# Relationship between free energy and chemical equilibrium expression

### Free Energy and the Equilibrium Constant - Chemistry LibreTexts

Learn the concepts of Relationship Between Free Energy And Equilibrium Constant with the help of study material for IIT JEE by askIITians. At constant temperature and pressure, the change in Gibbs free energy is defined as Δ G In chemistry, a spontaneous processes is one that occurs without the .. in the forward direction, backward direction, or if the reaction is at equilibrium. In this lesson, we learn the important connection between free energy and the equilibrium constant. We will begin by considering systems.

Although these relations are strictly correct only for perfect gases, we will see later that equations of similar form can be applied to many liquid solutions by substituting concentrations for pressures. The free energy change for the reaction is sum of the free energies of the products, minus that of the reactants: The free energy G is a quantity that becomes more negative during the course of any natural process.

Thus as a chemical reaction takes place, G only falls and will never become more positive. Eventually a point is reached where any further transformation of reactants into products would cause G to increase.

At this point G is at a minimum see the plot belowand no further net change can take place; the reaction is at equilibrium. The free energy of 1 mole of N2O4 1 is smaller than that of 2 moles of NO2 2 by 5.

## Standard change in free energy and the equilibrium constant

The straight diagonal line shows the free energy of all possible compositions if the two gases were prevented from mixing. The red curved line show the free energy of the actual reaction mixture. This passes through a minimum at 3 where 0. The difference 4 corresponds to the free energy of mixing of reactants and products which always results in an equilibrium mixture whose free energy is lower than that of either pure reactants or pure products.

It corresponds to the free energy change for a process that never really happens: Thus for the limiting cases of pure N2O4 or NO2 as far from the equilibrium state as the system can be!

Isomerization of butane The standard molar free energy change for this very simple reaction is —2. Notice particularly that The sum of the free energies of the two gases n-butane and iso-butane separately varies linearly with the composition of the mixture red line. The green curve adds the free energy of mixing to the above sum; its minimum defines the equilibrium composition.

The detailed calculations that lead to the values shown above can be found here. So, let's write down our equation that relates delta-G zero to K. Delta-G zero is negative This is equal to the negative, the gas constant is 8. So, we need to write over here, joules over moles of reaction. So, for this balanced equation, for this reaction, delta-G zero is equal to negative So, we say kilojoules, or joules, over moles of reaction just to make our units work out, here.

Temperature is in Kelvin, so we have K, so, we write K in here, Kelvin would cancel out, and then we have the natural log of K, our equilibrium constant, which is what we are trying to find.

So, let's get out the calculator and we'll start with the value for delta-G zero which is negative So, we're going to divide that by negative 8. And so, we get So, now we have So, how do we solve for K here? Well, we would take E to both sides. So, if we take E to the So, let's take E to the And since we're dealing with gases, if you wanted to put in a KP here, you could.

So, now we have an equilibrium constant, K, which is much greater than one.

And we got this value from a negative value for delta-G zero. So, let's go back up to here, and we see that delta-G zero, right, is negative. So, when delta-G zero is less than zero, so when delta-G zero is negative, what do we get for our equilibrium constant? We get that our equilibrium constant, K, is much greater than one. So, what does this tell us about our equilibrium mixture? This tells us that at equilibrium, the products are favored over the reactants, so the equilibrium mixture contains more products than reactants.

### Relationship Between Free Energy And Equilibrium Constant - Study Material for IIT JEE | askIITians

And we figured that out by using our value for delta-G zero. Let's do the same problem again, but let's say our reaction is at a different temperature. So now, our reaction is at Kelvin, so we're still trying to make ammonia here, and our goals is to find the equilibrium constant at this temperature.

**Free Energy and the Equilibrium Constant**

At Kelvin, the standard change in free energy, delta-G zero, is equal to zero. So, we write down our equation, delta-G zero is equal to negative RT, a natural log of the equilibrium constant, K. And this time, for delta-G zero, we're plugging in zero. So, zero is equal to, we know that R is the gas constant, and we know that the temperature here would be Kelvin. So, for everything on the right to be equal to zero, the natural log of K must be equal to zero.

## 19.7: Free Energy and the Equilibrium Constant

So, we have zero is equal to the natural log of K. And now, we're solving for K, we're finding the equilibrium constant. So, we take E to both sides. So, E to the zero is equal to E to the natural log of K. E to the natural log of K is just equal to K. So, K is equal to E to the zero, and E to the zero is equal to one. So, when delta-G zero is equal to zero, so let's write this down on here, so, when your standard change in free energy, delta-G zero, is equal to zero, K is equal to one.

And that means that at equilibrium, your products and your reactants are equally favored.

Let's do one more example. So, let's find the equilibrium constant again at another temperature.