"Thermodynamic derivatives are at the very heart of teqp. All models are defined in the form $\\alpha^r(T, \\rho, z)$, where $\\rho$ is the molar density, and z are mole fractions. There are exceptions for models for which the independent variables are in simulation units (Lennard-Jones and its ilk).\n",
"\n",
"Thereofore, to obtain the residual pressure, it is obtained as a derivative:\n",
"\n",
"$$ p^r = \\rho R T \\left( \\rho \\left(\\frac{\\partial \\alpha^r}{\\partial \\rho}\\right)_{T}\\right)$$\n",
"\n",
"and other residual thermodynamic properties are defined likewise.\n",
"1.93 µs ± 32.9 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)\n",
"2.13 µs ± 36.7 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n"
]
}
],
"source": [
"%timeit model.get_Ar01(300,300,z)\n",
"%timeit model.get_Ar04n(300,300,z)"
]
},
{
"cell_type": "markdown",
"id": "d05e9070",
"metadata": {},
"source": [
"Note: calling overhead is usually on the order of 1 microsecond"
]
},
{
"cell_type": "markdown",
"id": "502e9830",
"metadata": {},
"source": [
"## Virial coefficients\n",
"\n",
"Virial coefficients represent the thermodynamics of the interaction of two-, three-, ... bodies interacting with each other. They can be obtained rigorously if the potential energy surface of interaction is fully known. In general, such a surface can only be constructed for small rigid molecules. Many simple thermodynamic models do a poor job of predicting the thermodynamics captured by the virial coefficients. \n",
"teqp supports the virial coefficient directly, there is the ``get_B2vir`` for the second virial coefficient:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "720e120c",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-0.00023661263734465424"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model.get_B2vir(300, z)"
]
},
{
"cell_type": "markdown",
"id": "2fba4369",
"metadata": {},
"source": [
"And the ``get_Bnvir`` method that allows for the calculation of higher virial coefficients:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "92d01992",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{2: -0.0002366126373446542,\n",
" 3: 3.001768410777936e-08,\n",
" 4: -3.2409760373816355e-12,\n",
" 5: 3.9617816466337214e-16,\n",
" 6: -4.552923983836698e-20,\n",
" 7: 5.3759278511184914e-24}"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model.get_Bnvir(7, 300, z)"
]
},
{
"cell_type": "markdown",
"id": "d50998b1",
"metadata": {},
"source": [
"The ``get_Bnvir`` function was implemented because when doing automatic differentiation, all the intermediate derivatives are also retained."
]
}
],
"metadata": {
...
...
%% Cell type:markdown id:1afc4517 tags:
# Thermodynamic Derivatives
## Helmholtz energy derivatives
Thermodynamic derivatives are at the very heart of teqp. All models are defined in the form $\alpha^r(T, \rho, z)$, where $\rho$ is the molar density, and z are mole fractions. There are exceptions for models for which the independent variables are in simulation units (Lennard-Jones and its ilk).
Thereofore, to obtain the residual pressure, it is obtained as a derivative:
$$ p^r = \rho R T \left( \rho \left(\frac{\partial \alpha^r}{\partial \rho}\right)_{T}\right)$$
and other residual thermodynamic properties are defined likewise.
1.93 µs ± 32.9 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
2.13 µs ± 36.7 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
%% Cell type:markdown id:d05e9070 tags:
Note: calling overhead is usually on the order of 1 microsecond
%% Cell type:markdown id:502e9830 tags:
## Virial coefficients
Virial coefficients represent the thermodynamics of the interaction of two-, three-, ... bodies interacting with each other. They can be obtained rigorously if the potential energy surface of interaction is fully known. In general, such a surface can only be constructed for small rigid molecules. Many simple thermodynamic models do a poor job of predicting the thermodynamics captured by the virial coefficients.
