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Thermodynamic potentials
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Key thermodynamic potentials include:
• Internal Energy: the energy required to form a thermodynamic system, isolated from any external environment.
• Enthalpy: the energy needed to form a system and place it within an environment. Enthalpy essentially combines internal energy and the work needed to integrate the system into its surroundings.
• Helmholtz Free Energy: the portion of a systems internal energy that is available to perform work at constant temperature.
• Gibbs Free Energy: the part of the enthalpy of a system that can be used to perform work under constant temperature and pressure conditions.
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Mechanisms
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Thermodynamic potentials are quantities that provide comprehensive information about the energy status of a thermodynamic system under specific conditions. They are used to predict the direction of spontaneous processes, determine equilibrium states, and calculate the maximum work that can be extracted from a system.
The Internal Energy is the total energy contained within a system, encompassing both kinetic and potential energies at the molecular level. It accounts for the energy required to create the system. Internal energy changes when heat is added to or removed from the system and when work is done on or by the system.
The Enthalpy is useful in processes occurring at constant pressure. It represents the total heat content of a system and is used to calculate the heat exchange during chemical reactions and phase transitions at constant pressure.
The Helmholtz Free Energy is used for systems at constant temperature and volume. It measures the maximum amount of work that can be extracted from a system, excluding pressure-volume work, and indicates the spontaneity of processes at constant temperature and volume.
The Gibbs Free Energy is used for systems at constant temperature and pressure. It represents the maximum reversible work that can be performed by the system, excluding work done by pressure-volume changes. Gibbs free energy is crucial in predicting the direction of chemical reactions and phase transitions at constant temperature and pressure. A negative change in Gibbs free energy indicates a spontaneous process.
The Grand Potential is used in open systems where the number of particles can change. It is useful for analyzing systems in contact with a reservoir that can exchange both energy and particles.
Mechanisms
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Internal energy
Concept
If the absolute temperature ($T$) and the pressure ($p$) are kept constant, the variation of the internal energy ($dU$), which depends on the entropy variation ($dS$) and the volume Variation ($dV$), is expressed as:
| $ dU = T dS - p dV $ |
Integrating this results in the following expression in terms of the internal energy ($U$), the entropy ($S$), and the volume ($V$):
| $ U = T S - p V $ |
[1] "Über die quantitative und qualitative Bestimmung der Kräfte" (On the Quantitative and Qualitative Determination of Forces), Julius Robert von Mayer, Annalen der Chemie und Pharmacie, 1842
[2] "Über die Erhaltung der Kraft" (On the Conservation of Force), Hermann von Helmholtz, 1847
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Enthalpy
Concept
The enthalpy ($H$) refers to the energy contained within a system, including any energy required to create it. It is composed of the internal energy ($U$) and the work necessary to form the system, which is represented as $pV$ where the pressure ($p$) and the volume ($V$) are involved.
It is a function of the entropy ($S$) and the pressure ($p$), allowing it to be expressed as $H = H(S,p)$ and it follows the following mathematical relation:
| $ H = U + p V $ |
An article that can be considered as the origin of the concept, although it does not include the definition of the name, is:
[1] "Memoir on the Motive Power of Heat, Especially as Regards Steam, and on the Mechanical Equivalent of Heat," written by Benoît Paul Émile Clapeyron (1834).
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Helmholtz free energy
Concept
The helmholtz free fnergy ($F$) [1] refers to the energy contained within a system, but excludes the energy that cannot be used to perform work. In this sense, it represents the energy available to do work as long as it does not include the energy required to form the system. It is composed, therefore, of the internal energy ($U$), from which the thermal energy $ST$, where the entropy ($S$) and the absolute temperature ($T$) are involved, is subtracted.
This function depends on the absolute temperature ($T$) and the volume ($V$), allowing it to be expressed as $F = F(V,T)$, and it satisfies the following mathematical relation:
| $ F = U - T S $ |
[1] "Über die Thermodynamik chemischer Vorgänge" (On the thermodynamics of chemical processes.), Hermann von Helmholtz, Dritter Beitrag. Offprint from: ibid., 31 May, (1883)
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Gibbs free energy
Concept
The gibbs free energy ($G$) refers to the energy contained within a system, including the energy required for its formation, but excludes the energy that cannot be used to do work. In this sense, it represents the energy available to do work in a process that includes the energy required for its formation. It is composed, therefore, of the enthalpy ($H$), from which the thermal energy $ST$, where the entropy ($S$) and the absolute temperature ($T$) are involved, is subtracted.
This function depends on the absolute temperature ($T$) and the pressure ($p$), allowing it to be expressed as $G = G(T,p)$ and it satisfies the following mathematical relation:
| $ G = H - T S $ |
[1] "On the Equilibrium of Heterogeneous Substances," J. Willard Gibbs, Transactions of the Connecticut Academy of Arts and Sciences, 3: 108-248 (October 1875 May 1876)
[2] "On the Equilibrium of Heterogeneous Substances," J. Willard Gibbs, Transactions of the Connecticut Academy of Arts and Sciences, 3: 343-524 (May 1877 July 1878)
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Example of application of the method
Description
If we consider the internal energy $U(V,S)$, it depends on two variables:
• The volume $V$
• The entropy $S$
Therefore, its variation can be expressed using the relationship:
| $ df = \left(\displaystyle\frac{\partial f }{\partial x }\right)_ y dx + \left(\displaystyle\frac{\partial f }{\partial y }\right)_ x dy $ |
in the form:
$dU = \left(\displaystyle\frac{\partial U }{\partial V }\right)_ S dV + \left(\displaystyle\frac{\partial U }{\partial S }\right)_ V dS$
According to the first law of thermodynamics, we know that the variation of internal energy $dU$ is equal to:
From this, we can conclude that the slopes are the pressure $p$:
$\left(\displaystyle\frac{\partial U }{\partial V }\right)_ S = -p$
and the temperature $T$:
$\left(\displaystyle\frac{\partial U }{\partial S }\right)_ V = T$
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Ejemplo de potencial termodinámico
Description
To establish the relationships, thermodynamic potentials are introduced, which are potential energies that include or exclude certain forms of energy in a system, such as the energy associated with work $pV$ and the energy associated with entropy $TS$, which cannot be used to perform work.
In the case of enthalpy $H$, it corresponds to the internal energy of the system, which includes the movement of particles, but also incorporates the energy required to form the system, i.e., the work $pV$ done to establish it. Therefore, it is defined as:
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Model
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Parameters
Variables
Calculations
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Calculations
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Variable 
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Target : Equation To be used 
Equations
$ F = U - T S $
F = U - T * S
$ G = H - T S $
G = H - T * S
$ H = U + p V $
H = U + p * V
$ U = T S - p V $
U = T * S - p * V
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Internal Energy
Equation
The internal energy ($U$) is with the absolute temperature ($T$), the pressure ($p$), the entropy ($S$) and the volume ($V$) equal to:
| $ U = T S - p V $ |
If the absolute temperature ($T$) and the pressure ($p$) are kept constant, the variation of the internal energy ($dU$), which depends on the entropy variation ($dS$) and the volume Variation ($dV$), is expressed as:
| $ dU = T dS - p dV $ |
Integrating this results in the following expression in terms of the internal energy ($U$), the entropy ($S$), and the volume ($V$):
| $ U = T S - p V $ |
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Enthalpy
Equation
The enthalpy ($H$) is defined as the sum of the internal energy ($U$) and the formation energy. The latter corresponds to the work done in the formation, which is equal to $pV$ with the pressure ($p$) and the volume ($V$). Therefore, we have:
| $ H = U + p V $ |
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Helmholtz free Energy
Equation
The helmholtz free fnergy ($F$) is defined as the difference between the internal energy ($U$) and the energy that cannot be utilized to perform work. The latter corresponds to $ST$ with the entropy ($S$) and the absolute temperature ($T$). Therefore, we obtain:
| $ F = U - T S $ |
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Gibbs and Helmholtz free energy
Equation
The gibbs free energy ($G$) [1,2] represents the total energy, encompassing both the internal energy and the formation energy of the system. It is defined as the enthalpy ($H$), excluding the portion that cannot be used to perform work, which is represented by $TS$ with the absolute temperature ($T$) and the entropy ($S$). This relationship is expressed as follows:
| $ G = H - T S $ |
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