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Mechanisms
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Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy of the system, which is the energy required to create the system, and the energy needed to make space for it by displacing its environment. In other words, enthalpy accounts for the energy contained within the system and the energy required to displace the surrounding environment to accommodate the system.
Enthalpy is used to measure heat exchange in processes that occur at constant pressure. In such processes, the change in enthalpy directly represents the heat added to or removed from the system. It is also essential for understanding the energy dynamics of chemical reactions, indicating whether a reaction is exothermic (releases heat) or endothermic (absorbs heat). Enthalpy describes energy changes associated with phase transitions, such as melting, boiling, or sublimation, with specific values like the enthalpy of fusion for melting.
Mechanisms
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Enthalpy
Concept
The enthalpy ($H$) refers to the energy contained within a system, including any energy required to create it. It is composed of the internal energy ($U$) and the work necessary to form the system, which is represented as $pV$ where the pressure ($p$) and the volume ($V$) are involved.
It is a function of the entropy ($S$) and the pressure ($p$), allowing it to be expressed as $H = H(S,p)$ and it follows the following mathematical relation:
| $ H = U + p V $ |
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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Enthalpy differential
Concept
The enthalpy ($H$) explains how this behaves under variations in the pressure ($p$) and the entropy ($S$), which is expressed as:
| $ dH = T dS + V dp $ |
Under the variation of the pressure ($p$), it occurs with a positive slope that is equal to the volume ($V$).
Under the variation of the entropy ($S$), it occurs with a negative slope that is equal to the absolute temperature ($T$).
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Enthalpy and equation of state at constant entropy
Condition
The differential enthalpy ($dH$) is a function of the variations of the entropy variation ($dS$) and the pressure Variation ($dp$), 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 $ |
When compared to the equation for the differential enthalpy ($dH$):
| $ dH = T dS + V dp $ |
it follows that the slope of the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$) with respect to the variation of the volume ($V$) is:
| $ DH_{p,S} = V $ |
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Enthalpy and equation of state at constant pressure
Condition
The differential enthalpy ($dH$) is a function of the variations of the entropy variation ($dS$) and the pressure Variation ($dp$), 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 ($DH_{p,S}$), expressed as:
| $ dH = DH_{S,p} dS + DH_{p,S} dp $ |
Comparing this with the equation for the differential enthalpy ($dH$):
| $ dH = T dS + V dp $ |
we find that the slope of the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) with respect to the variation of the absolute temperature ($T$) is:
| $ DH_{S,p} = T $ |
ID:(572, 0)
Enthalpy and its relation of Maxwell
Concept
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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Equations
$ dH = DH_{S,p} dS + DH_{p,S} dp $
dH = DH_Sp * dS + DH_pS * dp
$ dH = T dS + V dp $
dH = T * dS + V * dp
$ DH_{p,S} = V $
DH_pS = V
$ DH_{S,p} = T $
DH_Sp = T
$ DT_{p,S} = DV_{S,p} $
DT_pS = DV_Sp
$ H = T S $
H = T * S
$ H = U + p V $
H = U + p * V
$ U = T S - p V $
U = T * S - p * V
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Internal Energy
Equation
The internal energy ($U$) is with the absolute temperature ($T$), the pressure ($p$), the entropy ($S$) and the volume ($V$) equal to:
| $ U = T S - p V $ |
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 ($dV$), is expressed as:
| $ dU = T dS - p dV $ |
Integrating this results in the following expression in terms of the internal energy ($U$), the entropy ($S$), and the volume ($V$):
| $ U = T S - p V $ |
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Enthalpy
Equation
The enthalpy ($H$) 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 $ |
ID:(3536, 0)
Enthalpy Ratio
Equation
The enthalpy ($H$) is reduced with the absolute temperature ($T$) and the entropy ($S$) to:
| $ H = T S $ |
The enthalpy ($H$) is defined using the internal energy ($U$), the pressure ($p$), and the volume ($V$) as:
| $ H = U + p V $ |
Considering the internal energy ($U$) as a function of the entropy ($S$) and the speed of the Oval Window ($v_2$) as:
| $ U = T S - p V $ |
it simplifies to:
| $ H = T S $ |
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Differential Enthalpy Relationship
Equation
The dependency of the differential enthalpy ($dH$) on the absolute temperature ($T$) and the entropy variation ($dS$), in addition to the volume ($V$) and the pressure Variation ($dp$), is given by:
| $ dH = T dS + V dp $ |
If we differentiate the definition of the enthalpy ($H$), which depends on the internal energy ($U$), the pressure ($p$), and the volume ($V$), given by:
| $ H = U + p V $ |
we obtain:
$dH = dU + Vdp + pdV$
using the differential enthalpy ($dH$), the internal energy differential ($dU$), the pressure Variation ($dp$), and the volume Variation ($dV$).
By differentiating the internal energy ($U$) with respect to the absolute temperature ($T$) and the entropy ($S$),
| $ U = T S - p V $ |
we get:
| $ dU = T dS - p dV $ |
with the internal energy differential ($dU$) and the entropy variation ($dS$).
Therefore, it finally results in:
| $ dH = T dS + V dp $ |
ID:(3473, 0)
Enthalpy and equation of state at constant pressure
Equation
Comparing the differential enthalpy ($dH$) it turns out that the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) is equal to the absolute temperature ($T$):
| $ DH_{S,p} = T $ |
The differential enthalpy ($dH$) is a function of the variations of the entropy variation ($dS$) and the pressure Variation ($dp$), 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 ($DH_{p,S}$), expressed as:
| $ dH = DH_{S,p} dS + DH_{p,S} dp $ |
Comparing this with the equation for the differential enthalpy ($dH$):
| $ dH = T dS + V dp $ |
we find that the slope of the partial derivative of enthalpy with respect to entropy at constant pressure ($DH_{S,p}$) with respect to the variation of the absolute temperature ($T$) is:
| $ DH_{S,p} = T $ |
ID:(3548, 0)
Enthalpy and equation of state at constant entropy
Equation
Comparing the differential enthalpy ($dH$) it turns out that the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$) is equal to the volume ($V$):
| $ DH_{p,S} = V $ |
The differential enthalpy ($dH$) is a function of the variations of the entropy variation ($dS$) and the pressure Variation ($dp$), 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 $ |
When compared to the equation for the differential enthalpy ($dH$):
| $ dH = T dS + V dp $ |
it follows that the slope of the partial derivative of enthalpy with respect to pressure at constant entropy ($DH_{p,S}$) with respect to the variation of the volume ($V$) is:
| $ DH_{p,S} = V $ |
ID:(3538, 0)
Enthalpy differential
Equation
The differential enthalpy ($dH$) is a function of the variations of the entropy variation ($dS$) and the pressure Variation ($dp$), 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 ($DH_{p,S}$), expressed as:
| $ dH = DH_{S,p} dS + DH_{p,S} dp $ |
Given that the enthalpy ($H$) depends on the entropy ($S$) and the pressure ($p$), the differential enthalpy ($dH$) can be calculated using:
$dH = \left(\displaystyle\frac{\partial H}{\partial S}\right)_p dS + \left(\displaystyle\frac{\partial H}{\partial p}\right)_S dp$
To simplify this expression, we introduce the notation for the derivative of the enthalpy ($H$) with respect to the entropy ($S$) while keeping the pressure ($p$) constant as:
$DH_{S,p} \equiv \left(\displaystyle\frac{\partial H}{\partial S}\right)_p$
and for the derivative of the enthalpy ($H$) with respect to the pressure ($p$) while keeping the entropy ($S$) constant as:
$DH_{p,S} \equiv \left(\displaystyle\frac{\partial H}{\partial p}\right)_S$
thus we can write:
| $ dH = DH_{S,p} dS + DH_{p,S} dp $ |
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Enthalpy and its relation of Maxwell
Equation
With the entropy ($S$), the volume ($V$), the absolute temperature ($T$) and the pressure ($p$) we obtain one of the so-called Maxwell relations:
| $ 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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