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Se obtienen mediante la función partición las distintas funciones y relaciones termodinámicas.

>Model

ID:(442, 0)


Helmholtz free energy with partition function
Description

As the derivative with respect to the volume of the free energy of Helmholtz at constant temperature is:

ID:(11725, 0)


Energía Líbre
Description

Se obtienen mediante la función partición las distintas funciones y relaciones termodinámicas.

Variables
Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$T$
T
Absolute temperature
K
$dF$
dF
Differential Helmholtz Free Energy
J
$S$
S
Entropy
J/K
$Z$
Z
Función Partición
-
$F$
F
Helmholtz Free Energy
J
$F$
F
Helmholtz free fnergy
J
$DS_{V,T}$
DS_VT
Partial derivative of entropy with respect to volume at constant temperature
J/m^3
$Dp_{T,V}$
Dp_TV
Partial derivative of pressure with respect to temperature at constant volume
m^3/K
$DF_{T,V}$
DF_TV
Partial derivative of the Helmholtz free energy with respect to temperature at constant volume
J/K
$DF_{V,T}$
DF_VT
Partial derivative of the Helmholtz free energy with respect to volume at constant temperature
J/m^3
$p$
p
Pressure
Pa
$T$
T
Temperatura
K
$dT$
dT
Temperature variation
K
$V$
V
Volume
m^3
$\Delta V$
DV
Volume Variation
m^3

Calculations

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Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

 Variable   Given   Calculate   Target :   Equation   To be used


Equations

The helmholtz free fnergy ($F$) is defined using the internal energy ($U$), the absolute temperature ($T$), and the entropy ($S$) as:

$ F = U - T S $



When we differentiate this equation, we obtain with the differential Helmholtz Free Energy ($dF$), the variation of the internal energy ($dU$), the entropy variation ($dS$), and the temperature variation ($dT$):

$dF = dU - TdS - SdT$



With the differential of internal energy and the variables the pressure ($p$) and the volume Variation ($\Delta V$),

$ dU = T dS - p dV $



we finally obtain:

$ dF =- S dT - p dV $

(ID 3474)

The differential Helmholtz Free Energy ($dF$) is a function of the variations of 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 ($DF_{V,T}$), expressed as:

$ dF = DF_{T,V} dT + DF_{V,T} dV $



Comparing this with the equation for the differential Helmholtz Free Energy ($dF$):

$ dF =- S dT - p dV $



and with the first law of thermodynamics, it follows that the partial derivative of the Helmholtz free energy with respect to temperature at constant volume ($DF_{T,V}$) is equal to negative the entropy ($S$):

$ DF_{T,V} =- S $

(ID 3550)

The differential Helmholtz Free Energy ($dF$) is a function of the variations of 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 equation for the differential Helmholtz Free Energy ($dF$):

$ dF =- S dT - p dV $



and with the first law of thermodynamics, it follows that the partial derivative of the Helmholtz free energy with respect to volume at constant temperature ($DF_{V,T}$) is equal to negative the pressure ($p$):

$ DF_{V,T} =- p $

(ID 3551)

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)

Given that the helmholtz free fnergy ($F$) depends on the absolute temperature ($T$) and the volume ($V$), the differential Helmholtz Free Energy ($dF$) can be calculated using:

$dF = \left(\displaystyle\frac{\partial F}{\partial T}\right)_V dT + \left(\displaystyle\frac{\partial F}{\partial V}\right)_T dV$



To simplify this expression, we introduce the notation for the derivative of the helmholtz free fnergy ($F$) with respect to the absolute temperature ($T$) while keeping the volume ($V$) constant as:

$DF_{T,V} \equiv \left(\displaystyle\frac{\partial F}{\partial T}\right)_V$



and for the derivative of the helmholtz free fnergy ($F$) with respect to the volume ($V$) while keeping the absolute temperature ($T$) constant as:

$DF_{V,T} \equiv \left(\displaystyle\frac{\partial F}{\partial V}\right)_T$



thus we can write:

$ dF = DF_{T,V} dT + DF_{V,T} dV $

(ID 8187)

Examples

As the derivative with respect to the volume of the free energy of Helmholtz at constant temperature is:

(ID 11725)

The dependency of the differential Helmholtz Free Energy ($dF$) on the entropy ($S$) and the temperature variation ($dT$), in addition to the pressure ($p$) and the volume Variation ($\Delta V$), is given by:

$ dF =- S dT - p dV $

(ID 3474)

The differential Helmholtz Free Energy ($dF$) is a function of the variations of 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 $

(ID 8187)

La derivada de la energ a interna en el volumen a entropia constante es

$ DF_{V,T} =\left(\displaystyle\frac{\partial F }{\partial V }\right)_ T $

(ID 12416)

La derivada de la energ a interna en el volumen a entropia constante es

$ DF_{T,V} =\left(\displaystyle\frac{\partial F }{\partial T }\right)_ V $

(ID 12417)

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)

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)

Como la derivada respecto del volumen de la energ a 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 presi n es con igual a

$\bar{p}=\displaystyle\frac{1}{\beta}\displaystyle\frac{\partial\ln Z}{\partial V}$



se tiene que la energ a libre de Helmholtz es con

$ F =-\displaystyle\frac{1}{ \beta }\ln Z $

(ID 3540)

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_{V,T} = Dp_{T,V} $

(ID 3554)

La derivada de la entrop a en el volumen a temperatura constante es

$ DS_{V,T} =\left(\displaystyle\frac{\partial S }{\partial V }\right)_ T $

(ID 12422)

La derivada de la presi n en la temperatura a volumen constante es

$ Dp_{T,V} =\left(\displaystyle\frac{\partial p }{\partial T }\right)_ V $

(ID 12420)

ID:(442, 0)