Editing Thermodynamic free energy (section)

==Application==
Just like the general concept of energy, free energy has a few definitions suitable for different conditions. In physics, chemistry, and biology, these conditions are thermodynamic parameters (temperature <math>T</math>, volume <math>V</math>, pressure <math>p</math>, etc.). Scientists have come up with several ways to define free energy. The mathematical expression of Helmholtz free energy is:

:<math>A = U-TS</math>

This definition of free energy is useful for gas-phase reactions or in physics when modeling the behavior of isolated systems kept at a constant volume. For example, if a researcher wanted to perform a combustion reaction in a bomb calorimeter, the volume is kept constant throughout the course of a reaction. Therefore, the heat of the reaction is a direct measure of the free energy change, <math>q=\Delta A</math>. In solution chemistry, on the other hand, most chemical reactions are kept at constant pressure. Under this condition, the heat <math>q</math> of the reaction is equal to the enthalpy change <math>\Delta H</math> of the system. Under constant pressure and temperature, the free energy in a reaction is known as Gibbs free energy <math>G</math>.

:<math>G = H-TS</math>

These functions have a minimum in chemical equilibrium, as long as certain variables (<math>T</math>, and <math>V</math> or <math>p</math>) are held constant. In addition, they also have theoretical importance in deriving [[Maxwell relations]]. Work other than {{nowrap|''p&thinsp;dV''}} may be added, e.g., for [[electrochemistry|electrochemical]] cells, or {{nowrap|''f&thinsp;dx''}} work in [[elastomer|elastic]] materials and in [[muscle]] contraction. Other forms of work which must sometimes be considered are [[stress (physics)|stress]]-[[strain (materials science)|strain]], [[magnetism|magnetic]], as in [[adiabatic process|adiabatic]] de[[magnetization]] used in the approach to [[absolute zero]], and work due to electric [[dipole|polarization]]. These are described by [[tensor]]s.

In most cases of interest there are internal [[degrees of freedom (physics and chemistry)|degrees of freedom]] and processes, such as [[chemical reaction]]s and [[phase transition]]s, which create entropy. Even for homogeneous "bulk" materials, the free energy functions depend on the (often suppressed) [[chemical compound|composition]], as do all proper [[thermodynamic potentials]] ([[extensive quantity|extensive functions]]), including the internal energy.
{{table of thermodynamic potentials|noU=1|noH=1|noO=1}}
<math>N_i</math> is the number of molecules (alternatively, [[mole (unit)|moles]]) of type <math>i</math> in the system. If these quantities do not appear, it is impossible to describe compositional changes. The [[differential (infinitesimal)|differential]]s for processes at uniform pressure and temperature are (assuming only <math>pV</math> work):

:<math>\mathrm dA = - p\,\mathrm dV - S\,\mathrm dT + \sum_i \mu_i \,\mathrm dN_i\,</math>

:<math>\mathrm dG = V\,\mathrm dp - S\,\mathrm dT + \sum_i \mu_i \,\mathrm dN_i\,</math>

where ''μ''<sub>''i''</sub> is the [[chemical potential]] for the ''i''th [[component (thermodynamics)|component]] in the system. The second relation is especially useful at constant <math>T</math> and <math>p</math>, conditions which are easy to achieve experimentally, and which approximately characterize [[life|living]] creatures. Under these conditions, it simplifies to

:<math>(\mathrm dG)_{T,p} = \sum_i \mu_i \,\mathrm dN_i\,</math>

Any decrease in the Gibbs function of a system is the upper limit for any [[isothermal process|isothermal]], [[isobaric process|isobaric]] work that can be captured in the surroundings, or it may simply be [[dissipation|dissipated]], appearing as <math>T</math> times a corresponding increase in the entropy of the system and/or its surrounding.

An example is [[surface free energy]], the amount of increase of free energy when the area of surface increases by every unit area.

The [[path integral Monte Carlo]] method is a numerical approach for determining the values of free energies, based on quantum dynamical principles.

===Work and free energy change===
For a reversible isothermal process, Δ''S'' = ''q''<sub>rev</sub>/''T'' and therefore the definition of ''A'' results in

: <math>\Delta A = \Delta U - T \Delta S = \Delta U - q_\text{rev} = w_\text{rev}</math> (at constant temperature)

This tells us that the change in free energy equals the reversible or maximum work for a process performed at constant temperature. Under other conditions, free-energy change is not equal to work; for instance, for a reversible adiabatic expansion of an ideal gas, {{nowrap|<math>\Delta A = w_\text{rev} - S \Delta T</math>.}} Importantly, for a heat engine, including the [[Carnot cycle]], the free-energy change after a full cycle is zero, {{nowrap|<math>\Delta_\text{cyc} A = 0 </math>,}} while the engine produces nonzero work. It is important to note that for heat engines and other thermal systems, the free energies do not offer convenient characterizations; internal energy and enthalpy are the preferred potentials for characterizing thermal systems.

===Free energy change and spontaneous processes===
According to the [[second law of thermodynamics]], for any process that occurs in a closed system, the [[Clausius theorem|inequality of Clausius]], Δ''S'' > ''q''/''T''<sub>surr</sub>, applies. For a process at constant temperature and pressure without non-''PV'' work, this inequality transforms into <math>\Delta G < 0 </math>. Similarly, for a process at constant temperature and volume, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy is a necessary condition for a process to be spontaneous; this is the most useful form of the second law of thermodynamics in chemistry. In chemical equilibrium at constant ''T'' and ''p'' without electrical work, d''G'' = 0.