If you like us, please share us on social media.
The latest UCD Hyperlibrary newsletter is now complete, check it out.

MindTouch
Copyright (c) 2006-2014 MindTouch Inc.
http://mindtouch.com

This file and accompanying files are licensed under the MindTouch Master Subscription Agreement (MSA).

At any time, you shall not, directly or indirectly: (i) sublicense, resell, rent, lease, distribute, market, commercialize or otherwise transfer rights or usage to: (a) the Software, (b) any modified version or derivative work of the Software created by you or for you, or (c) MindTouch Open Source (which includes all non-supported versions of MindTouch-developed software), for any purpose including timesharing or service bureau purposes; (ii) remove or alter any copyright, trademark or proprietary notice in the Software; (iii) transfer, use or export the Software in violation of any applicable laws or regulations of any government or governmental agency; (iv) use or run on any of your hardware, or have deployed for use, any production version of MindTouch Open Source; (v) use any of the Support Services, Error corrections, Updates or Upgrades, for the MindTouch Open Source software or for any Server for which Support Services are not then purchased as provided hereunder; or (vi) reverse engineer, decompile or modify any encrypted or encoded portion of the Software.

A complete copy of the MSA is available at http://www.mindtouch.com/msa

# 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).

### Introduction

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)

### Free energy of reaction (ΔG)

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:

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

### Standard-State Free Energy of Formation (ΔGfo)

• 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:

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

### Free energy and equilibrium constants

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).

### Free energy and cell potentials

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

### References

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

13:08, 13 Jan 2015

## Classifications

(not set)
(not set)

### Textbook Maps

An NSF funded Project