Question

Derive and explain the relationship between the Nernst equation, the equilibrium constant and other thermodynamic quantities such as enthalpy, Gibbs energy and entropy of reaction.

Answer 1

The K equilibrium is an important factor to calculate gibbs energy, nernst equation and also we can study the relationship of K eq with Enthalpy and Entropy

For a reaction we have

and equilibrium is defined as:

a, b, c and d are the stoichiometrical coefficients.

Now we know that it is related to free gibs energy

Gibbs energy of reaction in equilibrium is defined as:

R is gas constant

T is temperature

Keq is the equilibrium constant for a reaction

For spontaneity we have:

is negative for a spontaneus reaction

At this point we can se that for a spontaneus reactions if it is shifted towards the products the value of G will have a higher (negative) value.

Now let´s relate it to the cell potential,

The Cell potential E is the trend of chemical species in a redox reaction to aquire electrons, so when a reaction happens a flow of eletrons occur, it has to be related with gibbs for reversible reaction:

is positive reaction is spontaneus

E is the cell potencial for a rection

There is a matematical relation between E and

where n is the number of electrons for semirreactions

F is faraday number

If we relate this last 2 equations with the definition of Gibbs free energy we get:

are the values of gibbs and E at standard conditions (1 atm and 25C)

It has a negative sign because of the relation of spontaneity of Gibbs free energy and cell potential I explained.

If we divide this equiation by nF we get

Now let´s multiply by -1

This is the nernst equation.

In this case for a redox reaction, at the beggining of the reaction it has the higher value of cell potential, as the reaction proceeds the E value will be lower and lower until it gets to zero.

In the last equation we can see that as the reaction proceeds (products are formed) the Keq will grow also and the E will have a lower value until it reaches zero (reaction is complete), it cannot have a negative value

Now remember again

If we ignore

Now the formal definition of gibss is:

For a spontaneus process in order to be negative the Gibbs energy

making equal those 2 equations we get:

If we divide it by RT we get:

A differention equation with respect to the variable T leads you to the Vant Hoff equation.

But let´s explain what we have here:

The K of equilibrium can be described as a linear function of with as the slpe and the as the intercept.

So for a reversible reaction we now that for diferent temperatures the Equilibrium constant will change in relation with the Enthalpy of reaction and Entropy

For endothermic process

for an exothermic process:

You can apply la chetelier principle too.

For an exothermic reaction if you remove heat (decrease temperature) the equilibrium will naturally be displaced to the products(Keq will be higher) if you increase heat the equilibrium will displace toward the reactants.

The opposite is true for an endothermic reaction

These can be seen clearly with the equations that relates equilibrium with enthalpy and entropy