Equilibria (by a synthetic chemist)

Introduction

Many students view reactions in absolutes. The reaction either proceeds or it doesn't. This is especially true of those that focus on organic synthesis. There is a tendency to forget that all reactions are reversible, going forward to give products and backwards as the products return to starting materials. Reactions are a chemical equilibrium and it is just the position of this equilibrium that makes us think they have either failed or gone to completion. Even old synthetic chemists like myself, rarely consider these process unless something goes wrong with a reaction (although I would like to think that we intuitively include it in our reasoning) … or we have to teach it (which, thankfully, hasn’t happened in years!).

An understanding of chemical equilibria is vital to all synthetic chemists. The most obvious use is in understanding acids and bases. Acids/base chemistry is wonderful. Think about all the reactions that involve protonation and/or deprotonation. But even more important is the concepts covered in acid/base chemistry (predicting stability of various species so that you can predict relative strengths), which are vital for understanding a host of other reactions (think leaving groups etc.). Organic chemists really should spend a bit more time on this subject rather than leaving it in either general chemistry or physical chemistry courses.

The following summary is written from the perspective of a synthetic chemist who hasn’t studied this subject in decades. I will ignore discussion of activities and will undoubtedly become sloppy with my use of ° or ⦵ to denote standard conditions. This will make many chemists unhappy. I (don't) apologise for this, my goal is to get the general concepts across and my experience suggests students aren't as worried (or confused) by these inconsistencies as many of my colleagues would suggest.

An Introduction to Chemical Equilibria

At the start of a reaction, it is not in equilibrium, and the rate of conversion of one set of reactants into the other is greater than the reverse process. The reaction appears to be going in one direction.

Initial reaction appears to go in one direction — The initial reaction proceeds in a single direction (as indicated by the single black arrows) as there are no molecules of the product to react in reverse direction.

When the reaction reaches equilibrium, the rate of the forward reaction is identical to the rate of the reverse reaction. The reaction appears to have stopped. It hasn’t, both processes are still occurring, starting materials are becoming products and products are becoming starting materials, it is just the quantities of the reactants and products remains constant.

In a reaction at equilibrium the rate of forward and reverse reactions is the same — When at equilibrium, the rate of the forward reaction is the same as the rate of the reverse reaction (same number of single arrows going in both directions).

The rate of the forward and reverse reactions are now the same but the concentration, or amounts, of each reactant don’t have to be the same. Which side of the equilibrium is favored, whether the reaction appears to have gone forward (left-to-right) or backward (right-to-left) is determined by the position of the equilibrium.

At equilibrium the amonuts of reactants and products don't have to be the same, only the rate of formation — In a reaction at equilibrium, the rate of forward reaction and backward reaction are the same ***but*** the amounts of each reagent do not need to be the same. Here, there is more product than starting materials.

The Equilibrium Constant

A reaction can be generalized as:

where a = number of moles of A in the balanced equation (the stoichiometry or reaction coefficient); A is the substance; and the double arrow represents a reaction that can go in both directions (but should never be mistaken for the double-headed resonance arrow, which represents two snapshots of the same molecule and not a reaction).

The position of the equilibrium, effectively the amount of each substance, is given by the equilibrium constant K_c:

[A] = concentration of A in M (or mol/L).

This is a specific version of the equilibrium constant and is called the equilibrium constant of concentration (the subscript c means concentration). This is a simplification of the true equilibrium constant and is based on the concentration of the substances, not the activity. It holds true for dilute solutions. When using K_c, pure solids and liquids are ignored (not included in the equation). The normal explanation for this states that their concentration doesn't change as they are pure but this is assumption is incorrect. They are ignored as we define their activities as 1 and so they have no influence (hopefully that is vaguely correct and keeps my physical chemistry colleagues placated). As a result of this 'simplification', it is necessary to always include the state symbols of each reactant into the reaction equation (liquid = (l); solid = (s); gas = (g); aqueous solution = (aq)). Without these, it is impossible to determine which reactants must be included in the equilibrium constant.

The Position of the Equilibrium

If K_c > 1 then there is more product than starting material. The reaction is said to favour the right-hand side. If the value of K_c is 10² there will be 1% of the starting material remaining and the reaction can be said to have gone to completion. Higher values and there is even less starting material.

Reaction mixture when the equilibrium constant is greater than 1 — If Kc > 1, the reaction favors the products or right-hand side of the equation (there will be more product than starting materials).

Often you will see one arrow longer and bigger than the other. This indicates which side of the reaction is favored. But this type of diagram often leads to a misconception with students (and staff) viewing the two sides of the reaction as separate, and the arrow as some kind of barrier. The reality is that the reaction vessel is a mixture of components. It might be better represented by the diagram below where we show the original reagents before reaction, and then the equilibrium mixture. Our reaction equation represents the right-hand side of this diagram:

It is important to remember that an equilibrium reaction contains a mixture of both sides of the reaction equation, the amount of which is dependent on the equilibrium constant. If the equilibrium constant is > 10^2 then over 99% of the mixture is product.

When K_c = 1 there are equal amounts of the starting materials and the products. Again, it is important to understand that the substances are mixed together and not equal, but separate, quantities. Any reaction with a K_c between 10^–2 and 10² shows appreciable quantities of both reactant and product. This is probably what most synthetic chemists imagine when they think of an equilibrium process. For example, the K_c for esterification at room temperature (298 K) is approximately 4.

If Kc is close to 1 there will be equal amounts of the reactants and the products, neither side of the reaction is favored to any great extent. This is true of any reaction with a Kc between 10^-2 and 10^2.

Finally, if K_c < 1, the reaction favors the starting materials or proceeds to the left-hand side. Sometimes this is referred to as the reaction going backwards. If K_c < 10^–2 the reaction appears not to proceed and there will be less than 1% product formed. This is often represented by a larger left pointing arrow and a small one going forwards.

Reaction with small value of Kc — If Kc < 1 then the left-hand side, or starting materials is favored. Kc = 10^2 and only 1% products. The reaction does not appear to proceed.

What controls the reaction?

A reaction will proceed in the forwards direction, from starting materials to products, if it releases energy. Synthetic chemists often simplify this to the diagram below. This shows that the free energy G of the product is less than the free energy of the products. Or the Gibbs free-energy change of the reaction, 𝚫_rG°, is negative.

Products favored if there is a negative free energy change — Reaction favors formation of the products as it releases energy (𝚫G is negative).

But why do reactions give a mixture of starting materials and products when the products are energetically favored? If the products have less energy (or are more stable) shouldn’t the reaction simply go to completion with 100% conversion? The problem is in the definition of Gibbs free-energy. The free energy of the reaction changes as the concentration of each reactant changes. As the reaction proceeds so the free energy decreases until it reaches a minimum value. It takes energy to create more products or form more starting materials. An input of energy is disfavored. At this value 𝚫G = 0 and the reaction is at equilibrium. This is shown below:

Change in free-energy as concenttration of reactants alters — As reaction proceeds (in either direction) so the Gibbs free-energy falls until it reaches a minimum. At this point the reaction requires energy to go in either direction. It is now at equilibrium.

As both the equilibrium constant K_c and the change in Gibbs free-energy, 𝚫G, are related to the position of the equilibrium, or how far the reaction has progressed, it shouldn't be a surprise that there is a useful mathematical connection:

Relationship between equilibrium constant and Gibbs free-energy — The connection between the Gibbs free-energy change and the equilibrium constant.

In this, 𝚫G° means the change in standard Gibbs free-energy under standard conditions, which is normally a pure substance in its most stable state at one atmosphere pressure and a defined temperature. R is the gas constant (8.314 J K^–1 mol^–1 and T is the temperature in Kelvin.

From this equation, it is clear that if:

𝚫G° is negative, the products (the right-hand side of the equation) will be favoured (K_c > 1). The reaction goes forward or left to right.
𝚫G° is positive, the starting materials (the left-hand side of the equation) will be favoured (K_c < 1). The reaction goes backwards or in reverse.
𝚫G° is 0, the there will be a 1:1 mixture of starting materials and products (K_c = 1).

Gibbs free energy and the position of the equilibrium.

Being able to relate a change in energy to the ratio of starting materials:products or calculate enegry change by knowing the concentrations is useful. Most discussions concentrate on reactions and whether a reaction goes forward or backwards but I have always found it helpful when looking at the conformation of molecules. For example, if you know the concentration of both the axial and equatorial conformations of a cyclohexane derivative, you can determine the energy difference between them. Consider methylcyclohexane exists, it exists as a 95:5 mixture of the equatorial conformation to the axial conformation at room temperature. This means the equilibrium constant K_c is 19 in favour of equatorial conformation.

Determining the difference in energy of axial and equatorial methylcyclohexane from the ratio of concentrations — Determining the strain energy of axial methylcyclohexane based on the population of the two conformations.

Using the equation it is possible to determine that the equatorial conformation is 7.3 kJ mol^–1 more stable than the axial conformation. This is approximately the value given in the literature.

Use the difference in energy to determine ratio of each conformation — Determining the population of two conformations from the difference in strain energy.

Using the relationship between 𝚫G and K_c gives you a value for K_c of 3905. This is the same as a ratio of 3905:1, which can be converted to a percentage [(3905/3906) x 100 = 99.97%]. From this we can say that a tert-butyl substituent effectively blocks ring-flipping at room temperature, with approximately 0.03% of the molecule residing in this conformation at any one time.

What Determines the Gibbs Free-Energy Change?

If we return our attention to an equilibrium reaction, the earlier plot of composition of the reaction mixture versus change in Gibbs free-energy reveals that the Gibbs free-energy is not solely determined by the energy or stability of the reactants and products. If all the mattered was the strength of the bonds within the molecules, then each reaction would either proceed to completion or fail to give any product. Chemistry would be simple (but life impossible), and I wouldn’t be struggling to write this summary.

The Gibbs free-energy change 𝚫G° comprises of two terms. The first is the enthalpy change of reaction (𝚫H°) and the second is the entropy change of reaction (𝚫S°) as described:

Connecting Gibbs free-energy with enthalpy and entropy

The enthalpy change or heat of reaction is related to the stability of the starting materials and products. It is the difference or change in strength of the bonds made in the reaction and those that are broken. If 𝚫H is negative the reaction releases energy and is called exothermic. In an exothermic reaction the products (right-hand side of the reaction) have less energy and are more stable than the starting materials. If 𝚫H is positive, the reaction absorbs heat and is endothermic. The products are less stable with weaker bonds than the starting materials.

An exothermic process — An exothermic reaction releases heat (energy) and the products are more stable than the starting materials. The bonds of the product are stronger.

An endothermic process — An endothermic reaction takes in heat (energy). The products are less stable than the starting materials and they will have weaker bonds.

Definitions of entropy and the entropy change, 𝚫S, are a little more vague, and often lead to heated discussion (especially on the intertubes). The most common definition is the change in randomness or disorder. 𝚫S is positive if the reaction becomes more random or disordered, and is negative if randomness/disorder decreases. For example, disorder increases if a reaction results in a molecule splitting in two:

A simple definition of entropy — A simple view of entropy. In the top reaction, the number of molecules increases so the disorder increases and 𝚫S is positive. The opposite is shown at the bottom. In an addition reaction the number of molecules decreases and 𝚫S is negative.

Alternatively, in an addition reaction, where two molecules combine to give a single molecule, the disorder is said to decrease.

Entropy can also be described in terms of freedom of motion or a measure of energy dispersal/distribution, and any change in a reaction that leads to higher temperatures, more molecules, a larger degree of freedom of movement or an increase in volume will lead to an increase in entropy, and 𝚫S will be positive.

As previously mentioned, for a reaction to proceed, by which I mean the products or right-hand side are favoured over the starting materials at equilibrium, 𝚫_rG, must be negative. There are two ways this can be achieved:

If the reaction is exothermic then 𝚫H will be negative, and, as long as 𝚫S is not large and negative, 𝚫G will also be negative.
If the reaction is endothermic then 𝚫H will be positive. 𝚫G will only be negative if –T𝚫S is larger than 𝚫H. This requires a large positive entropy change (𝚫S) and/or a high reaction temperature (T).

Favoured reactions — There are two ways to achieve a negative 𝚫G and thus a favourable reaction.

I know I promised not to get bogged down with standard reaction conditions and 𝚫G vs. 𝚫G° but I also know how vicious the intertubes gets when someone reveals weakness (by which I mean, makes a honest mistake), so I should mention position of the equilibrium and spontaneity of a reaction. 𝚫G°can be used to determine the position of the equilibrium and hence whether the reaction will progress to the right or left. It should not be used to determine whether a reaction is spontaneous (will proceed or not). For that, use 𝚫G. The reason for this distinction is that it is very rare for a reaction to occur under standard conditions (there is more to this argument, but the person who explained it to me, one of the few physical chemists I have met that would correct my ignorance/sloppiness without making me feel like an idiot, has left for a rival university and I can’t remember everything she told me … I may have misremember much of it as well. All mistakes are definitely mine).

A Word of Warning

This entire summary is about equilibria. 𝚫G and K_c tell you about the position of the equilibrium, which side of the reaction is favoured. They tell you nothing about the rate of the reaction or how fast the reaction reaches equilibrium. The products of a particular reaction can be favoured to the exclsuion of all starting material so that a chemist would say that the reaction goes to completion, but it might take seconds or thousands of years for the reaction to reach that point. The classic example of this is petrol. Oxidation of petrol (burning it in air) to give carbon dioxide and water is a favorable process and the equilibrium constant is massive. The reaction favors the right-hand side completely. But, luckily for us, the reaction doesn't occur without a push otherwise we could never fill our petrol tanks at the garage (let's avoid arguments about whether we should be doing this!).

Conclusion

All reactions are at equilibrium, meaning the rate of the forward and backward reactions are the same. This is not always apparent, as the equilibrium may favor either the reactants or the products to such an extent that the reaction appears to proceed in a single direction.

The position of the equilibrium constant is related to the difference in Gibbs free-energy between the starting materials and the products. If the energy of the products is less that the starting materials, or 𝚫G is negative, the reaction favors formation of the products or the equilibrium is said to favor the right-hand side.

The Gibbs free-energy change of a reaction is related to the enthalpy change of reaction (𝚫H) and the entropy change of reaction (𝚫S). Enthalpy represents the difference in stability of the starting materials and products. It is related to the strength of the bonds made and broken during the reaction. A reaction is exothermic if the bonds of the product are stronger than those of the starting material and endothermic if they are weaker. Entropy can be considered a measure of disorder or randomness of the reaction. 𝚫S is negative if the reaction leads to less disorder or positive if it leads to more disorder.

For 𝚫G to be negative, and the forward reaction favored, a reaction should either be exothermic (𝚫H is negative) and 𝚫S is not large or negative or, if it is endothermic (𝚫H is positive) then there must be a large positive 𝚫S and/or high reaction temperature.

Summary of a Summary

The equilibrium constant Kc of a reaction.

Position of equilibrium — The position of the equilibrium. The relationship between Gibbs free-energy change and the equilibrium constant.

Gibbs free-energy and enthalpy and entropy — The components of the Gibbs free-energy change of a reaction, enthalpy (stability or bond strength) and entropy (disorder/randomness as a simplification).

In time, some practice questions will appear HERE … honest.