Chemical Equilibrium Constant computed

It shows the main ideas for calculating chemical equilibrium constants from spectroscopic and other molecular data. Let's assume an isomerization reaction A Û B. A and B shall have the same properties except for a different formation enthalpy at T = 0 K, and different frequency of one molecular vibration. For A the formation enthalpy is taken as 0 (= reference), for B +400 cm^-1 (= 4.77 kJ/mol, DE°). The vibrational frequency in question is 475 and 150 cm^-1, respectively. We start at a temperature of 173 K in steps of 100 K and look on the screen shot at the last temperature of 1773 K with a summary for the whole temperature range.

The first frame shows the equilibrium partition of the species A and B, respectively, on their proper ladder of energy levels. This is determined uniquely by the properties of each molecular species and the total energy E of each ensemble. The length of the yellow and cyan bars is proportional to the relative number of B or A molecules sitting on a level in equilibrium at 1773 K.

Now we put both species into the same container in frame two. They can freely exchange quanta while colliding, thus jumping around on their respective ladder. If they are at the same temperature, defining the same average energy per particle, as before, the same distribution obtains as for each species alone.

In the third frame we add a catalyst to make the isomerization reaction go. Now in addition to quanta also the identity of the species may be exchanged in collisions: A particle on the A-ladder is an A, one on the B-ladder is a B molecule. The chemical reaction makes them jump from one level system to the other and back billions of times. This process induces two merged ladders at the exact distance of the difference in formation enthalpy DE°. This produces gains and losses from 50:50 starting populations - depicted by different colors - such that a smooth equilibrium distribution over the merged ladders is formed. For every temperature a different, but time independent population ratio - an equilibrium constant - is obtained as shown by the red and blue bars in the inset for every temperature, and the numerical value of Kp = [B]/[A] given below the graph. The bars correspond to the populations on the A- and B-levels.

In the last frame the usual plot of ln(Kp) versus 1000/T is shown with the calculated Kp values. As you might recall - equation of van't Hoff -, the slope of the straight line allows to compute the reaction enthalpy.

You can change interactively all parameters. Here, we see that the reaction is enthalpy driven at low temperatures (the A ladder is bottommost, A is more stable), at high temperature it is entropy driven, since the finer spaced B ladder offers many more possibilities to arrange particles, given a certain average energy defined by the temperature. Therefore, the A species dominates at low, the B isomer at high temperature. That's also a demonstration of the principle of Le Châtelier: At higher temperature the equilibrium shifts towards the product, that consumes heat. By changing the formation enthalpy (e.g. inverting its sign) and the vibrational frequencies you can simulate any mix of enthalpy or entropy driven reactions. You will also find out, that the straight line in the 4th frame does not stay 'straight' with all combinations of the parameters! That is no surprise, however, because neither the van't Hoff equation produces a straight line if the reaction enthalpy is changing in the temperature range under observation.