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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 $ |
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Gibbs free energy as differential
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The dependency of the variation of Gibbs Free Energy ($dG$) on the entropy ($S$) and the temperature variation ($dT$), in addition to the volume ($V$) and the pressure Variation ($dp$), is given by:
| $ 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 $ |
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Gibbs Free Energy Differential
Equation
The differential of the Gibbs free energy ($dG$) is a function of the variations of 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 $ |
Given that the gibbs free energy ($G$) depends on the absolute temperature ($T$) and the pressure ($p$), the variation of Gibbs Free Energy ($dG$) can be calculated using:
$dG = \left(\displaystyle\frac{\partial G}{\partial T}\right)_p dT + \left(\displaystyle\frac{\partial G}{\partial p}\right)_T dp$
To simplify this expression, we introduce the notation for the derivative of the gibbs free energy ($G$) with respect to the absolute temperature ($T$) while keeping the pressure ($p$) constant as:
$DG_{T,p} \equiv \left(\displaystyle\frac{\partial G}{\partial T}\right)_p$
and for the derivative of the gibbs free energy ($G$) with respect to the pressure ($p$) while keeping the absolute temperature ($T$) constant as:
$DG_{p,T} \equiv \left(\displaystyle\frac{\partial G}{\partial p}\right)_T$
thus we can write:
| $ 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
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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 $ |
The differential of the Gibbs free energy ($dG$) is a function of the variations of 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 ($DG_{p,T}$), expressed as:
| $ dG = DG_{T,p} dT + DG_{p,T} dp $ |
Comparing this with the equation for the variation of Gibbs Free Energy ($dG$):
| $ dG =- S dT + V dp $ |
and with the first law of thermodynamics, it follows that the partial derivative of the Gibbs free energy with respect to temperature at constant pressure ($DG_{T,p}$) is equal to negative the entropy ($S$):
| $ DG_{T,p} =- S $ |
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Gibbs Free Energy and Equation of State by Constant Temperature
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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 $ |
The differential of the Gibbs free energy ($dG$) is a function of the variations of 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 ($DG_{p,T}$), expressed as:
| $ dG = DG_{T,p} dT + DG_{p,T} dp $ |
Comparing this with the equation for the variation of Gibbs Free Energy ($dG$):
| $ dG =- S dT + V dp $ |
and with the first law of thermodynamics, it follows 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
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With the entropy ($S$), the volume ($V$), the absolute temperature ($T$) and the pressure ($p$) we obtain one of the so-called Maxwell relations:
| $ 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
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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
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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