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Gibbs free energy
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The Gibbs free energy represents the portion of a system's enthalpy that is available to perform work.
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Mechanisms
Definition
The Gibbs free energy is a thermodynamic potential that represents the maximum amount of work that can be performed by a system at constant temperature and pressure. It quantifies the maximum usable work, excluding work done by pressure-volume changes, that a system can perform when it transitions from one state to another under these conditions.
The change in Gibbs free energy during a process indicates whether the process is spontaneous. A negative change in Gibbs free energy means the process is spontaneous, while a positive change means the process is non-spontaneous. When the change is zero, the system is in equilibrium. At equilibrium, the Gibbs free energy of the system is minimized, helping to determine the equilibrium position of chemical reactions and the stability of different phases.
Gibbs free energy is also used to analyze phase transitions, such as melting, boiling, and sublimation, under constant temperature and pressure. The point at which the Gibbs free energy of different phases becomes equal marks the phase transition. This principle helps in understanding and predicting the behavior of substances under varying conditions.
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Gibbs free energy
Image
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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Gibbs free energy and equation of state at constant pressure
Note
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
Quote
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 Differential
Exercise
The gibbs free energy ($G$) explains how this responds to variations in the absolute temperature ($T$) and the pressure ($p$), expressed as:
| $ dG =- S dT + V dp $ |
When the absolute temperature ($T$) varies, a positive slope equal to the entropy ($S$) is observed.
When the pressure ($p$) varies, a negative slope equal to the volume ($V$) is produced.
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Gibbs free energy and its Maxwell relation
Equation
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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Gibbs free energy
Storyboard
The Gibbs free energy represents the portion of a system's enthalpy that is available to perform work.
Variables
Calculations
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Equation
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Equations
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 ($\Delta V$), is expressed as:
Integrating this results in the following expression in terms of the internal energy ($U$), the entropy ($S$), and the volume ($V$):
The gibbs free energy ($G$) with the internal energy ($U$), the entropy ($S$), the absolute temperature ($T$), the pressure ($p$), and the volume ($V$) is represented as:
And with the substitution of the internal energy ($U$),
We obtain:
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:
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 ($S$):
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 ($V$):
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:
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:
Examples
The Gibbs free energy is a thermodynamic potential that represents the maximum amount of work that can be performed by a system at constant temperature and pressure. It quantifies the maximum usable work, excluding work done by pressure-volume changes, that a system can perform when it transitions from one state to another under these conditions.
The change in Gibbs free energy during a process indicates whether the process is spontaneous. A negative change in Gibbs free energy means the process is spontaneous, while a positive change means the process is non-spontaneous. When the change is zero, the system is in equilibrium. At equilibrium, the Gibbs free energy of the system is minimized, helping to determine the equilibrium position of chemical reactions and the stability of different phases.
Gibbs free energy is also used to analyze phase transitions, such as melting, boiling, and sublimation, under constant temperature and pressure. The point at which the Gibbs free energy of different phases becomes equal marks the phase transition. This principle helps in understanding and predicting the behavior of substances under varying conditions.
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:
[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)
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 ($S$):
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 ($V$):
The gibbs free energy ($G$) explains how this responds to variations in the absolute temperature ($T$) and the pressure ($p$), expressed as:
When the absolute temperature ($T$) varies, a positive slope equal to the entropy ($S$) is observed.
When the pressure ($p$) varies, a negative slope equal to the volume ($V$) is produced.
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:
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:
The gibbs free energy ($G$) [1,2] corresponds to the internal energy ($U$), which includes the energy required to form the system $pV$, involving the pressure ($p$) and the volume ($V$). From this total energy, we subtract the energy that cannot be used to perform work, represented as $TS$, with the absolute temperature ($T$) and the entropy ($S$) being significant contributors. This relationship is expressed as follows:
The internal energy ($U$) is with the absolute temperature ($T$), the pressure ($p$), the entropy ($S$) and the volume ($V$) equal to:
The gibbs free energy ($G$) reduces to:
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:
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 ($S$):
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 ($V$):
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:
With the entropy ($S$), the volume ($V$), the absolute temperature ($T$) and the pressure ($p$) we obtain one of the so-called Maxwell relations:
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