beans on the left:

we find K = 1  for both of the conditions above.  This value of K is not unexpected, since there is really no difference in the beans on the left or the right.  If there was a difference in energy, or if the beans could change their size or shape as they cross the barrier (maybe two of them stick together when they are on the right side), we might expect the value of K to be something other than unity.

This example, in which there is no energy difference, illustrates a fundamental part of the entropic effect which is related to concentration and is called the cratic part.  It is connected to the way we write the equilibrium constant - the right-hand side of the equation above.  There can be another contribution to the entropy which is related to the probabilities of the beans being in different energy states.  This part of the entropy effect (the intrinsic part) and the difference in energy contribute to the value of the equilibrium constant - the left-hand side of the equation above.

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Looking again at the situation in which there is a difference in height between the two parts of the box (an energy difference), the equilibrium constant is written in the same way, but now it is easier for a bean to jump from left-to-right than from right-to-left.  This results in a greater concentration on the right side when the jumps from left-to-right are equal to the jumps from right-to-left (equilibrium),

and now the value of K is greater, K~ 2 .  The difference in energy affects the value of the equilibrium constant.  In general, the lower the energy of the products (right compartment) relative to the reactants (left compartment), the greater is the value of K.  However, in Chemical Reactions, there are a few rare situations in which the contributions from the intrinsic entropy term to the value of K may be greater than the contribution of the energy term, and the value of K may be smaller than unity, even though the products have lower energy than the reactants.

It is important to bring in the effect of temperature before this discussion deteriorates into a lot of mathematics and thermodynamic quantities.  Temperature effects will also be discussed in the following section on Kinetics.  The basic idea is that increasing the temperature makes the beans more excited and they jump around with more energy, increasing the probability of jumps that can clear the barrier.  The height of the barrier then becomes less important in controlling how fast the molecules are transferred from one side to the other.  The difference in height between the two boxes also becomes less important in determining the value of the equilibrium constant.

The difference in energy between products and reactants is called the standard enthalpy of reaction,  (enthalpy is a slightly modified definition of energy).  In this example,  represents the difference in energy of a single bean on the right-hand side and one on the left.

The difference in the intrinsic entropy between products and reactants is called the standard entropy of reaction, .  The standard entropy of reaction arises from the different ways in which the collections of molecules (beans) can exhibit their energy.  This energy is in the motions of the atoms within the molecules, and in the movements of the molecules within the container.  In general, if two reactant molecules (beans) combine to form a single product molecule (bean), the standard entropy of reaction is negative, because beans that are stuck together cannot be distributed in as many ways as the individual beans.  The maximum probability for beans that are stuck together is less than the maximum probability for individual beans, so the individual beans have more intrinsic entropy.  However, collections of molecules with higher energy usually have more ways of exhibiting that energy, and this also contributes to the intrinsic entropy.

The standard enthalpy of reaction and the standard entropy of reaction have opposite effects on the value of the equilibrium constant.  A negative value for  leads to a larger value of K, while a negative value of  leads to a smaller value of K.

These effects of the standard enthalpy of reaction and the standard entropy of reaction are combined in the standard (Gibbs) free energy of reaction, 

.

This may be re-written as:

This is a rather complex relationship, but  is seen to contribute negatively to the value of K and  contributes positively, as stated above.  It is important to note that the absolute temperature (T) appears in the denominator of the second exponential term.  The way that K changes with temperature is controlled by the value of .  If  is positive (endothermic reaction), the value of K will increase as the temperature increases.  The opposite effect is observed if the  is negative (exothermic reaction).

The standard entropy has contributions from intrinsic effects, such as the nature of the bonding between the atoms which compose the molecule, but there are also contributions from the distribution of energy states (that quivering of the beans mentioned at the top of the page) that separate the standard enthalpy from the ground-state energy. Standard entropies are also referred to a specific concentration.  For gases and for pure liquids and solids, this standard concentration is simply one mole of the material in the volume that it occupies at 1 bar pressure (or 1 atmosphere in older texts) at the specified temperature.  For dissolved substances, the standard concentration is one mole of solute in a liter of solution (concentrations in Molarity, M), or one mole of the solute in the volume occupied by the solute and one kilogram of the solvent at the specified temperature and 1 bar pressure (concentrations in Molality, m).

The standard free energy is a combination of the standard enthalpy and the standard entropy, and thus contains contributions from the energy states and the intrinsic entropy.  It is also referred to the specific concentration defined for the standard entropy.

 

When we write a reaction such as:

(of course we would be using chemical formulas instead of beans)

and talk about the standard enthalpy of reaction, standard entropy of reaction, and standard free energy of reaction, we are NOT referring to the equilibrium condition.  We are talking about the standard state reaction, which might be depicted as:

It is clear that the blue bean has less enthalpy than the sum of the two red beans, so this is an exothermic reaction.  It should also be clear that the combined freedom of the two red beans gives them more entropy.  The combination of these two effects in the Gibbs free energy of reaction gives an equilibrium constant

which is larger than it would be if the energies were the same, and smaller than it would be if the entropies were the same.

 

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