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== Thermodynamic Potential ==

'''Thermodynamic potentials''' are scalar quantities used in thermodynamics to describe the equilibrium and spontaneous behavior of physical systems. They are functions of state variables such as temperature, pressure, volume, and entropy, and provide criteria for spontaneous processes and equilibrium under different constraints.

=== Overview ===
Thermodynamic potentials combine the system's internal energy with other thermodynamic parameters, allowing us to analyze changes in energy under various conditions such as constant volume, pressure, temperature, or entropy.

The four most common thermodynamic potentials are:  
* Internal Energy (<math>U</math>)  
* Enthalpy (<math>H</math>)  
* Helmholtz Free Energy (<math>F</math> or <math>A</math>)  
* Gibbs Free Energy (<math>G</math>)

Each potential is defined to be useful under specific experimental or natural conditions.

=== Definitions ===

* '''Internal Energy (<math>U</math>)''': The total energy contained within the system.

<math>
dU = TdS - PdV
</math>

where:  
* <math>T</math> = temperature  
* <math>S</math> = entropy  
* <math>P</math> = pressure  
* <math>V</math> = volume

* '''Enthalpy (<math>H</math>)''': Useful for processes at constant pressure, defined as:

<math>
H = U + PV
</math>

Differential form:

<math>
dH = TdS + VdP
</math>

* '''Helmholtz Free Energy (<math>F</math> or <math>A</math>)''': Useful for processes at constant volume and temperature, defined as:

<math>
F = U - TS
</math>

* '''Gibbs Free Energy (<math>G</math>)''': Useful for processes at constant pressure and temperature, defined as:

<math>
G = H - TS = U + PV - TS
</math>

=== Physical Significance ===
- Thermodynamic potentials help determine the direction of spontaneous processes.  
- The potential that decreases at constant constraints (e.g., constant T and P or constant T and V) indicates the tendency toward equilibrium.

For example:  
- At constant temperature and volume, the Helmholtz free energy <math>F</math> decreases in spontaneous processes.  
- At constant temperature and pressure, the Gibbs free energy <math>G</math> decreases.

=== Relationships Between Potentials ===
Using Legendre transformations, thermodynamic potentials are related as follows:

<math>
H = U + PV
</math>

<math>
F = U - TS
</math>

<math>
G = H - TS = U + PV - TS
</math>

These relations allow switching between natural variables suitable for different experimental conditions.

=== Applications ===
* Predicting spontaneity and equilibrium of chemical reactions  
* Designing thermodynamic cycles in engines and refrigerators  
* Analyzing phase transitions  
* Understanding biochemical processes and energy transfer

=== References ===
* Callen, H. B. (1985). ''Thermodynamics and an Introduction to Thermostatistics''. Wiley.  
* Atkins, P., & de Paula, J. (2010). ''Physical Chemistry''. Oxford University Press.

[[Category:Chemistry]]