Thermodynamics | Gibbs Free Energy and Third Law of Thermodynamics

Gibbs Free Energy (G)

• There are two types of energy in a system: (a) Entropy (Waste energy) and (b) Gibbs free energy (Useful energy).
• The energy in the system which can be converted to useful work is known as Gibbs Free energy.
• It is a state function. So, its value depends only on initial and final state of the system.
• It is an extensive property.

Mathematical Definition of Gibbs Free Energy

• Mathematically, we can define Gibbs Free Energy, G as:

$$G=H-TS$$

Here, H = Enthalpy of the system

T = Temperature and S = Entropy of the system

$$dG=dH-TdS-SdT$$

• If the process is isothermal:

$$dG=dH-TdS$$

Integrating both sides, we get:

$$\Delta G=\Delta H-T\Delta S$$

• This equation is also known as Gibbs Helmoltz Equation.

Spontaneity of a process

• Case 1: If ΔG < 0, process will be spontaneous.
• Case 2: If ΔG > 0, process will be non-spontaneous.
• Case 3: If ΔG = 0, system will be in equilibrium.

Standard Change in Gibbs Free Energy (ΔG^o)

Change in Gibbs Free Energy at 1 bar pressure and 298 K temperature is known as standard change in Gibbs Free Energy (ΔG^o).

At standard conditions:

$$\Delta G^o=\Delta H^o-T\Delta S^o$$

Here, ΔH^o = Change in enthalpy of the system at 1 bar pressure and 298 K

and, ΔS^o = Change in entropy of the system at 1 bar pressure and 298 K

Change in Gibbs Free Energy for isothermal reversible process

Mathematical definition of G is given by:

$$G=H-TS$$

$$dG=dH-TdS$$

$$dG=d(U+PV)-TdS$$

$$dG=dU+PdV+VdP-TdS$$

At constant temperature, dU = 0.

$$dG=PdV+VdP-TdS$$

For an isothermal reversible process, we can write using first law of thermodynamics:

$$dU=dq+dW$$

$$0=dq-PdV$$

$$PdV=dq=TdS$$

Putting this value of PdV in the expression of dG:

$$dG=TdS+VdP-TdS$$

$$dG=VdP$$

Integrating both sides, we get:

$$\underset{G_1}{\overset{G_2}{\int }}dG=\underset{P_1}{\overset{P_2}{\int }}VdP$$

$$(G_2-G_1)=\underset{P_1}{\overset{P_2}{\int }}\frac{nRT}{P}dP$$

$$\Delta G=nRTln\frac{P_2}{P_1}$$

Since, pressure is inversely proportional to volume, we can write:

$$\Delta G=nRTln\frac{V_1}{V_2}$$

Gibbs Free Energy and Reaction Quotient

• Mathematically, Gibbs Free Energy is related to reaction quotient by the following equation:

$$\Delta G=\Delta G^o+RTln{Q}_c$$

Here, ΔG = Change in Gibbs Free Energy

ΔG^o = Change in Standard Gibbs Free Energy

T = Temperature

R = Universal Gas Constant

Q_c = Reaction Quotient

• At equilibrium, ΔG = 0 and Q = K_c (equilibrium constant):

$$0=\Delta G^o+RTln{K}_c$$

$$\Delta G^o=-RTln{K}_c$$

Third Law of Thermodynamics

• The third law of thermodynamics states that the entropy of a solid approaches zero at absolute zero temperature (0 K).

• In other words, the entropy of a perfect crystalline substance becomes zero at absolute zero temperature (0 K).

$$S_0=0$$

• The third law of thermodynamics can be used to calculate the absolute entropy of a substance at a given temperature.

At any temperature T:

$$\Delta S=S_T-S_0$$

$$\Delta S=S_T-0$$

$$Or,S_T=\Delta S$$

Here, S_T = entropy at temperature T