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

Gibbs Free Energy

Gibbs free energy, denoted G, combines enthalpy and entropy into a single value. The change in free energy, ΔG, is equal to the sum of the enthalpy plus the product of the temperature and entropy of the system. ΔG can predict the direction of the chemical reaction under two conditions: (1) constant temperature and (2) constant pressure. If ΔG is positive, then the reaction is nonspontaneous (it requires the input of external energy to occur) and if it is negative, then it is spontaneous (occurs without external energy input).

In 1875, Josiah Gibbs introduced a thermodynamic quantity combining enthalpy and entropy into a single value called Gibbs free energy. This quantity is the energy associated with a chemical reaction that can be used to do work, and is the sum of its enthalpy (H) and the product of the temperature and the entropy (S) of the system. This quantity is defined as follows:

\[ G= H-TS\]

or more completely as

\[ G= U+PV-TS\]

where

- U = internal energy (SI unit: joule)
- P = pressure (SI unit: pascal)
- V = volume (SI unit: \(m^3\))
- T = temperature (SI unit: kelvin)
- S = entropy (SI unit: joule/kelvin)
- H = enthalpy (SI unit: joule)

In chemical reactions involving the changes in thermodynamic quantities, a variation on this equation is often encountered:

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

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

The sign of ΔG indicates the direction of a chemical reaction:

**Terminology**

- If \(\Delta H^o\) < 0 and \(\Delta S^o\) > 0, then the reaction is spontaneous (\(\Delta G^o\) < 0 ) at any temperature.
- If \(\Delta H^o\) > 0 and entropy \(\Delta S^o\) < 0, then the reaction is nonspontaneous (\(\Delta G^o\) > 0 ) at any temperature.

Standard-state free energy of reaction ( \(\Delta G^o\)) is defined as the free energy of reaction at standard state conditions:

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

- The partial pressure of any gas involved in the reaction is 0.1 MPa.
- The concentrations of all aqueous solutions are 1 M.
- Measurements are generally taken at a temperature of 25° C (298 K).

The standard-state free energy of formation is the change in free energy that occurs when a compound is formed from 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 constituent elements at standard-state conditions.

The standard-state free energy of reaction can be calculated from the standard-state free energies of formation. 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:

\[ \Delta G^o = \sum \Delta G^o_{f_{products}} - \sum \Delta G^o_{f_{reactants}} \]

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

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

- \(\Delta G\) = free energy at any moment
- \(\Delta G^o\) = standard-state free energy
- R is the ideal gas constant = 8.314 J/mol-K
- T is the absolute temperature (Kelvin)
- \(\ln Q\) is natural logarithm of the
**reaction quotient**

At equilibrium, ΔG = 0 and Q=K. Thus the equation can be arranged into:

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

This equation is particularly interesting as it relates the free energy difference under standard conditions to the properties of a system at equilibrium (which is very rarely at standard conditions).

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

\[ E = E^o - \frac {RT}{nF} \ln Q \]

- \(E\) = 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:

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

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

which is similar to:

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

By juxtaposing these two equations:

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

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

it can be concluded that:

\[ \Delta G = -nFE\]

Therefore,

\[ \Delta G^o = -nFE^o \]

Example 1 |
---|

These values can be substituted 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

Last modified

13:08, 13 Jan 2015

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