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ChemistryEdu Logo 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:

\[ΔG = ΔH - TΔS\]
  • 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 (ΔGo)#

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

At standard conditions:

\[ΔG^o = ΔH^o - TΔS^o\]

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

and, ΔSo = 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:

\[ \int\limits_{G_1}^{G_2} dG = \int\limits_{P_1}^{P_2} VdP\]
\[ (G_2 - G_1) = \int\limits_{P_1}^{P_2}{nRT \over P}dP\]
\[ΔG = nRTln{P_2 \over P_1}\]

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

\[ΔG = nRTln{V_1 \over V_2}\]

Gibbs Free Energy and Reaction Quotient#

  • Mathematically, Gibbs Free Energy is related to reaction quotient by the following equation:
\[ΔG = ΔG^o + RTlnQ_c\]

Here, ΔG = Change in Gibbs Free Energy

ΔGo = Change in Standard Gibbs Free Energy

T = Temperature

R = Universal Gas Constant

Qc = Reaction Quotient

  • At equilibrium, ΔG = 0 and Q = Kc (equilibrium constant):
\[0 = ΔG^o + RTlnK_c\]
\[ΔG^o = -RTlnK_c\]

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:

\[ΔS = S_T - S_0\]
\[ΔS = S_T - 0\]
\[Or,\ S_T = ΔS\]

Here, ST = entropy at temperature T