User:
Energía Líbre
Storyboard
Se obtienen mediante la función partición las distintas funciones y relaciones termodinámicas.
ID:(442, 0)
Helmholtz free energy with partition function
Image
As the derivative with respect to the volume of the free energy of Helmholtz at constant temperature is:
ID:(11725, 0)
Differential relation Helmholtz Free Energy
Equation
Since the Helmholtz free energy depends on the temperature $T$ and volume $V$, the differential is obtained as:
| $ F = U - T S $ |
where:
 $ dF =- S dT - p dV $ |
If we differentiate the definition of Helmholtz free energy:
| $ F = U - T S $ |
we obtain:
$dF = dU - TdS - SdT$
With the differential of internal energy:
| $ dU = T dS - p dV $ |
we can conclude that:
| $ dF =- S dT - p dV $ |
ID:(3474, 0)
Differential of Helmholtz Free Energy
Equation
Given that the helmholtz free fnergy ($F$) is a function of the absolute temperature ($T$) and the volume ($V$), we can express the differential enthalpy ($dH$) as follows:
$dF=\left(\displaystyle\frac{\partial F}{\partial T}\right)_VdT+\left(\displaystyle\frac{\partial F}{\partial V}\right)_TdV$
This allows us to define the differential Helmholtz Free Energy ($dF$) in terms of the slopes the partial derivative of the Helmholtz free energy with respect to temperature at constant volume ($DF_{T,V}$) and the partial derivative of the Helmholtz free energy with respect to volume at constant temperature ($DF_{V,T}$):
| $ dF = DF_{T,V} dT + DF_{V,T} dV $ |
ID:(8187, 0)
Calculo de la derivada parcial de la energía libre de Helmholtz en el volumen a temperatura constante
Equation
La derivada de la energía interna en el volumen a entropia constante es
| $ DF_{V,T} =\left(\displaystyle\frac{\partial F }{\partial V }\right)_ T $ |
ID:(12416, 0)
Calculo de la derivada parcial de la energía libre de Helmholtz en la temperatura a volumen constante
Equation
La derivada de la energía interna en el volumen a entropia constante es
| $ DF_{T,V} =\left(\displaystyle\frac{\partial F }{\partial T }\right)_ V $ |
ID:(12417, 0)
Helmholtz Free Energy and Equation of State at Constant Volume
Equation
The differential Helmholtz Free Energy ($dF$) is a function of the variations in the absolute temperature ($T$) and the volume ($V$), as well as the slopes the partial derivative of the Helmholtz free energy with respect to temperature at constant volume ($DF_{T,V}$) and the partial derivative of the Helmholtz free energy with respect to volume at constant temperature ($DF_{V,T}$), which is expressed as:
| $ dF = DF_{T,V} dT + DF_{V,T} dV $ |
Comparing this with the first law of thermodynamics, it turns out that the partial derivative of the Helmholtz free energy with respect to temperature at constant volume ($DF_{T,V}$) is equal to minus the entropy ($S$):
| $ DF_{T,V} =- S $ |
ID:(3550, 0)
Helmholtz Free Energy and Equation of State at Constant Temperature
Equation
The differential Helmholtz Free Energy ($dF$) is a function of the variations in the absolute temperature ($T$) and the volume ($V$), as well as the slopes the partial derivative of the Helmholtz free energy with respect to temperature at constant volume ($DF_{T,V}$) and the partial derivative of the Helmholtz free energy with respect to volume at constant temperature ($DF_{V,T}$), which is expressed as:
| $ dF = DF_{T,V} dT + DF_{V,T} dV $ |
Comparing this with the first law of thermodynamics, it turns out that the partial derivative of the Helmholtz free energy with respect to volume at constant temperature ($DF_{V,T}$) is equal to minus the pressure ($p$):
| $ DF_{V,T} =- p $ |
ID:(3551, 0)
Helmholtz free energy with partition function
Equation
Como la derivada respecto del volumen de la energía libre de Helmholtz a temperatura constante es con partial derivative of the Helmholtz free energy with respect to volume at constant temperature $J/m^3$ and pressure $Pa$
| $ DF_{V,T} =- p $ |
y la presión es con igual a
| $\bar{p}=\displaystyle\frac{1}{\beta}\displaystyle\frac{\partial\ln Z}{\partial V}$ |
se tiene que la energía libre de Helmholtz es con
| $ F =-\displaystyle\frac{1}{ \beta }\ln Z $ |
ID:(3540, 0)
Helmholtz Free Energy and its Relation of Maxwell
Equation
Since the helmholtz free fnergy ($F$) is an exact differential, it means that you can first vary the absolute temperature ($T$) and then the volume ($V$), 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:
$D(DF_{T,V})_{V,T}=D(DF_{V,T})_{T,V}$
If you replace the differential with the corresponding variable, you obtain the relationship involving the entropy ($S$) and the pressure ($p$):
| $ DS_{V,T} = Dp_{T,V} $ |
Since the differential Helmholtz Free Energy ($dF$) is an exact differential, we should note that the helmholtz free fnergy ($F$) with respect to the absolute temperature ($T$) and the volume ($V$) must be independent of the order in which the function is derived:
$D(DF_{T,V})_{V,T}=D(DF{V,T})_{T,V}$
Using the relationship between the slope the partial derivative of the Helmholtz free energy with respect to temperature at constant volume ($DF_{T,V}$) and the entropy ($S$)
| $ DF_{T,V} =- S $ |
,
and the relationship between the slope the partial derivative of the Helmholtz free energy with respect to volume at constant temperature ($DF_{V,T}$) and the pressure ($p$)
| $ DF_{V,T} =- p $ |
,
we can conclude that:
| $ DS_{V,T} = Dp_{T,V} $ |
ID:(3554, 0)
Calculo de la derivada parcial de la entropía en el volumen a temperatura constante
Equation
La derivada de la entropía en el volumen a temperatura constante es
| $ DS_{V,T} =\left(\displaystyle\frac{\partial S }{\partial V }\right)_ T $ |
ID:(12422, 0)
Calculo de la derivada parcial de la presión en la temperatura a volumen constante
Equation
La derivada de la presión en la temperatura a volumen constante es
| $ Dp_{T,V} =\left(\displaystyle\frac{\partial p }{\partial T }\right)_ V $ |
ID:(12420, 0)
0
Video
Video: Helmholtz Free Energy