E27 Is it possible to predict whether a reaction goes to completion or to an equilibrium mixture ?

Very many chemists ask the question: “When different species react with one another, why is the reaction so often incomplete?”

In the context of the familiar NO₂-dimerization reaction, why does the reaction end before all the NO₂ is consumed?

Up to now only equilibrium reactions have been considered. What characterizes a reaction which goes to completion ? If chlorinegas is mixed with an excess of hydrogengas, an explosive reaction occurs in which the chlorine is completely consumed:

This and many other reactions which go to completion are exothermic, making. negative. However, there are endothermic reactions (positive ) which also go to completion, even at room temperature e.g.

There must be a more general criterion, which determines whether or not a reaction goes to completion. Without a detailed derivation, the relationship which determines qualitatively and quantitatively the driving force for a particular reactant mixture is:

where H and S are the enthalpy and entropy increases associated with the reaction to completion of the reactants respectively in a closed reaction vessel at an absolute temperature T in kelvin.

G is known as the Gibbs free energy change (increase).
An entropy increase in a chemical reaction occurs when the new condition has a higher degree of disorder. This is the case for a reaction in which molecules are formed, which are less well ordered (e.g. as a result of phase changes) or in which the temperature has increased.

Both NO₂-dimerization and NH₃-synthesis result in an increase in order!

In general the initially posed rhetorical question can be answered as follows : “a chemical reaction in a closed reaction vessel proceeds in that direction in which the Gibbs free energy of the reaction mixture decreases”.
This is thus the direction in which heat is given up to the surroundings and/or the degree of disorder increases.

According to this criterion, four types of chemical reaction can be identified depending on the H/S combination:

In general, reactions of type 1 always go to completion, regardless of reaction temperature. For reactions of type 2 and 3 the position of the equilibrium is determined by the reaction temperature, the reaction in most cases not going to completion.

NO₂-dimerization and NH₃-synthesis are examples of exothermic equilibrium reactions. Endothermic equilibrium reactions of type 3 only attain a high degree of conversion at high temperatures.

Reactions of type 4 are always associated with a positive DG, regardless of reaction temperature. Such reactions are not necessarily impossible, but they are never spontaneous (self-sustaining). Energy must be continuously supplied to keep such reactions going, the electrolysis of water for example.

Reactions proceeding with a decrease in Gibbs free energy, on the other hand, proceed without externally supplied energy.

Little has been said about the equilibrium conditions in type 2 and type 3 reactions.

The equilibrium constant K at a given temperature is related to the molar Gibbs free energy at that temperature, by the expression:

Where R is 8.314 J/mol.K and T is the thermodynamic (absolute) temperature in kelvin.

The quantity (the molar Gibbs free energy change at normal temperature and pressure) can be calculated with the appropriate thermodynamic data for the reactants and the products.

The enthalpy and entropy of formation at normal temperature and pressure for N₂O₄ and NO₂ are summarized in the table below:

Where the subscript f stands for formation.

and K is 9.2.

The K-value obtained is a K_p-value, if the thermodynamic data concern a gas phase reaction.

K_c can be derived from K_p by converting the partial pressures of the gases into concentrations, as described below. The well known relationship: pV = nRT, where p is the pressure (in Pa), V is the volume (in m³), n is the amount of gas (in mol), R is the molar gas constant and T is the absolute temperature (in kelvin), relates pressure to concentration, as can be seen by rearranging the expression as follows:

If we fill in the values of the constants, then we can write down the following expression:

Where B = 0.08206 L.bar/mol.kelvin. (For the sake of simplicity we leave out the units here.) This last equation can be used to convert K_p into K_c . The following relationship is generally valid:

Where u is the difference in stoichiometric numbers between the reaction products and the reactants. In the case of the NO₂-dimerization:

For NO₂-dimerization this expression gives a K_c value of 225, which agrees very well with the experimental value of 222.

How does one predict, whether a particular reaction goes to completion or to an equilibrium mixture?

It is evident from the above that thermodynamic quantities such as and K do not indicate whether or not a reaction goes to completion, is fast or is slow etc...

They do give an idea of how the degree of conversion can be optimized or, after possible stimulation, can be made self-sustaining.

The relationship between the degree of conversion of a reaction and the K-value depends upon the initial composition. In addition, the expression for the equilibrium constant, K is also important.

Notwithstanding this, the table below can be used in a qualitative manner to relate and K to the likely type of reaction.

BEWARE

It is not possible to predict kinetic behaviour on the basis of such thermodynamic data. Thus at room temperature the NO₂-dimerization is not only self-sustaining, but the reaction starts spontaneously. On the other hand, N₂ and H₂ do not react at room temperature, requiring a high temperature even in the presence of a catalyst to start the reaction even though the -value for the ammonia synthesis is more negative.

Illustration 27 shows on the left how the Gibbs free energy G°, changes during the dimerization of NO₂.
Starting with 2 moles of NO₂ or 1 mole of N₂O₄ at room temperature and atmospheric pressure, G° decreases as NO₂ is converted into N₂O₄ and also as N₂O₄ is dissociated into NO₂ despite G° being higher for two mol of NO₂ than for a mole of N₂O₄. This occurs because the entropy increases due to the formation of a mixture, socalled “mixing entropy”.

This mixing entropy, represented by , is positive because in a mixture there is more disorder than for a pure species. For this mixing process is thus also negative. In this way, in this particular case, a minimum in DG° is attained with a mixture containing a relatively large amount of dimer. The equilibrium is thus shifted to the right i.e. in favour of dimer formation. In this equilibrium mixture the dimerization and dissociation reactions have stopped and a stable minimum in G° has been reached.

However, the dependence of G upon the quantities of the components can vary considerably upon changing the temperature or pressure from those of the standard conditions. Such a situation is shown on the right of illustration E27 for experiment D from illustration E06. At a pressure of 0.1 bar, the degree of conversion (0.5) is considerably lower.