Differentials

Equation

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Thermodynamics is the science of 'small steps', where one explores the behavior of a physical system by making variations on known functions f. To do this:

- 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 :

$ df = \left(\displaystyle\frac{\partial f }{\partial x }\right)_ y dx + \left(\displaystyle\frac{\partial f }{\partial y }\right)_ x dy $

$df$
Diferencial de la función termodinámica
$-$
9381
$dx$
Diferencial de la primera variable termodinámica
$-$
9382
$dy$
Diferencial de la segunda variable termodinámica
$-$
9383
$f$
Función termodinámica
$-$
9380
$x$
Primera variable termodinámica
$-$
9354
$y$
Segunda variable termodinámica
$-$
9355



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'.

ID:(12388, 0)



Definition of the first derivative of the Gibbs potential

Equation

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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:

$ g_x =\displaystyle\frac{\partial g }{\partial x }$

$g$
Energía libre de Gibbs molar
$J/mol$
9356
$g_x$
Primera derivada de la energía libre de Gibbs molar
$J/mol$
9357
$x$
Primera variable termodinámica
$-$
9354

ID:(12356, 0)



Second derivative of the Gibbs potential

Equation

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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:

$ g_{xy} =\displaystyle\frac{\partial^2 g }{\partial x \partial y }$

$g$
Energía libre de Gibbs molar
$J/mol$
9356
$x$
Primera variable termodinámica
$-$
9354
$g_{xy}$
Segunda derivada de la energía libre de Gibbs molar
$J/mol$
9358
$y$
Segunda variable termodinámica
$-$
9355

ID:(12357, 0)



Example of application of the method

Description

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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 $T$:

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

ID:(12389, 0)



Ejemplo de potencial termodinámico

Description

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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:

ID:(12390, 0)



Molar enthalpy

Equation

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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

$ h = \displaystyle\frac{ H }{ M_{mol} }$

$H$
Entalpía
$J$
9384
$h$
Entalpía molar
$J/mol$
9374
$M_m$
Masa molar
$kg/mol$
9393

where $M_m$ is the molar mass.

ID:(12391, 0)



Molar enthalpy differentiation

Equation

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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:

$ dh = \left(\displaystyle\frac{\partial h }{\partial s }\right)_{ p , i } ds + \left(\displaystyle\frac{\partial h }{\partial p }\right)_{ p , i } dp + \left(\displaystyle\frac{\partial h }{\partial i }\right)_{ p , s } di $

$dh$
Variación de la entalpía molar
$J/mol$
9386
$ds$
Variación de la entropía
$J/mol K$
9387
$dp$
Variación de la presión
$Pa$
9388
$di$
Variación de la salinidad
$-$
9389
$Dh_s$
Variation of molar enthalpy with entropy
$K$
9928
$Dh_p$
Variation of molar enthalpy with pressure
$m^3/kg$
9927
$Dh_i$
Variation of molar enthalpy with salinity
$J/mol$
9929

ID:(12392, 0)



Molar enthalpy change

Equation

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It has been determined that the molar enthalpy $h$ varies as a function of molar entropy $s$, pressure $p$, and salinity $i$ as follows:

$ dh =T ds + \alpha dp + \mu di$

$\mu$
Pendiente de la salinidad
$J/mol$
9394
$T$
Temperatura
$K$
9343
$dh$
Variación de la entalpía molar
$J/mol$
9386
$ds$
Variación de la entropía
$J/mol K$
9387
$dp$
Variación de la presión
$Pa$
9388
$di$
Variación de la salinidad
$-$
9389
$\alpha$
Volumen especifico
$m^3/kg$
9396

ID:(12393, 0)



Slope of enthalpy in entropy

Equation

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The slope of the molar enthalpy $h$ with respect to entropy is equal to the temperature $T$:

$ T = \displaystyle\frac{\partial h }{\partial S }$

$T$
Temperatura
$K$
9343
$Dh_s$
Variation of molar enthalpy with entropy
$K$
9928

If we compare the differentiation of enthalpy

$ dh = \left(\displaystyle\frac{\partial h }{\partial s }\right)_{ p , i } ds + \left(\displaystyle\frac{\partial h }{\partial p }\right)_{ p , i } dp + \left(\displaystyle\frac{\partial h }{\partial i }\right)_{ p , s } di $



with its variation

$ dh =T ds + \alpha dp + \mu di$



we can conclude that

$ T = \displaystyle\frac{\partial h }{\partial S }$

ID:(12394, 0)



Pendiente de la presión

Equation

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The slope of the molar enthalpy $h$ with respect to entropy is equal to the pressure $p$:

$ \alpha = \displaystyle\frac{\partial h }{\partial p } $

$Dh_p$
Variation of molar enthalpy with pressure
$m^3/kg$
9927
$\alpha$
Volumen especifico
$m^3/kg$
9396

If we compare the differentiation of the enthalpy

$ dh = \left(\displaystyle\frac{\partial h }{\partial s }\right)_{ p , i } ds + \left(\displaystyle\frac{\partial h }{\partial p }\right)_{ p , i } dp + \left(\displaystyle\frac{\partial h }{\partial i }\right)_{ p , s } di $



with its variation

$ dh =T ds + \alpha dp + \mu di$



we can conclude that

$ \alpha = \displaystyle\frac{\partial h }{\partial p } $

ID:(12395, 0)



Slope of enthalpy in salinity

Equation

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The slope of the molar enthalpy $h$ with respect to entropy is equal to salinity $s$:

$ \mu = \displaystyle\frac{\partial h }{\partial i }$

$\mu$
Pendiente de la salinidad
$J/mol$
9394
$Dh_i$
Variation of molar enthalpy with salinity
$J/mol$
9929

If we compare the differentiation of the enthalpy

$ dh = \left(\displaystyle\frac{\partial h }{\partial s }\right)_{ p , i } ds + \left(\displaystyle\frac{\partial h }{\partial p }\right)_{ p , i } dp + \left(\displaystyle\frac{\partial h }{\partial i }\right)_{ p , s } di $



with its variation

$ dh =T ds + \alpha dp + \mu di$



we can conclude that

$ \mu = \displaystyle\frac{\partial h }{\partial i }$

.

ID:(12396, 0)



Adiabatic lapse rate

Equation

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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:

$ \Gamma \equiv \displaystyle\frac{\partial T }{\partial p }$

$\Gamma$
Tasa de lapso adiabático
$K/Pa$
9377
$DT_p$
Variation of temperature with pressure
$K/Pa$
9930

ID:(12397, 0)



Adiabatic lapse rate and enthalpy

Equation

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The adiabatic lapse rate can be calculated using the effective volume $\alpha$ and the specific heat at constant pressure $c_p$ as follows:

$ \Gamma = \displaystyle\frac{ \alpha }{ c_p }$

$c_p$
Capacidad calorica a presión constante
$J/kg K$
9395
$\Gamma$
Tasa de lapso adiabático
$K/Pa$
9377
$\alpha$
Volumen especifico
$m^3/kg$
9396

The adiabatic lapse rate, given by

$ \Gamma \equiv \displaystyle\frac{\partial T }{\partial p }$



can be expressed in terms of enthalpy using the relationship

$ T = \displaystyle\frac{\partial h }{\partial S }$



and the relationship

$ \alpha = \displaystyle\frac{\partial h }{\partial p } $



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

$ \Gamma = \displaystyle\frac{ \alpha }{ c_p }$

ID:(12398, 0)



Variation of entropy with temperature

Equation

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The molar entropy varies with temperature according to the following relationship:

$ \left(\displaystyle\frac{\partial s }{\partial T }\right)_{ p , i } = \displaystyle\frac{ c_p }{ T }$

$c_p$
Capacidad calorica a presión constante
$J/kg K$
9395
$Ds_T$
Variation of molar entropy with temperature
$J/mol K$
9932

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

$ T = \displaystyle\frac{\partial h }{\partial S }$



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

$ \left(\displaystyle\frac{\partial s }{\partial T }\right)_{ p , i } = \displaystyle\frac{ c_p }{ T }$

ID:(12399, 0)



Adiabatic lapse rate without entropy

Equation

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The adiabatic lapse rate can be calculated using the equation:

$ \Gamma = \displaystyle\frac{ T }{ c_p } \displaystyle\frac{ \partial \alpha }{ \partial T } $

$c_p$
Specific heat at constant pressure
$J/kg K$
9426
$\Gamma$
Tasa de lapso adiabático
$K/Pa$
9377
$T$
Temperatura
$K$
9343
$D\alpha_T$
Variation of specific volume with temperature
$m^3/kg K$
9931

With the adiabatic lapse rate given by

$ \Gamma = \displaystyle\frac{ \alpha }{ c_p }$



we have

$ \left(\displaystyle\frac{\partial s }{\partial T }\right)_{ p , i } = \displaystyle\frac{ c_p }{ T }$



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

$ \Gamma = \displaystyle\frac{ T }{ c_p } \displaystyle\frac{ \partial \alpha }{ \partial T } $

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.

ID:(12400, 0)



Adiabatic lapse rate and properties

Equation

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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:

$ \Gamma = \displaystyle\frac{ T }{ c_p }\displaystyle\frac{ k_T }{ \rho }$

$\rho$
Densidad
$kg/m^3$
9371
$c_p$
Specific heat at constant pressure
$J/kg K$
9426
$\Gamma$
Tasa de lapso adiabático
$K/Pa$
9377
$T$
Temperatura
$K$
9343
$k_T$
Thermic dilatation coefficient
$1/K$
9361

With the definition of the specific volume

$ \alpha = \displaystyle\frac{1}{ \rho }$



and the relationship for thermal expansion given by

$ k_T =\displaystyle\frac{1}{ \alpha }\left(\displaystyle\frac{ \partial\alpha }{ \partial T }\right)_{ p , S }$



the derivative of the specific volume with respect to the adiabatic lapse rate, expressed as

$ \Gamma = \displaystyle\frac{ T }{ c_p } \displaystyle\frac{ \partial \alpha }{ \partial T } $



can be expressed as

$ \Gamma = \displaystyle\frac{ T }{ c_p }\displaystyle\frac{ k_T }{ \rho }$

ID:(12401, 0)