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Enthalpy and partition function
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The enthalpy can be calculated from the partition function if it is remembered that this is equal to the internal energy and the pressure times the volume:
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Enthalpy $H(S,p)$
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If we need to take into account the energy required to form the system in addition to the internal energy, we must consider the enthalpy ($H$).
the enthalpy ($H$) [1] is defined as the sum of the internal energy ($U$) and the formation energy. The latter corresponds to the work done in the formation, which is equal to $pV$ with the pressure ($p$) and the volume ($V$).
Therefore, we have:
 $ H = U + p V $ |
the enthalpy ($H$) is a function of the entropy ($S$) and the pressure ($p$).
An article that can be considered as the origin of the concept, although it does not include the definition of the name, is:
[1] "Memoir on the Motive Power of Heat, Especially as Regards Steam, and on the Mechanical Equivalent of Heat," written by Benoît Paul Émile Clapeyron (1834).
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Differential Enthalpy Relationship
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Since the enthalpy ($H$) is a function of the internal energy ($U$), the pressure ($p$), and the volume ($V$) according to the equation:
| $ H = U + p V $ |
and this equation depends solely on the entropy ($S$) and the pressure ($p$), we can show that its partial derivative with respect to the differential enthalpy ($dH$) is equal to:
| $ dH = T dS + V dp $ |
If we differentiate the enthalpy function:
| $ H = U + p V $ |
we obtain:
$dH = dU + Vdp + pdV$
With the differential of the internal energy:
| $ dU = T dS - p dV $ |
we can conclude:
| $ dH = T dS + V dp $ |
where the entropy variation ($dS$), the pressure Variation ($dp$), and the absolute temperature ($T$) are also considered.
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Calculo de la derivada parcial de la entalpia en la entropia a presión constante
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La derivada de la entalpia en la entropia a presión constante es
| $ DH_{S,p} =\left(\displaystyle\frac{\partial H }{\partial S }\right)_ p $ |
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Calculo de la derivada parcial de la entalpia en la presión a entropía constante
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La derivada de la entalpia en la presión a entropia constante es
| $ DH_{p,S} =\left(\displaystyle\frac{\partial H }{\partial p }\right)_ S $ |
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Enthalpy differential
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Given that the enthalpy ($H$) is a function of the entropy ($S$) and the pressure ($p$), we can express the differential enthalpy ($dH$) as follows:
$dH=\left(\displaystyle\frac{\partial H}{\partial S}\right)_pdS+\left(\displaystyle\frac{\partial H}{\partial p}\right)_Sdp$
This allows us to define the differential enthalpy ($dH$) in terms of the slopes the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) and the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$):
| $ dH = DH_{S,p} dS + DH_{p,S} dp $ |
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Enthalpy and equation of state at constant pressure
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The differential enthalpy ($dH$) is a function of the variations in the entropy ($S$) and the pressure ($p$), as well as the slopes the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) and the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$), which is expressed as:
| $ dH = DH_{S,p} dS + DH_{p,S} dp $ |
Comparing this with the first law of thermodynamics, it turns out that the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) is equal to minus the volume ($V$):
| $ DH_{S,p} = T $ |
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Enthalpy and equation of state at constant entropy
Equation
The differential enthalpy ($dH$) is a function of the variations in the entropy ($S$) and the pressure ($p$), as well as the slopes the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) and the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$), which is expressed as:
| $ dH = DH_{S,p} dS + DH_{p,S} dp $ |
Comparing this with the first law of thermodynamics, it turns out that the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$) is equal to minus the absolute temperature ($T$):
| $ DH_{p,S} = V $ |
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Enthalpy and partition function
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La entalpía se logra calcular de la función partición si se recuerda que esta es igual a la energía interna y a la presión por el volumen que con enthalpy $J$, internal energy $J$, pressure $Pa$ and volume $m^3$ es:
| $ H = U + p V $ |
Como la energía interna es con igual a
| $U=-\displaystyle\frac{\partial\ln Z}{\partial\beta}$ |
y con la presión es
| $\bar{p}=\displaystyle\frac{1}{\beta}\displaystyle\frac{\partial\ln Z}{\partial V}$ |
se tiene que con es
| $ H =-\displaystyle\frac{\partial \ln Z }{\partial \beta }+\displaystyle\frac{ V }{ \beta }\displaystyle\frac{\partial \ln Z }{\partial V }$ |
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Enthalpy and its relation of Maxwell
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Since the enthalpy ($H$) is an exact differential, it means that you can first vary the entropy ($S$) and then the pressure ($p$), 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:
$D(DH_{S,p})_{p,S}=D(DH_{p,S})_{S,p}$
If you replace the differential with the corresponding variable, you obtain the relationship involving the absolute temperature ($T$) and the volume ($V$):
| $ DT_{p,S} = DV_{S,p} $ |
Since the differential enthalpy ($dH$) is an exact differential, we should note that the enthalpy ($H$) with respect to the entropy ($S$) and the pressure ($p$) must be independent of the order in which the function is derived:
$D(DH_{S,p}){p,S}=D(DH{p,S})_{S,p}$
Using the relationship between the slope the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) and the absolute temperature ($T$)
| $ DH_{S,p} = T $ |
,
and the relationship between the slope the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$) and the volume ($V$)
| $ DH_{p,S} = V $ |
,
we can conclude that:
| $ DT_{p,S} = DV_{S,p} $ |
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Calculo de la derivada parcial de la temperatura en la presión a entropía constante
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La derivada de la temperatura en la presión a entropia constante es
| $ DT_{p,S} =\left(\displaystyle\frac{\partial T }{\partial p }\right)_ S $ |
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Calculo de la derivada parcial del volumen en la entropia a presión constante
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La derivada el volumen en la entropia a presión constante es
| $ DV_{S,p} =\left(\displaystyle\frac{\partial V }{\partial S }\right)_ p $ |
ID:(12029, 0)
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