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Example of application of the method
Definition 
If we consider the internal energy $U(V,S)$, it depends on two variables:
• The volume $V$
• The entropy $S$
Therefore, its variation can be expressed using the relationship:
| $ df = \left(\displaystyle\frac{\partial f }{\partial x }\right)_ y dx + \left(\displaystyle\frac{\partial f }{\partial y }\right)_ x dy $ |
in the form:
$dU = \left(\displaystyle\frac{\partial U }{\partial V }\right)_ S dV + \left(\displaystyle\frac{\partial U }{\partial S }\right)_ V dS$
According to the first law of thermodynamics, we know that the variation of internal energy $dU$ is equal to:
From this, we can conclude that the slopes are the pressure $p$:
$\left(\displaystyle\frac{\partial U }{\partial V }\right)_ S = -p$
and the temperature $T$:
$\left(\displaystyle\frac{\partial U }{\partial S }\right)_ V = T$
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Ejemplo de potencial termodinámico
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To establish the relationships, thermodynamic potentials are introduced, which are potential energies that include or exclude certain forms of energy in a system, such as the energy associated with work $pV$ and the energy associated with entropy $TS$, which cannot be used to perform work.
In the case of enthalpy $H$, it corresponds to the internal energy of the system, which includes the movement of particles, but also incorporates the energy required to form the system, i.e., the work $pV$ done to establish it. Therefore, it is defined as:
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Forma de trabajar en termodinámica
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to 
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then, select the variable: 
to 
Calculations
Variable
Given
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Equation
To be used
Equations
If we compare the differentiation of enthalpy
with its variation
we can conclude that
If we compare the differentiation of the enthalpy
with its variation
we can conclude that
If we compare the differentiation of the enthalpy
with its variation
we can conclude that
The adiabatic lapse rate, given by
can be expressed in terms of enthalpy using the relationship
and the relationship
as
$\Gamma =\displaystyle\frac{\partial T}{\partial p}=\displaystyle\frac{\partial}{\partial p}\displaystyle\frac{\partial h}{\partial s}=\displaystyle\frac{\partial}{\partial s}\displaystyle\frac{\partial h}{\partial p}=\displaystyle\frac{\partial \alpha}{\partial s}$
therefore, the adiabatic lapse rate is
Since the heat capacity at constant pressure is defined through enthalpy as
$c_p=\left(\displaystyle\frac{\partial h }{\partial T }\right)_{ p , i }$
we have
which implies
$c_p=\displaystyle\frac{\partial h}{\partial T}=\displaystyle\frac{\partial h}{\partial s}\displaystyle\frac{\partial s}{\partial T}=T\displaystyle\frac{\partial s}{\partial T}$
thus, we have the relationship
With the adiabatic lapse rate given by
we have
that the adiabatic lapse rate can be written as
$\Gamma=\displaystyle\frac{\partial \alpha }{\partial s }=\displaystyle\frac{\partial \alpha }{\partial T }\displaystyle\frac{\partial T }{\partial s }=\displaystyle\frac{ T }{ c_p }\displaystyle\frac{\partial \alpha }{\partial T }$
we have
With the definition of the specific volume
and the relationship for thermal expansion given by
the derivative of the specific volume with respect to the adiabatic lapse rate, expressed as
can be expressed as
Examples
Thermodynamics is the science of 'small steps', where one explores the behavior of a physical system by making variations on known functions
- The dependence of a function on parameters (e.g., $x$ and $y$) is determined, that is, $f(x, y)$.
- Each of these parameters is varied (e.g., $dx$ and $dy$), and the corresponding slope of the variation is identified.
- The aim is to find the relationship between the slope and the already established relationships within thermodynamics.
Mathematically, this is expressed as
The expression
$D_{x, y}f\equiv\left(\displaystyle\frac{\partial f }{\partial x }\right)_ y$
represents the slope in the x-direction with the other variables held constant (in this case, y). It is read as 'partial derivative of f with respect to x, with y held constant'.
In order to calculate the various parameters, it is necessary to be able to differentiate the Gibbs potential, which corresponds to the slopes of this function with respect to pressure or temperature.
In general, the Gibbs potential factors, denoted as $g_x$, are defined with $x$ representing the variable and $g$ representing the molar Gibbs free energy, as follows:
For the calculation of various parameters, it is necessary to be able to take second-order derivatives of the Gibbs potential, which corresponds to the curvatures of this function with respect to pressure and/or temperature.
In general, the factors of the Gibbs potential are defined as follows:
If we consider the internal energy $U(V,S)$, it depends on two variables:
• The volume $V$
• The entropy $S$
Therefore, its variation can be expressed using the relationship:
in the form:
$dU = \left(\displaystyle\frac{\partial U }{\partial V }\right)_ S dV + \left(\displaystyle\frac{\partial U }{\partial S }\right)_ V dS$
According to the first law of thermodynamics, we know that the variation of internal energy $dU$ is equal to:
From this, we can conclude that the slopes are the pressure $p$:
$\left(\displaystyle\frac{\partial U }{\partial V }\right)_ S = -p$
and the temperature $T$:
$\left(\displaystyle\frac{\partial U }{\partial S }\right)_ V = T$
To establish the relationships, thermodynamic potentials are introduced, which are potential energies that include or exclude certain forms of energy in a system, such as the energy associated with work $pV$ and the energy associated with entropy $TS$, which cannot be used to perform work.
In the case of enthalpy $H$, it corresponds to the internal energy of the system, which includes the movement of particles, but also incorporates the energy required to form the system, i.e., the work $pV$ done to establish it. Therefore, it is defined as:
In addition to the thermodynamic potential itself, its molar version can be defined by simply dividing its magnitude by the molar mass. In the case of enthalpy $H$, this is defined as
where $M_m$ is the molar mass.
The enthalpy depends on the pressure $p$, entropy $h$, and in our case, also on the salt concentration $i$. Therefore, the respective differences $dh$, $dp$, and $di$ must satisfy:
It has been determined that the molar enthalpy $h$ varies as a function of molar entropy $s$, pressure $p$, and salinity $i$ as follows:
The slope of the molar enthalpy $h$ with respect to entropy is equal to the temperature $T$:
The slope of the molar enthalpy $h$ with respect to entropy is equal to the pressure $p$:
The slope of the molar enthalpy $h$ with respect to entropy is equal to salinity $s$:
The stability of seawater is characterized by the so-called adiabatic lapse rate, which is directly related to the problem of temperature and salinity gradients that can destabilize the marine water column.
The adiabatic lapse rate is defined as:
The adiabatic lapse rate can be calculated using the effective volume $\alpha$ and the specific heat at constant pressure $c_p$ as follows:
The molar entropy varies with temperature according to the following relationship:
The adiabatic lapse rate can be calculated using the equation:
where $T$ is the temperature, $c_p$ is the specific heat capacity at constant pressure, and $\partial\alpha/\partial T$ is the variation of the relative volume with respect to temperature.
The adiabatic lapse rate can be calculated using the temperature $T$, the specific heat capacity at constant pressure $c_p$, the thermal expansion coefficient $k_T$, and the density $\rho$, as follows:
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