User:
Realistic models of the ocean equation of state
Storyboard 
Variables
Calculations
to 
,
then, select the variable: 
to 
Calculations
Variable
Given
Calculate
Target :
Equation
To be used
Equations
The equation for the specific Gibbs energy as a function of temperature and pressure, $g(T,p)$, can be expressed as a polynomial in the reduced temperature $\tau$ and reduced pressure $\pi$:
which can be written as:
To account for the specific Gibbs free energy of oceanic water, it is necessary to consider the portion of the specific Gibbs free energy that corresponds to the effect of salinity as a function of temperature and pressure, $g(T,p,i)$. This can be expressed as a polynomial in the reduced temperature $\tau$,
reduced pressure $\pi$,
and reduced salinity $\xi$,
which is calculated as follows:
The derivative of Gibbs free energy $G$ with respect to pressure $p$ is equal to the volume $V$:
If we divide the equation by mass, we obtain the same relationship but with the specific Gibbs free energy $g$ and density $\rho$:
$\displaystyle\frac{\partial g}{\partial p}=\displaystyle\frac{1}{M}\displaystyle\frac{\partial G}{\partial p}=\displaystyle\frac{V}{M}=\displaystyle\frac{1}{\rho}$
In other words,
The derivative of Gibbs free energy $G$ with respect to temperature $T$ is equal to minus the entropy $S$:
If we divide it by mass, we obtain the same relationship but with the molar Gibbs free energy $g$ and molar entropy $s$:
$\displaystyle\frac{\partial g}{\partial T}=\displaystyle\frac{1}{M}\displaystyle\frac{\partial G}{\partial T}=\displaystyle\frac{S}{M}=s$
In other words,
Since the Gibbs free energy is
we can rearrange it to obtain the enthalpy using
which gives us
$H = G + TS = G - T\displaystyle\frac{\partial G}{\partial T}$
If we divide the equation by the mass, we obtain the specific version:
Given that the Gibbs free energy is
we can solve for the internal energy using
and
which gives us
$U = G + TS + pV = G - T\displaystyle\frac{\partial G}{\partial T} - p\displaystyle\frac{\partial G}{\partial p}$
If we divide the equation by mass, we obtain the specific version:
Since the Gibbs free energy is
we can solve for the Helmholtz free energy using
which gives us
$F = G + pV = G - p\displaystyle\frac{\partial G}{\partial p}$
If we divide the equation by the mass, we obtain the specific version:
With the specific Gibbs energy of ocean water given by:
and its corresponding second derivative:
it is possible to estimate the specific heat capacity at constant pressure for a given temperature, pressure, and salinity. It is calculated using the following expression:
Since the coefficient of thermal expansion is defined as
we can calculate it using the relationship
The coefficient of thermal expansion can be calculated as
$k_T=\displaystyle\frac{1}{V}D_T V=\displaystyle\frac{D_{pT} G}{D_p G}$
If we multiply and divide the expression by the mass, we can convert the Gibbs free energy into the specific Gibbs free energy, and the relationship becomes
Since the compressibility coefficient is defined as
we can calculate the compressibility coefficient using the relationship
The compressibility coefficient can be calculated as
$k_p=\displaystyle\frac{1}{V}D_p V=\displaystyle\frac{D_{pp} G}{D_p G}$
If we multiply and divide the expression by the mass, we can convert the Gibbs free energy into the specific Gibbs free energy, and the relationship becomes
Since the haline contraction coefficient is defined by the equation:
and considering the relationship:
we can calculate the compressibility coefficient using:
$k_i=\displaystyle\frac{1}{\alpha}D_p \alpha=\displaystyle\frac{D_{ip} g}{D_p g}$
Therefore, we obtain:
Examples
The reduced temperature is a normalized scale calculated using two reference temperatures in order to obtain values between 0 and 1.
Therefore, using $T_0$ as a base temperature, $T_r$ as the temperature range, and $T$ as the temperature in question, the reduced temperature can be defined as:
The reduced pressure is a normalized scale calculated using two reference pressures in order to obtain values between 0 and 1.
Therefore, with $p_0$ as a base pressure, $p_r$ as the pressure range, and $p$ as the pressure, the reduced pressure can be defined as:
The reduced salinity is a normalized scale calculated using a reference salinity in order to obtain values around 1.
Therefore, using $i_r$ as the salinity range and $i$ as the salinity in question, the reduced salinity can be defined as:
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:
The density of Gibbs free energy, or specifically Gibbs free energy, as a function of temperature and pressure $g(T,p)$, can be expressed as a polynomial in reduced temperature $\tau$ and reduced pressure $\pi$, which is written as follows:
where $g_r$ is the value of the specific Gibbs free energy for the reference temperature and pressure.
To account for the specific Gibbs free energy of oceanic water, it is necessary to consider the portion of the Gibbs free energy that corresponds to the effect of salinity as a function of temperature and pressure, $g(T,p,i)$. This can be expressed as a polynomial in the reduced temperature $\tau$, reduced pressure $\pi$, and reduced salinity $\xi$, which is calculated as follows:
where $g_r$ is the value of the specific Gibbs free energy for the reference temperature and pressure.
The specific Gibbs free energy of the ocean can be calculated as the sum of the specific Gibbs free energy of water $g_w(T,p)$ and the specific Gibbs free energy of salt $g_i(T,p,i)$:
where the latter depends on the salinity $i$.
The derivative of Gibbs free energy $G$ with respect to pressure $p$ is equal to volume $V$. Therefore, dividing Gibbs free energy by mass gives us the specific Gibbs free energy $g$. Similarly, when we perform this operation with volume, we obtain the inverse of density $\rho$. Hence, the relationship between the derivative of Gibbs free energy and density is expressed as follows:
The derivative of Gibbs free energy $G$ with respect to temperature $T$ is equal to minus the entropy $S$. Therefore, if we divide Gibbs free energy $G$ by mass $M$, we obtain the specific Gibbs free energy $g$. Similarly, when we perform this operation with entropy, we obtain the specific entropy $s$. Hence, the relationship between the derivative of Gibbs free energy and entropy is expressed as follows:
The specific enthalpy $h$ can be calculated from the specific Gibbs free energy $g$ and its derivative $g_T$ using:
where $T$ is the temperature.
The specific internal energy $u$ can be calculated from the specific Gibbs free energy $g$ and its derivatives with respect to temperature $g_T$ and pressure $g_p$ using:
where $T$ is the temperature and $p$ is the pressure.
The specific Helmholtz free energy $f$ can be calculated from the specific Gibbs free energy $g$ and its derivative with respect to pressure $g_p$ using:
where $p$ is the pressure.
The second derivative of the molar Gibbs free energy with respect to temperature allows to calculate the specific heat capacity at constant pressure of ocean water using
The thermal expansion coefficient is calculated as the derivative of volume with respect to pressure, divided by the volume. Since volume is related to the derivative of Gibbs free energy with respect to pressure, we can show that:
Since the derivative of the molar Gibbs free energy $g$ with respect to pressure $p$ is equal to the volume $V$, we can show that the isothermal compressibility is given by:
Con la energ a especifica de Gibbs del agua oce nica con
y con la segunda derivada correspondiente con
con la primera derivada correspondiente con
se puede estimar la compresibilidad isotermal que existe para una temperatura, presi n y salinidad dadas. Con
Given that the derivative of the specific Gibbs free energy, denoted as $g$, with respect to pressure $p$ is equal to the inverse of density $\rho$, we can show that the haline contraction coefficient is equal to:
Con la energ a especifica de Gibbs del agua oce nica con
y con las derivadas correspondientes con
se puede estimar el potencial qu mico que existe para una temperatura, presi n y salinidad dadas. Con
Con la energ a especifica de Gibbs del agua oce nica con
y con la segunda derivada correspondiente con
con la primera derivada correspondiente con
se puede estimar la compresibilidad isotermal que existe para una temperatura, presi n y salinidad dadas. Con
Con la energ a especifica de Gibbs del agua oce nica con
y con la primera derivada correspondiente con
se puede estimar la compresibilidad isotermal que existe para una temperatura, presi n y salinidad dadas. Con
ID:(1647, 0)