ChemWiki: The Dynamic Chemistry E-textbook > Physical Chemistry > Thermodynamics > State Functions > Free Energy > Gibbs Free Energy > Gibbs Free Energy

Gibbs free energy combines enthalpy and entropy into a single value. The change of free energy is equal to the sum of its enthalpy plus the product of the temperature and entropy of the system. ΔG can also predict the direction of the chemical reaction. If ΔG is positive, then the reaction is non-spontaneous. If it is negative, then it is spontaneous.

In 1875, Josiah Gibbs introduced a thermodynamic quantity combining enthalpy and entropy into a single value called Gibbs free energy. This quantity can be defined as:

\[ G= H-TS\]

\[ G= U+PV-TS\]

where

- U = internal energy (SI unit: joule)
- P = pressure (SI unit: pascal)
- V = volume (SI unit: m3)
- T = temperature (SI unit: kelvin)
- S = entropy (SI unit: joule/kelvin)
- H = enthalpy (SI unit: joule)

Gibbs Free Energy (G) - The energy associated with a chemical reaction that can be used to do work. The free energy of a system is the sum of its enthalpy (H) plus the product of the temperature (Kelvin) and the entropy (S) of the system:

\[ G=H-TS \]

In chemical reactions involving the changes in thermodynamic quantities we often use another variation of this equation:

\[ \Delta G \qquad \qquad = \qquad \qquad \Delta H \qquad \qquad - \qquad \qquad T \Delta S \]

\[ \text {change in free energy} \qquad \text {change in enthalpy} \qquad \text {(temperature) change in entropy}\]

The sign of delta G can allow us to predict the direction of a chemical reaction:

**Terminology**:

- If \(\Delta H^0\) < 0 and \(\Delta S^0\) > 0 then the reaction will be SPONTANEOUS (\(\Delta G^0\) < 0 ) at any temperature.
- If \(\Delta H^0\) > 0 and entropy \(\Delta S^0\) < 0 then the reaction will be NONSPONTANEOUS (\(\Delta G^0\) > 0 ) at any temperature.

Standard-state free energy of reaction ( \(\Delta \)G)

- The free energy of reaction at standard state conditions:

\[ \Delta G^0 = \Delta H^0 - T \Delta S^0\]

- The partial pressures of any gases involved in the reaction is 0.1 MPa.
- The concentrations of all aqueous solutions are 1 M.

Measurements are also generally taken at a temperature of 25C (298 K). The change in free energy that occurs when a compound is formed form its elements in their most thermodynamically stable states at standard-state conditions. In other words, it is the difference between the free energy of a substance and the free energies of its elements in their most thermodynamically stable states at standard-state conditions.

The standard-state free energy of reaction can be calculated from the standard-state free energies of formation as well. It is the sum of the free energies of formation of the products minus the sum of the free energies of formation of the reactants:

\[ \Delta G^0 = \sum \Delta G^0_{f_{produts}} - \sum \Delta G^0_{f_{reactants}} \]

The following equation relates the standard-state free energy of reaction with the free energy of reaction at any moment in time during a reaction (not necessarily at standard-state conditions):

\[ \Delta G = \Delta G^0 + RT \ln Q \]

- ΔG = free energy at any moment
- ΔG = standard-state free energy
- R = ideal gas constant = 8.314 J/mol-K
- T = temperature (Kelvin)
- lnQ = natural log of the
**reaction quotient**

At Equilibrium ΔG = 0. Thus the equation can be arranged into:

\[ \Delta G^0 = - RT \ln K \]

The **Nernst equation** relates the standard-state cell potential with the cell potential of the cell at any moment in time:

\[ E = E^0 - \frac {RT}{nF} \ln Q_c \]

- E = cell potential in volts (Joules per Coulomb)
- n = moles of electrons
- F = Faraday's constant: 96,485 Coulombs per mole of electrons

By rearranging this equation we obtain:

\[ E = E^0 - \frac {RT}{nF} \ln Q \text {multiply the entire equation by nF} \]

\[ nFE = nFE^0 - RT \ln Q \]

which is similiar to:

\[ \Delta G = \Delta G^0 + RT \ln Q\]

By showing these two equations:

\[ nFE = nFE^0 - RT \ln Q \]

\[ \Delta G = \Delta G^0 + RT \ln Q \]

We can conclude that:

\[ \Delta G = -nFE \text {and} \Delta G^0 = -nFE^0 \]

Example 1 |
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Now, can plug in these values we've calculated into the free energy equation |

- General Chemistry: Principles of Modern Applications 9th Edition (pages 792-795)
- SAT Subject Test: Chemistry 6th Edition (page 66)
- http://en.wikipedia.org/wiki/Gibbs_free_energy

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