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Energía Libre de Gibbs
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Se obtienen mediante la función partición las distintas funciones y relaciones termodinámicas.
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Gibbs free energy with partition function
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To calculate the Gibbs function of the partition function, it is enough to see how the enthalpy and the entropy of it are constructed. How do you have to
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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 $ |
Where the absolute temperature ($T$) and the entropy ($S$) play a significant role.
[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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Gibbs free energy as differential
Equation
As the gibbs free energy ($G$) [1,2] depends on the enthalpy ($H$), the entropy ($S$), and the absolute temperature ($T$):
| $ G = H - T S $ |
The dependence on the gibbs free energy ($G$) with respect to the pressure ($p$) is obtained, and from the absolute temperature ($T$), we obtain the differential:
| $ dG =- S dT + V dp $ |
The gibbs free energy ($G$) as a function of the enthalpy ($H$), the entropy ($S$), and the absolute temperature ($T$) is expressed as:
| $ G = H - T S $ |
The value of the differential of the Gibbs free energy ($dG$) is determined using the differential enthalpy ($dH$), the temperature variation ($dT$), and the entropy variation ($dS$) through the equation:
$dG=dH-SdT-TdS$
Since the differential enthalpy ($dH$) is related to the volume ($V$) and the pressure Variation ($dp$) as follows:
| $ dH = T dS + V dp $ |
It follows that the differential enthalpy ($dH$), the entropy variation ($dS$), and the pressure Variation ($dp$) are interconnected in the following manner:
| $ dG =- S dT + V dp $ |
[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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Gibbs Free Energy Differential
Equation
Given that the gibbs free energy ($G$) is a function of the absolute temperature ($T$) and the pressure ($p$), we can express the differential of the Gibbs free energy ($dG$) as follows:
$dG=\left(\displaystyle\frac{\partial G}{\partial T}\right)_pdT+\left(\displaystyle\frac{\partial G}{\partial p}\right)_Tdp$
This allows us to define the differential of the Gibbs free energy ($dG$) in terms of the slopes the partial derivative of the Gibbs free energy with respect to pressure at constant temperature ($DG_{p,T}$) and the partial derivative of the Gibbs free energy with respect to temperature at constant pressure ($DG_{T,p}$):
| $ dG = DG_{T,p} dT + DG_{p,T} dp $ |
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Calculo de la derivada parcial de la energía libre de Gibbs en la temperatura a presión constante
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La derivada de la energía interna en el volumen a entropia constante es
| $ DG_{T,p} =\left(\displaystyle\frac{\partial G }{\partial T }\right)_ p $ |
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Calculo de la derivada parcial de la energía libre de Gibbs en la presión a temperatura constante
Equation
La derivada de la energía interna en el volumen a entropia constante es
| $ DG_{p,T} =\left(\displaystyle\frac{\partial G }{\partial p }\right)_ T $ |
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Gibbs free energy and equation of state at constant pressure
Equation
The differential of the Gibbs free energy ($dG$) is a function of the variations in the absolute temperature ($T$) and the pressure ($p$), as well as the slopes the partial derivative of the Gibbs free energy with respect to temperature at constant pressure ($DG_{T,p}$) and the partial derivative of the Gibbs free energy with respect to pressure at constant temperature ($DG_{p,T}$), which is expressed as:
| $ dG = DG_{T,p} dT + DG_{p,T} dp $ |
Comparing this with the first law of thermodynamics, it turns out that the partial derivative of the Gibbs free energy with respect to temperature at constant pressure ($DG_{T,p}$) is equal to minus the entropy ($S$):
| $ DG_{T,p} =- S $ |
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Gibbs Free Energy and Equation of State by Constant Temperature
Equation
The differential of the Gibbs free energy ($dG$) is a function of the variations in the absolute temperature ($T$) and the pressure ($p$), as well as the slopes the partial derivative of the Gibbs free energy with respect to temperature at constant pressure ($DG_{T,p}$) and the partial derivative of the Gibbs free energy with respect to pressure at constant temperature ($DG_{p,T}$), which is expressed as:
| $ dG = DG_{T,p} dT + DG_{p,T} dp $ |
Comparing this with the first law of thermodynamics, it turns out that the partial derivative of the Gibbs free energy with respect to pressure at constant temperature ($DG_{p,T}$) is equal to the volume ($V$):
| $ DG_{p,T} = V $ |
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Gibbs free energy with partition function
Equation
Para calcular la función de Gibbs de la función partición basta ver como se construye la entalpía y la entropía de esta misma. Como se tiene que con absolute temperature $K$, enthalpy $J$, entropy $J/K$ and gibbs free energy $J$
| $ G = H - T S $ |
con
| $ H =-\displaystyle\frac{\partial \ln Z }{\partial \beta }+\displaystyle\frac{ V }{ \beta }\displaystyle\frac{\partial \ln Z }{\partial V }$ |
con
| $ S = k_B ( \ln Z + \beta U )$ |
y con
| $ k_B T \equiv\displaystyle\frac{1}{ \beta }$ |
se tiene que con
| $ G =-\displaystyle\frac{1}{ \beta }\ln Z +\displaystyle\frac{ V }{ \beta }\displaystyle\frac{\partial\ln Z }{\partial V }$ |
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Gibbs free energy and its Maxwell relation
Equation
Since the gibbs free energy ($G$) is an exact differential, it means that you can first vary the absolute temperature ($T$) and then the pressure ($p$), or in the reverse order, and the result will be the same. This can be expressed by taking derivatives of slopes in different orders, and there will be no difference:
$D(DG_{T,p})_{p,T}=D(DG_{p,T})_{T,p}$
If you replace the differential with the corresponding variable, you obtain the relationship involving the entropy ($S$) and the volume ($V$):
| $ DS_{p,T} = -DV_{T,p} $ |
Since the differential of the Gibbs free energy ($dG$) is an exact differential, it implies that the gibbs free energy ($G$) with respect to the absolute temperature ($T$) and the pressure ($p$) must be independent of the order in which the function is derived:
$D(DG_{T,p}){p,T}=D(DG{p,T})_{T,p}$
Using the relationship for the slope the partial derivative of the Gibbs free energy with respect to pressure at constant temperature ($DG_{p,T}$) with respect to the volume ($V$)
| $ DG_{p,T} = V $ |
,
and the relationship for the slope the partial derivative of the Gibbs free energy with respect to temperature at constant pressure ($DG_{T,p}$) with respect to the entropy ($S$)
| $ DG_{T,p} =- S $ |
,
we can conclude that:
| $ DS_{p,T} = -DV_{T,p} $ |
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Calculo de la derivada parcial de la entropía en la presión a temperatura constante
Equation
La derivada de la entropía en la presión a temperatura constante es
| $ DS_{p,T} =\left(\displaystyle\frac{\partial S }{\partial p }\right)_ T $ |
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Calculo de la derivada parcial del volumen en la temperatura a presión constante
Equation
La derivada el volumen en la temperatura a presión constante es
| $ DV_{T,p} =\left(\displaystyle\frac{\partial V }{\partial T }\right)_ p $ |
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Video: Gibbs Free Energy