The concept of free energy tells us*—*when does a process occur spontaneously? The second law of thermodynamics introduced to us the spontaneity of any process by using Clausius Inequality:

dS > đq/T_{surrounding}

The above inequality suggests that for an irreversible spontaneous process, the entropy always increases.

But to find out, whether a process is spontaneous or not, we have to estimate the entropy changes for both the system and surroundings. This is quite a tedious job. Moreover, it involves the estimation of some non-state variables as well.

This is even more difficult to predict in the case of chemical reactions. For example, look at the following reactions:

We have different conditions for each reaction (in terms of enthalpy and entropy). In our 3^{rd }reaction, both the changes for enthalpy and entropy are positive.

For the second reaction, both the changes for enthalpy and entropy are negative.

For the first reaction, the change in enthalpy is negative while the change in entropy is positive.

So how could we tell which of the above process is spontaneous under ambient conditions?

We must have a quantity that helps us determine the spontaneity of the above reactions in a systematic way. This is the situation where “Free energy” plays its role. We would arrive at the mathematical interpretation of the free energy in the upcoming text.

**Free Energy**

Let us combine the first and the second laws of thermodynamics.

According to the first law, we have:

dU = đq + đw

From the second law, we have:

dS > đq/T_{surrounding}

⇒ đq < T_{surr}dS & đw = -pdV

Combining both these laws written above, we get:

dU < T_{surr}dS – pdV

⇒ dU- T_{surr}dS + pdV <0 … (1)

The above inequality represents the condition for any spontaneous change. All the variables included in this inequality are state functions. It also contains the parameters like temperature and pressure, which we can control to obtain the required results.

Now, we would use this inequality to look at different conditions and understand what do they tell us?

**Helmholtz Free Energy**

Let us consider a process that has constant temperature and volume. We know that for a constant volume process, work done is zero, i.e. pdV = 0.

Therefore inequality (1) suggests:

dU – TdS <0

Since the temperature is constant, its differential would be zero. So we can write,

d(U-TS) <0

Here, we define a new quantity (A), which we call Helmholtz free energy.

A = U – TS

⇒ d(A) _{const. T & V} <0

This has very important implications. Since it tells us that for a spontaneous process, the Helmholtz free energy of the system decreases. Hence, it tells us about the lower limit of any spontaneous change that can occur.

It also tells the maximum amount of work that we can extract when a process occurs at a constant temperature and volume. Since ΔU is the sum of heat and work transactions with the surroundings. And the difference between U-TS is the amount of useful energy available for work, whereas “TS” is the amount of heat/energy transferred.

**Gibbs Free Energy**

Now we consider another condition in which temperature and pressure are held constant to see the implications of the inequality (1).

From (1),

dU + pdV -TdS <0

Since pressure and temperature are constant. Therefore, we can write:

d (U +PV-TS) <0

We know that enthalpy “H = U + PV “

⇒ d (H – TS) <0

This “H – TS = G” is called the Gibbs free energy.

The inequality corresponds to the following three conditions in particular:

If ΔG <0, The process would be spontaneous.

If ΔG = 0, The process has achieved equilibrium (where no further change occurs)

If ΔG >0, The process would not be spontaneous.

Now, we can calculate the Gibbs Free Energy of those three reactions we discussed earlier. Calculating the “G” for those reactions under the ambient conditions tells us, that all of them are spontaneous at room temperature.

An important thing to consider here is that G=0 doesn’t tell us whether a reaction is completed or not. We can’t say that we have zero concentration on the reactants side or we have achieved 100% product. It just tells us that we have reached a state of equilibrium where no further conversion is possible under the same conditions.

However, we can measure the concentration of different species and tell how much conversion took place by performing some lab tests.

Another important implication of Gibbs Free Energy is that the actual equilibrium depends upon both the enthalpy and entropy. We have seen in the reactions discussed previously that signs of these quantities can vary in a number of ways and even though the processes can be spontaneous.

This is because the energy may favour the reaction in one direction*—*say towards products which have low energy. Simultaneously the entropy might be favouring towards the direction which has more disorder.