^{*Warning: Long discussion ahead*}

Gibbs free energy is defined as

G = U + PV – TS

However, it is more useful to consider the following form of the equation for a system at constant pressure and temperature:

ΔG = ΔU + PΔV – TΔS

where

• ΔG is the change in Gibbs free energy,
• PΔV is the work done by the system, and
• T is the temperature of the system in Kelvin, and
• ΔS is the change in entropy of the system.

Even though this section is labelled under the Second Law, we will go a little bit into the First Law here. Recall the equation of the First Law:

ΔU = q + W

where

• ΔU is the total change in internal energy of the closed system,
• q is the heat added to the system, and
• W is the work done on the system.

Work done on the system is the negative of work done by the system, so we can rewite this equation as

ΔU = q – W_by

or more cleanly,

q = ΔU + W_by

Now, let’s consider the volume aspect of a system. When a system expands, it is usually a gas within the system doing the work. We can represent this work done by

W_by = PΔV

so we get

q = ΔU + PΔV

Recall that the enthalpy change for an exothermic reaction is negative, while for an endothermic reaction, it is positive. In other words, this is just the energy supplied to the reaction system. So,

ΔH = ΔU + PΔV

Giving us the familiar equation

ΔG = ΔH – TΔS

The Same Equation, A Different Derivation

Though the equation above is certainly valid, some of you may be uneasy about the way we arrived at it. So here is another way to show the validity of the statement.

Consider a system known as the “chemical reaction” and another system known as the “surroundings”. The energy loss by the surroundings is the negative of the energy loss by the system (due to the isolated nature of the universe). Let us also assume that the temperature of the universe doesn’t change. The change in entropy of the surroundings is defined as

ΔS_surroundings = -ΔH/T

where

• ΔH is the change in enthalpy for a given system, and
• T is the temperature of the universe in Kelvin.

Now, if we plug this into the equation

ΔS_universe = ΔS_system + ΔS_surroundings

we will get

ΔS_universe = ΔS_system – ΔH/T

Multiply this equation by -T to get

-TΔS_universe = -TΔS_system + ΔH

Now, consider the change in Gibbs free energy of the universe. The universe has no volume difference and no net internal energy changes (since it is an isolated system), so by the definition of a change in Gibbs free energy, we have

ΔG_universe = ΔU_universe + P_universeΔV_universe – TΔS_universe = 0 + 0 – TΔS_universe = -TΔS_universe

Therefore,

ΔG_universe = ΔH – TΔS_system

as desired.

Spontaneity of Reactions

The second derivation provided is somewhat clearer about the nature of ΔG, which we will consider in this section. At least read the last equation above before starting on this segment.

The spontaneity of a chemical reaction depends on the expected change in entropy of the universe. One consequence of the Second Law of Thermodynamics is that if entropy of the universe decreases, then work must be done by some external system to make this happen. Or conversely, any spontaneous process, requiring no external agent, must lead to an increase in entropy. The universe, on the other hand, is an isolated system. No work can be done on it. Therefore, if a chemical reaction is spontaneous, we must have

ΔS_universe > 0

and so,

ΔG_universe = -TΔS_universe < 0

i.e.

ΔG_universe = ΔH – TΔS_system < 0

Similarly, if a chemical reaction is not spontaneous, it must lead to a decrease in the entropy of the universe.

ΔS_universe < 0

and so,

ΔG_universe = -TΔS_universe > 0

i.e.

ΔG_universe = ΔH – TΔS_system > 0

The sharper readers will notice that the equality case has been omitted. This is simply because if a change to the system brings about no net change to the entropy of the universe, then the system is neither forward-feasible nor backward-feasible. In other words, the system is in stable equilibium.