The gibbs free energy ($G$) as a function of the enthalpy ($H$), the entropy ($S$), and the absolute temperature ($T$) is expressed as:

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:

It follows that the differential enthalpy ($dH$), the entropy variation ($dS$), and the pressure Variation ($dp$) are interconnected in the following manner:

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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:

Comparing this with the equation for the variation of Gibbs Free Energy ($dG$):

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 ($S$):

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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:

Comparing this with the equation for the variation of Gibbs Free Energy ($dG$):

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 ($V$):

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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 ($V$)

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 ($S$)

we can conclude that:

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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:

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