Instruction:
Here you have a function in Python 3, explain what it does
def Cpm(self):
        r'''Molar heat capacity of the chemical at its current phase and
        temperature, in units of [J/mol/K].

        Utilizes the object oriented interfaces
        :obj:`thermo.heat_capacity.HeatCapacitySolid`,
        :obj:`thermo.heat_capacity.HeatCapacityLiquid`,
        and :obj:`thermo.heat_capacity.HeatCapacityGas` to perform the
        actual calculation of each property.

        Examples
        --------
        >>> Chemical('cubane').Cpm
        137.05489206785944
        >>> Chemical('ethylbenzene', T=550, P=3E6).Cpm
        294.18449553310046
        '''
        return phase_select_property(phase=self.phase, s=self.Cpsm, l=self.Cplm, g=self.Cpgm)

Response: A method to get the chemical system properties for a given phase.


Instruction:
Make a summary of the following Python 3 code
def rho(self):
        r'''Mass density of the chemical at its current phase and
        temperature and pressure, in units of [kg/m^3].

        Utilizes the object oriented interfaces
        :obj:`thermo.volume.VolumeSolid`,
        :obj:`thermo.volume.VolumeLiquid`,
        and :obj:`thermo.volume.VolumeGas` to perform the
        actual calculation of each property. Note that those interfaces provide
        output in units of m^3/mol.

        Examples
        --------
        >>> Chemical('decane', T=550, P=2E6).rho
        498.67008448640604
        '''
        return phase_select_property(phase=self.phase, s=self.rhos, l=self.rhol, g=self.rhog)

Response: Mass density of the chemical at its current phase and temperature and pressure.


Instruction:
Write a Python 3 function for
returning the Z value of the chemical.

Response:
def Z(self):
        r'''Compressibility factor of the chemical at its current phase and
        temperature and pressure, [dimensionless].

        Examples
        --------
        >>> Chemical('MTBE', T=900, P=1E-2).Z
        0.9999999999079768
        '''
        Vm = self.Vm
        if Vm:
            return Z(self.T, self.P, Vm)
        return None


Instruction:
Can you generate a brief explanation for the following Python 3 code
def SG(self):
        r'''Specific gravity of the chemical, [dimensionless].

        For gas-phase conditions, this is calculated at 15.6 °C (60 °F) and 1
        atm for the chemical and the reference fluid, air.
        For liquid and solid phase conditions, this is calculated based on a
        reference fluid of water at 4°C at 1 atm, but the with the liquid or
        solid chemical's density at the currently specified conditions.

        Examples
        --------
        >>> Chemical('MTBE').SG
        0.7428160596603596
        '''
        phase = self.phase
        if phase == 'l':
            return self.SGl
        elif phase == 's':
            return self.SGs
        elif phase == 'g':
            return self.SGg
        rho = self.rho
        if rho is not None:
            return SG(rho)
        return None

Response: Returns the SG object for the current state of the object.


Instruction:
Write a Python 3 function that can
return the isobaric expansion of the chemical at its current phase and temperature.

Response:
def isobaric_expansion(self):
        r'''Isobaric (constant-pressure) expansion of the chemical at its
        current phase and temperature, in units of [1/K].

        .. math::
            \beta = \frac{1}{V}\left(\frac{\partial V}{\partial T} \right)_P

        Examples
        --------
        Radical change  in value just above and below the critical temperature
        of water:

        >>> Chemical('water', T=647.1, P=22048320.0).isobaric_expansion
        0.34074205839222449

        >>> Chemical('water', T=647.2, P=22048320.0).isobaric_expansion
        0.18143324022215077
        '''
        return phase_select_property(phase=self.phase, l=self.isobaric_expansion_l, g=self.isobaric_expansion_g)


Instruction:
How would you explain what the following Python 3 function does
def mu(self):
        r'''Viscosity of the chemical at its current phase, temperature, and
        pressure in units of [Pa*s].

        Utilizes the object oriented interfaces
        :obj:`thermo.viscosity.ViscosityLiquid` and
        :obj:`thermo.viscosity.ViscosityGas` to perform the
        actual calculation of each property.

        Examples
        --------
        >>> Chemical('ethanol', T=300).mu
        0.001044526538460911
        >>> Chemical('ethanol', T=400).mu
        1.1853097849748217e-05
        '''
        return phase_select_property(phase=self.phase, l=self.mul, g=self.mug)

Response: Returns the mu of the chemical at the current phase temperature and pressure in units of [ Pa * s ).


Instruction:
Explain what the following Python 3 code does
def k(self):
        r'''Thermal conductivity of the chemical at its current phase,
        temperature, and pressure in units of [W/m/K].

        Utilizes the object oriented interfaces
        :obj:`thermo.thermal_conductivity.ThermalConductivityLiquid` and
        :obj:`thermo.thermal_conductivity.ThermalConductivityGas` to perform
        the actual calculation of each property.

        Examples
        --------
        >>> Chemical('ethanol', T=300).kl
        0.16313594741877802
        >>> Chemical('ethanol', T=400).kg
        0.026019924109310026
        '''
        return phase_select_property(phase=self.phase, s=None, l=self.kl, g=self.kg)

Response: Returns the key that identifies the chemical system in which the chemical is in the current phase.


Instruction:
Given the following Python 3 function, write the documentation
def nu(self):
        r'''Kinematic viscosity of the the chemical at its current temperature,
        pressure, and phase in units of [m^2/s].

        .. math::
            \nu = \frac{\mu}{\rho}

        Examples
        --------
        >>> Chemical('argon').nu
        1.3846930410865003e-05
        '''
        return phase_select_property(phase=self.phase, l=self.nul, g=self.nug)

Response: Return the nu of the chemical.


Instruction:
Here you have a function in Python 3, explain what it does
def Pr(self):
        r'''Prandtl number of the chemical at its current temperature,
        pressure, and phase; [dimensionless].

        .. math::
            Pr = \frac{C_p \mu}{k}

        Examples
        --------
        >>> Chemical('acetone').Pr
        4.183039103542709
        '''
        return phase_select_property(phase=self.phase, l=self.Prl, g=self.Prg)

Response: Return the number of the chemical at its current temperature pressure and phase.


Instruction:
Can you tell what is the following Python 3 function doing
def Poynting(self):
        r'''Poynting correction factor [dimensionless] for use in phase
        equilibria methods based on activity coefficients or other reference
        states. Performs the shortcut calculation assuming molar volume is
        independent of pressure.

        .. math::
            \text{Poy} =  \exp\left[\frac{V_l (P-P^{sat})}{RT}\right]

        The full calculation normally returns values very close to the
        approximate ones. This property is defined in terms of
        pure components only.

        Examples
        --------
        >>> Chemical('pentane', T=300, P=1E7).Poynting
        1.5743051250679803

        Notes
        -----
        The full equation shown below can be used as follows:

        .. math::
            \text{Poy} = \exp\left[\frac{\int_{P_i^{sat}}^P V_i^l dP}{RT}\right]

        >>> from scipy.integrate import quad
        >>> c = Chemical('pentane', T=300, P=1E7)
        >>> exp(quad(lambda P : c.VolumeLiquid(c.T, P), c.Psat, c.P)[0]/R/c.T)
        1.5821826990975127
        '''
        Vml, Psat = self.Vml, self.Psat
        if Vml and Psat:
            return exp(Vml*(self.P-Psat)/R/self.T)
        return None

Response: Calculates the Poynting correction factor for use in phase INDRA reference INDRA objects.


Instruction:
Can you generate a brief explanation for the following Python 3 code
def ITS90_68_difference(T):
    r'''Calculates the difference between ITS-90 and ITS-68 scales using a
    series of models listed in [1]_, [2]_, and [3]_.

    The temperature difference is given by the following equations:

    From 13.8 K to 73.15 K:

    .. math::
        T_{90} - T_{68} = a_0 + \sum_{i=1}^{12} a_i[(T_{90}/K-40)/40]^i

    From 83.8 K to 903.75 K:

    .. math::
        T_{90} - T_{68} = \sum_{i=1}^8 b_i[(T_{90}/K - 273.15)/630]^i

    From 903.75 K to 1337.33 K:

    .. math::
        T_{90} - T_{68} = \sum_{i=0}^5 c_i[T_{90}/^\circ C]^i

    Above 1337.33 K:

    .. math::
        T_{90} - T_{68} = -1.398\cdot 10^{-7}\left(\frac{T_{90}}{K}\right)^2

    Parameters
    ----------
    T : float
        Temperature, ITS-90, or approximately ITS-68 [K]

    Returns
    -------
    dT : float
        Temperature, difference between ITS-90 and ITS-68 at T [K]

    Notes
    -----
    The conversion is straightforward when T90 is known. Theoretically, the
    model should be solved numerically to convert the reverse way. However,
    according to [4]_, the difference is under 0.05 mK from 73.15 K to
    903.15 K, and under 0.26 mK up to 1337.33 K.

    For temperatures under 13.8 K, no conversion is performed.

    The first set of coefficients are:
    -0.005903, 0.008174, -0.061924, -0.193388, 1.490793, 1.252347, -9.835868,
    1.411912, 25.277595, -19.183815, -18.437089, 27.000895, -8.716324.

    The second set of coefficients are:
    0, -0.148759, -0.267408, 1.08076, 1.269056, -4.089591, -1.871251,
    7.438081, -3.536296.

    The third set of coefficients are:
    7.8687209E1, -4.7135991E-1, 1.0954715E-3, -1.2357884E-6, 6.7736583E-10,
    -1.4458081E-13.
    These last coefficients use the temperature in degrees Celcius. A slightly
    older model used the following coefficients but a different equation over
    the same range:
    -0.00317, -0.97737, 1.2559, 2.03295, -5.91887, -3.23561, 7.23364,
    5.04151. The model for these coefficients was:

    .. math::
        T_{90} - T_{68} = c_0 + \sum_{i=1}^7 c_i[(T_{90}/K - 1173.15)/300]^i

    For temperatures larger than several thousand K, the differences have no
    meaning and grows quadratically.

    Examples
    --------
    >>> ITS90_68_difference(1000.)
    0.01231818956580355

    References
    ----------
    .. [1] Bedford, R. E., G. Bonnier, H. Maas, and F. Pavese. "Techniques for
       Approximating the International Temperature Scale of 1990." Bureau
       International Des Poids et Mesures, Sfievres, 1990.
    .. [2] Wier, Ron D., and Robert N. Goldberg. "On the Conversion of
       Thermodynamic Properties to the Basis of the International Temperature
       Scale of 1990." The Journal of Chemical Thermodynamics 28, no. 3
       (March 1996): 261-76. doi:10.1006/jcht.1996.0026.
    .. [3] Goldberg, Robert N., and R. D. Weir. "Conversion of Temperatures
       and Thermodynamic Properties to the Basis of the International
       Temperature Scale of 1990 (Technical Report)." Pure and Applied
       Chemistry 64, no. 10 (1992): 1545-1562. doi:10.1351/pac199264101545.
    .. [4] Code10.info. "Conversions among International Temperature Scales."
       Accessed May 22, 2016. http://www.code10.info/index.php%3Foption%3Dcom_content%26view%3Darticle%26id%3D83:conversions-among-international-temperature-scales%26catid%3D60:temperature%26Itemid%3D83.
    '''
    ais = [-0.005903, 0.008174, -0.061924, -0.193388, 1.490793, 1.252347,
           -9.835868, 1.411912, 25.277595, -19.183815, -18.437089, 27.000895,
           -8.716324]
    bis = [0, -0.148759, -0.267408, 1.08076, 1.269056, -4.089591, -1.871251,
           7.438081, -3.536296]
#    cis = [-0.00317, -0.97737, 1.2559, 2.03295, -5.91887, -3.23561, 7.23364,
#           5.04151]
    new_cs = [7.8687209E1, -4.7135991E-1, 1.0954715E-3, -1.2357884E-6,
              6.7736583E-10, -1.4458081E-13]
    dT = 0
    if T < 13.8:
        dT = 0
    elif T >= 13.8 and T <= 73.15:
        for i in range(13):
            dT += ais[i]*((T - 40.)/40.)**i
    elif T > 73.15 and T < 83.8:
        dT = 0
    elif T >= 83.8 and T <= 903.75:
        for i in range(9):
            dT += bis[i]*((T - 273.15)/630.)**i
    elif T > 903.75 and T <= 1337.33:
        # Revised function exists, but does not match the tabulated data
        # for i in range(8):
        #    dT += cis[i]*((T - 1173.15)/300.)**i
        for i in range(6):
            dT += new_cs[i]*(T-273.15)**i
    elif T > 1337.33:
        dT = -1.398E-7*T**2

    return dT

Response: r A function that calculates the difference between ITS - 90 and ITS -68 at T.


Instruction:
Make a summary of the following Python 3 code
def GWP(CASRN, AvailableMethods=False, Method=None):
    r'''This function handles the retrieval of a chemical's Global Warming
    Potential, relative to CO2. Lookup is based on CASRNs. Will automatically
    select a data source to use if no Method is provided; returns None if the
    data is not available.

    Returns the GWP for the 100yr outlook by default.

    Parameters
    ----------
    CASRN : string
        CASRN [-]

    Returns
    -------
    GWP : float
        Global warming potential, [(impact/mass chemical)/(impact/mass CO2)]
    methods : list, only returned if AvailableMethods == True
        List of methods which can be used to obtain GWP with the
        given inputs

    Other Parameters
    ----------------
    Method : string, optional
        The method name to use. Accepted methods are IPCC (2007) 100yr',
        'IPCC (2007) 100yr-SAR', 'IPCC (2007) 20yr', and 'IPCC (2007) 500yr'.
        All valid values are also held in the list GWP_methods.
    AvailableMethods : bool, optional
        If True, function will determine which methods can be used to obtain
        the GWP for the desired chemical, and will return methods
        instead of the GWP

    Notes
    -----
    All data is from [1]_, the official source. Several chemicals are available
    in [1]_ are not included here as they do not have a CAS.
    Methods are 'IPCC (2007) 100yr', 'IPCC (2007) 100yr-SAR',
    'IPCC (2007) 20yr', and 'IPCC (2007) 500yr'.

    Examples
    --------
    Methane, 100-yr outlook

    >>> GWP(CASRN='74-82-8')
    25.0

    References
    ----------
    .. [1] IPCC. "2.10.2 Direct Global Warming Potentials - AR4 WGI Chapter 2:
       Changes in Atmospheric Constituents and in Radiative Forcing." 2007.
       https://www.ipcc.ch/publications_and_data/ar4/wg1/en/ch2s2-10-2.html.
    '''
    def list_methods():
        methods = []
        if CASRN in GWP_data.index:
            methods.append(IPCC100)
            if not pd.isnull(GWP_data.at[CASRN, 'SAR 100yr']):
                methods.append(IPCC100SAR)
            methods.append(IPCC20)
            methods.append(IPCC500)
        methods.append(NONE)
        return methods
    if AvailableMethods:
        return list_methods()
    if not Method:
        Method = list_methods()[0]

    if Method == IPCC100:
        return float(GWP_data.at[CASRN, '100yr GWP'])
    elif Method == IPCC100SAR:
        return float(GWP_data.at[CASRN, 'SAR 100yr'])
    elif Method == IPCC20:
        return float(GWP_data.at[CASRN, '20yr GWP'])
    elif Method == IPCC500:
        return float(GWP_data.at[CASRN, '500yr GWP'])
    elif Method == NONE:
        return None
    else:
        raise Exception('Failure in in function')

Response: r Returns the GWP for the given CASRN.


Instruction:
How would you explain what the following Python 3 function does
def ODP(CASRN, AvailableMethods=False, Method=None):
    r'''This function handles the retrieval of a chemical's Ozone Depletion
    Potential, relative to CFC-11 (trichlorofluoromethane). Lookup is based on
    CASRNs. Will automatically select a data source to use if no Method is
    provided; returns None if the data is not available.

    Returns the ODP of a chemical according to [2]_ when a method is not
    specified. If a range is provided in [2]_, the highest value is returned.

    Parameters
    ----------
    CASRN : string
        CASRN [-]

    Returns
    -------
    ODP : float or str
        Ozone Depletion potential, [(impact/mass chemical)/(impact/mass CFC-11)];
        if method selected has `string` in it, this will be returned as a
        string regardless of if a range is given or a number
    methods : list, only returned if AvailableMethods == True
        List of methods which can be used to obtain ODP with the
        given inputs

    Other Parameters
    ----------------
    Method : string, optional
        The method name to use. Accepted methods are 'ODP2 Max', 'ODP2 Min',
        'ODP2 string', 'ODP2 logarithmic average', and methods for older values
        are 'ODP1 Max', 'ODP1 Min', 'ODP1 string', and 'ODP1 logarithmic average'.
        All valid values are also held in the list ODP_methods.
    Method : string, optional
        A string for the method name to use, as defined by constants in
        ODP_methods
    AvailableMethods : bool, optional
        If True, function will determine which methods can be used to obtain
        the ODP for the desired chemical, and will return methods
        instead of the ODP

    Notes
    -----
    Values are tabulated only for a small number of halogenated hydrocarbons,
    responsible for the largest impact. The original values of ODP as defined
    in the Montreal Protocol are also available, as methods with the `ODP1`
    prefix.

    All values are somewhat emperical, as actual reaction rates of chemicals
    with ozone depend on temperature which depends on latitude, longitude,
    time of day, weather, and the concentrations of other pollutants.

    All data is from [1]_. Several mixtures listed in [1]_ are not included
    here as they are not pure species.
    Methods for values in [2]_ are 'ODP2 Max', 'ODP2 Min', 'ODP2 string',
    'ODP2 logarithmic average',  and methods for older values are 'ODP1 Max',
    'ODP1 Min', 'ODP1 string', and 'ODP1 logarithmic average'.

    Examples
    --------
    Dichlorotetrafluoroethane, according to [2]_.

    >>> ODP(CASRN='76-14-2')
    0.58

    References
    ----------
    .. [1] US EPA, OAR. "Ozone-Depleting Substances." Accessed April 26, 2016.
       https://www.epa.gov/ozone-layer-protection/ozone-depleting-substances.
    .. [2] WMO (World Meteorological Organization), 2011: Scientific Assessment
       of Ozone Depletion: 2010. Global Ozone Research and Monitoring
       Project-Report No. 52, Geneva, Switzerland, 516 p.
       https://www.wmo.int/pages/prog/arep/gaw/ozone_2010/documents/Ozone-Assessment-2010-complete.pdf
    '''
    def list_methods():
        methods = []
        if CASRN in ODP_data.index:
            if not pd.isnull(ODP_data.at[CASRN, 'ODP2 Max']):
                methods.append(ODP2MAX)
            if not pd.isnull(ODP_data.at[CASRN, 'ODP1 Max']):
                methods.append(ODP1MAX)
            if not pd.isnull(ODP_data.at[CASRN, 'ODP2 Design']):
                methods.append(ODP2LOG)
            if not pd.isnull(ODP_data.at[CASRN, 'ODP1 Design']):
                methods.append(ODP1LOG)
            if not pd.isnull(ODP_data.at[CASRN, 'ODP2 Min']):
                methods.append(ODP2MIN)
            if not pd.isnull(ODP_data.at[CASRN, 'ODP1 Min']):
                methods.append(ODP1MIN)
            if not pd.isnull(ODP_data.at[CASRN, 'ODP2']):
                methods.append(ODP2STR)
            if not pd.isnull(ODP_data.at[CASRN, 'ODP1']):
                methods.append(ODP1STR)
        methods.append(NONE)
        return methods
    if AvailableMethods:
        return list_methods()
    if not Method:
        Method = list_methods()[0]

    if Method == ODP2MAX:
        return float(ODP_data.at[CASRN, 'ODP2 Max'])
    elif Method == ODP1MAX:
        return float(ODP_data.at[CASRN, 'ODP1 Max'])
    elif Method == ODP2MIN:
        return float(ODP_data.at[CASRN, 'ODP2 Min'])
    elif Method == ODP1MIN:
        return float(ODP_data.at[CASRN, 'ODP1 Min'])
    elif Method == ODP2LOG:
        return float(ODP_data.at[CASRN, 'ODP2 Design'])
    elif Method == ODP1LOG:
        return float(ODP_data.at[CASRN, 'ODP1 Design'])
    elif Method == ODP2STR:
        return str(ODP_data.at[CASRN, 'ODP2'])
    elif Method == ODP1STR:
        return str(ODP_data.at[CASRN, 'ODP1'])
    elif Method == NONE:
        return None
    else:
        raise Exception('Failure in in function')

Response: r Returns the ODP of a chemical with the given CASRN.


Instruction:
Can you generate the documentation for the following Python 3 function
def TRC_Antoine_extended(T, Tc, to, A, B, C, n, E, F):
    r'''Calculates vapor pressure of a chemical using the TRC Extended Antoine
    equation. Parameters are chemical dependent, and said to be from the
    Thermodynamics Research Center (TRC) at Texas A&M. Coefficients for various
    chemicals can be found in [1]_.

    .. math::
        \log_{10} P^{sat} = A - \frac{B}{T + C} + 0.43429x^n + Ex^8 + Fx^{12}

        x = \max \left(\frac{T-t_o-273.15}{T_c}, 0 \right)

    Parameters
    ----------
    T : float
        Temperature of fluid, [K]
    A, B, C, n, E, F : floats
        Regressed coefficients for the Antoine Extended (TRC) equation,
        specific for each chemical, [-]

    Returns
    -------
    Psat : float
        Vapor pressure calculated with coefficients [Pa]

    Notes
    -----
    Assumes coefficients are for calculating vapor pressure in Pascal.
    Coefficients should be consistent with input temperatures in Kelvin;

    Examples
    --------
    Tetrafluoromethane, coefficients from [1]_, at 180 K:

    >>> TRC_Antoine_extended(180.0, 227.51, -120., 8.95894, 510.595, -15.95,
    ... 2.41377, -93.74, 7425.9)
    706317.0898414153

    References
    ----------
    .. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
       New York: McGraw-Hill Professional, 2000.
    '''
    x = max((T - to - 273.15)/Tc, 0.)
    return 10.**(A - B/(T+C) + 0.43429*x**n + E*x**8 + F*x**12)

Response: r Calculates the vapor pressure of a Tetra - Fluoromethane in the Antoine Extended system.


Instruction:
Given the following Python 3 function, write the documentation
def Wagner_original(T, Tc, Pc, a, b, c, d):
    r'''Calculates vapor pressure using the Wagner equation (3, 6 form).

    Requires critical temperature and pressure as well as four coefficients
    specific to each chemical.

    .. math::
        \ln P^{sat}= \ln P_c + \frac{a\tau + b \tau^{1.5} + c\tau^3 + d\tau^6}
        {T_r}

        \tau = 1 - \frac{T}{T_c}

    Parameters
    ----------
    T : float
        Temperature of fluid, [K]
    Tc : float
        Critical temperature, [K]
    Pc : float
        Critical pressure, [Pa]
    a, b, c, d : floats
        Parameters for wagner equation. Specific to each chemical. [-]

    Returns
    -------
    Psat : float
        Vapor pressure at T [Pa]

    Notes
    -----
    Warning: Pc is often treated as adjustable constant.

    Examples
    --------
    Methane, coefficients from [2]_, at 100 K.

    >>> Wagner_original(100.0, 190.53, 4596420., a=-6.00435, b=1.1885,
    ... c=-0.834082, d=-1.22833)
    34520.44601450496

    References
    ----------
    .. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
       New York: McGraw-Hill Professional, 2000.
    .. [2] McGarry, Jack. "Correlation and Prediction of the Vapor Pressures of
       Pure Liquids over Large Pressure Ranges." Industrial & Engineering
       Chemistry Process Design and Development 22, no. 2 (April 1, 1983):
       313-22. doi:10.1021/i200021a023.
    '''
    Tr = T/Tc
    tau = 1.0 - Tr
    return Pc*exp((a*tau + b*tau**1.5 + c*tau**3 + d*tau**6)/Tr)

Response: A function that calculates the vapor pressure using the Wagner equation.


Instruction:
Given the following Python 3 function, write the documentation
def boiling_critical_relation(T, Tb, Tc, Pc):
    r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
    CSP relationship as in [1]_; requires a chemical's critical temperature
    and pressure as well as boiling point.

    The vapor pressure is given by:

    .. math::
        \ln P^{sat}_r = h\left( 1 - \frac{1}{T_r}\right)

        h = T_{br} \frac{\ln(P_c/101325)}{1-T_{br}}

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tb : float
        Boiling temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Pc : float
        Critical pressure of fluid [Pa]

    Returns
    -------
    Psat : float
        Vapor pressure at T [Pa]

    Notes
    -----
    Units are Pa. Formulation makes intuitive sense; a logarithmic form of
    interpolation.

    Examples
    --------
    Example as in [1]_ for ethylbenzene

    >>> boiling_critical_relation(347.2, 409.3, 617.1, 36E5)
    15209.467273093938

    References
    ----------
    .. [1] Reid, Robert C..; Prausnitz, John M.;; Poling, Bruce E.
       The Properties of Gases and Liquids. McGraw-Hill Companies, 1987.
    '''
    Tbr = Tb/Tc
    Tr = T/Tc
    h = Tbr*log(Pc/101325.)/(1 - Tbr)
    return exp(h*(1-1/Tr))*Pc

Response: r Calculates the vapor pressure of a single local system at arbitrary temperatures using a chemical s critical temperature and pressure as well as boiling point.


Instruction:
Make a summary of the following Python 3 code
def Ambrose_Walton(T, Tc, Pc, omega):
    r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
    CSP relationship by [1]_; requires a chemical's critical temperature and
    acentric factor.

    The vapor pressure is given by:

    .. math::
        \ln P_r=f^{(0)}+\omega f^{(1)}+\omega^2f^{(2)}

        f^{(0)}=\frac{-5.97616\tau + 1.29874\tau^{1.5}- 0.60394\tau^{2.5}
        -1.06841\tau^5}{T_r}

        f^{(1)}=\frac{-5.03365\tau + 1.11505\tau^{1.5}- 5.41217\tau^{2.5}
        -7.46628\tau^5}{T_r}

        f^{(2)}=\frac{-0.64771\tau + 2.41539\tau^{1.5}- 4.26979\tau^{2.5}
        +3.25259\tau^5}{T_r}

        \tau = 1-T_{r}

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Pc : float
        Critical pressure of fluid [Pa]
    omega : float
        Acentric factor [-]

    Returns
    -------
    Psat : float
        Vapor pressure at T [Pa]

    Notes
    -----
    Somewhat more accurate than the :obj:`Lee_Kesler` formulation.

    Examples
    --------
    Example from [2]_; ethylbenzene at 347.25 K.

    >>> Ambrose_Walton(347.25, 617.15, 36.09E5, 0.304)
    13278.878504306222

    References
    ----------
    .. [1] Ambrose, D., and J. Walton. "Vapour Pressures up to Their Critical
       Temperatures of Normal Alkanes and 1-Alkanols." Pure and Applied
       Chemistry 61, no. 8 (1989): 1395-1403. doi:10.1351/pac198961081395.
    .. [2] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
       New York: McGraw-Hill Professional, 2000.
    '''
    Tr = T/Tc
    tau = 1 - T/Tc
    f0 = (-5.97616*tau + 1.29874*tau**1.5 - 0.60394*tau**2.5 - 1.06841*tau**5)/Tr
    f1 = (-5.03365*tau + 1.11505*tau**1.5 - 5.41217*tau**2.5 - 7.46628*tau**5)/Tr
    f2 = (-0.64771*tau + 2.41539*tau**1.5 - 4.26979*tau**2.5 + 3.25259*tau**5)/Tr
    return Pc*exp(f0 + omega*f1 + omega**2*f2)

Response: A function that calculates the vapor pressure of a single class in a given temperature.


Instruction:
Explain what the following Python 3 code does
def Sanjari(T, Tc, Pc, omega):
    r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
    CSP relationship by [1]_. Requires a chemical's critical temperature,
    pressure, and acentric factor. Although developed for refrigerants,
    this model should have some general predictive ability.

    The vapor pressure of a chemical at `T` is given by:

    .. math::
        P^{sat} = P_c\exp(f^{(0)} + \omega f^{(1)} + \omega^2 f^{(2)})

        f^{(0)} = a_1 + \frac{a_2}{T_r} + a_3\ln T_r + a_4 T_r^{1.9}

        f^{(1)} = a_5 + \frac{a_6}{T_r} + a_7\ln T_r + a_8 T_r^{1.9}

        f^{(2)} = a_9 + \frac{a_{10}}{T_r} + a_{11}\ln T_r + a_{12} T_r^{1.9}

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Pc : float
        Critical pressure of fluid [Pa]
    omega : float
        Acentric factor [-]

    Returns
    -------
    Psat : float
        Vapor pressure, [Pa]

    Notes
    -----
    a[1-12] are as follows:
    6.83377, -5.76051, 0.90654, -1.16906,
    5.32034, -28.1460, -58.0352, 23.57466,
    18.19967, 16.33839, 65.6995, -35.9739.

    For a claimed fluid not included in the regression, R128, the claimed AARD
    was 0.428%. A re-calculation using 200 data points from 125.45 K to
    343.90225 K evenly spaced by 1.09775 K as generated by NIST Webbook April
    2016 produced an AARD of 0.644%. It is likely that the author's regression
    used more precision in its coefficients than was shown here. Nevertheless,
    the function is reproduced as shown in [1]_.

    For Tc=808 K, Pc=1100000 Pa, omega=1.1571, this function actually declines
    after 770 K.

    Examples
    --------
    >>> Sanjari(347.2, 617.1, 36E5, 0.299)
    13651.916109552498

    References
    ----------
    .. [1] Sanjari, Ehsan, Mehrdad Honarmand, Hamidreza Badihi, and Ali
       Ghaheri. "An Accurate Generalized Model for Predict Vapor Pressure of
       Refrigerants." International Journal of Refrigeration 36, no. 4
       (June 2013): 1327-32. doi:10.1016/j.ijrefrig.2013.01.007.
    '''
    Tr = T/Tc
    f0 = 6.83377 + -5.76051/Tr + 0.90654*log(Tr) + -1.16906*Tr**1.9
    f1 = 5.32034 + -28.1460/Tr + -58.0352*log(Tr) + 23.57466*Tr**1.9
    f2 = 18.19967 + 16.33839/Tr + 65.6995*log(Tr) + -35.9739*Tr**1.9
    return Pc*exp(f0 + omega*f1 + omega**2*f2)

Response: r Sanjari model for the base class.


Instruction:
How would you explain what the following Python 3 function does
def Edalat(T, Tc, Pc, omega):
    r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
    CSP relationship by [1]_. Requires a chemical's critical temperature,
    pressure, and acentric factor. Claimed to have a higher accuracy than the
    Lee-Kesler CSP relationship.

    The vapor pressure of a chemical at `T` is given by:

    .. math::
        \ln(P^{sat}/P_c) = \frac{a\tau + b\tau^{1.5} + c\tau^3 + d\tau^6}
        {1-\tau}

        a = -6.1559 - 4.0855\omega

        b = 1.5737 - 1.0540\omega - 4.4365\times 10^{-3} d

        c = -0.8747 - 7.8874\omega

        d = \frac{1}{-0.4893 - 0.9912\omega + 3.1551\omega^2}

        \tau = 1 - \frac{T}{T_c}

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Pc : float
        Critical pressure of fluid [Pa]
    omega : float
        Acentric factor [-]

    Returns
    -------
    Psat : float
        Vapor pressure, [Pa]

    Notes
    -----
    [1]_ found an average error of 6.06% on 94 compounds and 1106 data points.

    Examples
    --------
    >>> Edalat(347.2, 617.1, 36E5, 0.299)
    13461.273080743307

    References
    ----------
    .. [1] Edalat, M., R. B. Bozar-Jomehri, and G. A. Mansoori. "Generalized
       Equation Predicts Vapor Pressure of Hydrocarbons." Oil and Gas Journal;
       91:5 (February 1, 1993).
    '''
    tau = 1. - T/Tc
    a = -6.1559 - 4.0855*omega
    c = -0.8747 - 7.8874*omega
    d = 1./(-0.4893 - 0.9912*omega + 3.1551*omega**2)
    b = 1.5737 - 1.0540*omega - 4.4365E-3*d
    lnPr = (a*tau + b*tau**1.5 + c*tau**3 + d*tau**6)/(1.-tau)
    return exp(lnPr)*Pc

Response: r Returns a base class that calculates the vapor pressure of a chemical at arbitrary temperatures using the Lee - Kesler CSP relationship.


Instruction:
Can you generate a brief explanation for the following Python 3 code
def check_sufficient_inputs(self):
        '''Method to an exception if none of the pairs (T, P), (T, V), or
        (P, V) are given. '''
        if not ((self.T and self.P) or (self.T and self.V) or (self.P and self.V)):
            raise Exception('Either T and P, or T and V, or P and V are required')

Response: Method to an exception if neither T P or V are given.


Instruction:
Implement a Python 3 function for
solving for a single species.

Response:
def solve(self):
        '''First EOS-generic method; should be called by all specific EOSs.
        For solving for `T`, the EOS must provide the method `solve_T`.
        For all cases, the EOS must provide `a_alpha_and_derivatives`.
        Calls `set_from_PT` once done.
        '''
        self.check_sufficient_inputs()

        if self.V:
            if self.P:
                self.T = self.solve_T(self.P, self.V)
                self.a_alpha, self.da_alpha_dT, self.d2a_alpha_dT2 = self.a_alpha_and_derivatives(self.T)
            else:
                self.a_alpha, self.da_alpha_dT, self.d2a_alpha_dT2 = self.a_alpha_and_derivatives(self.T)
                self.P = R*self.T/(self.V-self.b) - self.a_alpha/(self.V*self.V + self.delta*self.V + self.epsilon)
            Vs = [self.V, 1j, 1j]
        else:
            self.a_alpha, self.da_alpha_dT, self.d2a_alpha_dT2 = self.a_alpha_and_derivatives(self.T)
            Vs = self.volume_solutions(self.T, self.P, self.b, self.delta, self.epsilon, self.a_alpha)
        self.set_from_PT(Vs)


Instruction:
How would you implement a function in Python 3 that
counts the number of real volumes in `Vs`, and determines what to do. If there is only one real volume, the method `set_properties_from_solution` is called with it. If there are two real volumes, `set_properties_from_solution` is called once with each volume. The phase is returned by `set_properties_from_solution`, and the volumes is set to either `V_l` or `V_g` as appropriate. Parameters ---------- Vs : list[float] Three possible molar volumes, [m^3/mol]

Response:
def set_from_PT(self, Vs):
        '''Counts the number of real volumes in `Vs`, and determines what to do.
        If there is only one real volume, the method
        `set_properties_from_solution` is called with it. If there are
        two real volumes, `set_properties_from_solution` is called once with
        each volume. The phase is returned by `set_properties_from_solution`,
        and the volumes is set to either `V_l` or `V_g` as appropriate.

        Parameters
        ----------
        Vs : list[float]
            Three possible molar volumes, [m^3/mol]
        '''
        # All roots will have some imaginary component; ignore them if > 1E-9
        good_roots = []
        bad_roots = []
        for i in Vs:
            j = i.real
            if abs(i.imag) > 1E-9 or j < 0:
                bad_roots.append(i)
            else:
                good_roots.append(j)

        if len(bad_roots) == 2:
            V = good_roots[0]
            self.phase = self.set_properties_from_solution(self.T, self.P, V, self.b, self.delta, self.epsilon, self.a_alpha, self.da_alpha_dT, self.d2a_alpha_dT2)
            if self.phase == 'l':
                self.V_l = V
            else:
                self.V_g = V
        else:
            # Even in the case of three real roots, it is still the min/max that make sense
            self.V_l, self.V_g = min(good_roots), max(good_roots)
            [self.set_properties_from_solution(self.T, self.P, V, self.b, self.delta, self.epsilon, self.a_alpha, self.da_alpha_dT, self.d2a_alpha_dT2) for V in [self.V_l, self.V_g]]
            self.phase = 'l/g'


Instruction:
Can you generate the documentation for the following Python 3 function
def set_properties_from_solution(self, T, P, V, b, delta, epsilon, a_alpha,
                                     da_alpha_dT, d2a_alpha_dT2, quick=True):
        r'''Sets all interesting properties which can be calculated from an
        EOS alone. Determines which phase the fluid is on its own; for details,
        see `phase_identification_parameter`.

        The list of properties set is as follows, with all properties suffixed
        with '_l' or '_g'.

        dP_dT, dP_dV, dV_dT, dV_dP, dT_dV, dT_dP, d2P_dT2, d2P_dV2, d2V_dT2,
        d2V_dP2, d2T_dV2, d2T_dP2, d2V_dPdT, d2P_dTdV, d2T_dPdV, H_dep, S_dep,
        beta, kappa, Cp_minus_Cv, V_dep, U_dep, G_dep, A_dep, fugacity, phi,
        and PIP.

        Parameters
        ----------
        T : float
            Temperature, [K]
        P : float
            Pressure, [Pa]
        V : float
            Molar volume, [m^3/mol]
        b : float
            Coefficient calculated by EOS-specific method, [m^3/mol]
        delta : float
            Coefficient calculated by EOS-specific method, [m^3/mol]
        epsilon : float
            Coefficient calculated by EOS-specific method, [m^6/mol^2]
        a_alpha : float
            Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]
        da_alpha_dT : float
            Temperature derivative of coefficient calculated by EOS-specific
            method, [J^2/mol^2/Pa/K]
        d2a_alpha_dT2 : float
            Second temperature derivative of coefficient calculated by
            EOS-specific method, [J^2/mol^2/Pa/K**2]
        quick : bool, optional
            Whether to use a SymPy cse-derived expression (3x faster) or
            individual formulas

        Returns
        -------
        phase : str
            Either 'l' or 'g'

        Notes
        -----
        The individual formulas for the derivatives and excess properties are
        as follows. For definitions of `beta`, see `isobaric_expansion`;
        for `kappa`, see isothermal_compressibility; for `Cp_minus_Cv`, see
        `Cp_minus_Cv`; for `phase_identification_parameter`, see
        `phase_identification_parameter`.

        First derivatives; in part using the Triple Product Rule [2]_, [3]_:

        .. math::
            \left(\frac{\partial P}{\partial T}\right)_V = \frac{R}{V - b}
            - \frac{a \frac{d \alpha{\left (T \right )}}{d T}}{V^{2} + V \delta
            + \epsilon}

            \left(\frac{\partial P}{\partial V}\right)_T = - \frac{R T}{\left(
            V - b\right)^{2}} - \frac{a \left(- 2 V - \delta\right) \alpha{
            \left (T \right )}}{\left(V^{2} + V \delta + \epsilon\right)^{2}}

            \left(\frac{\partial V}{\partial T}\right)_P =-\frac{
            \left(\frac{\partial P}{\partial T}\right)_V}{
            \left(\frac{\partial P}{\partial V}\right)_T}

            \left(\frac{\partial V}{\partial P}\right)_T =-\frac{
            \left(\frac{\partial V}{\partial T}\right)_P}{
            \left(\frac{\partial P}{\partial T}\right)_V}

            \left(\frac{\partial T}{\partial V}\right)_P = \frac{1}
            {\left(\frac{\partial V}{\partial T}\right)_P}

            \left(\frac{\partial T}{\partial P}\right)_V = \frac{1}
            {\left(\frac{\partial P}{\partial T}\right)_V}

        Second derivatives with respect to one variable; those of `T` and `V`
        use identities shown in [1]_ and verified numerically:

        .. math::
            \left(\frac{\partial^2  P}{\partial T^2}\right)_V =  - \frac{a
            \frac{d^{2} \alpha{\left (T \right )}}{d T^{2}}}{V^{2} + V \delta
            + \epsilon}

            \left(\frac{\partial^2  P}{\partial V^2}\right)_T = 2 \left(\frac{
            R T}{\left(V - b\right)^{3}} - \frac{a \left(2 V + \delta\right)^{
            2} \alpha{\left (T \right )}}{\left(V^{2} + V \delta + \epsilon
            \right)^{3}} + \frac{a \alpha{\left (T \right )}}{\left(V^{2} + V
            \delta + \epsilon\right)^{2}}\right)

            \left(\frac{\partial^2 T}{\partial P^2}\right)_V = -\left(\frac{
            \partial^2 P}{\partial T^2}\right)_V \left(\frac{\partial P}{
            \partial T}\right)^{-3}_V

            \left(\frac{\partial^2 V}{\partial P^2}\right)_T = -\left(\frac{
            \partial^2 P}{\partial V^2}\right)_T \left(\frac{\partial P}{
            \partial V}\right)^{-3}_T

            \left(\frac{\partial^2 T}{\partial V^2}\right)_P = -\left[
            \left(\frac{\partial^2 P}{\partial V^2}\right)_T
            \left(\frac{\partial P}{\partial T}\right)_V
            - \left(\frac{\partial P}{\partial V}\right)_T
            \left(\frac{\partial^2 P}{\partial T \partial V}\right) \right]
            \left(\frac{\partial P}{\partial T}\right)^{-2}_V
            + \left[\left(\frac{\partial^2 P}{\partial T\partial V}\right)
            \left(\frac{\partial P}{\partial T}\right)_V
            - \left(\frac{\partial P}{\partial V}\right)_T
            \left(\frac{\partial^2 P}{\partial T^2}\right)_V\right]
            \left(\frac{\partial P}{\partial T}\right)_V^{-3}
            \left(\frac{\partial P}{\partial V}\right)_T

            \left(\frac{\partial^2 V}{\partial T^2}\right)_P = -\left[
            \left(\frac{\partial^2 P}{\partial T^2}\right)_V
            \left(\frac{\partial P}{\partial V}\right)_T
            - \left(\frac{\partial P}{\partial T}\right)_V
            \left(\frac{\partial^2 P}{\partial T \partial V}\right) \right]
            \left(\frac{\partial P}{\partial V}\right)^{-2}_T
            + \left[\left(\frac{\partial^2 P}{\partial T\partial V}\right)
            \left(\frac{\partial P}{\partial V}\right)_T
            - \left(\frac{\partial P}{\partial T}\right)_V
            \left(\frac{\partial^2 P}{\partial V^2}\right)_T\right]
            \left(\frac{\partial P}{\partial V}\right)_T^{-3}
            \left(\frac{\partial P}{\partial T}\right)_V

        Second derivatives with respect to the other two variables; those of
        `T` and `V` use identities shown in [1]_ and verified numerically:

        .. math::
            \left(\frac{\partial^2 P}{\partial T \partial V}\right) = - \frac{
            R}{\left(V - b\right)^{2}} + \frac{a \left(2 V + \delta\right)
            \frac{d \alpha{\left (T \right )}}{d T}}{\left(V^{2} + V \delta
            + \epsilon\right)^{2}}

           \left(\frac{\partial^2 T}{\partial P\partial V}\right) =
            - \left[\left(\frac{\partial^2 P}{\partial T \partial V}\right)
            \left(\frac{\partial P}{\partial T}\right)_V
            - \left(\frac{\partial P}{\partial V}\right)_T
            \left(\frac{\partial^2 P}{\partial T^2}\right)_V
            \right]\left(\frac{\partial P}{\partial T}\right)_V^{-3}

            \left(\frac{\partial^2 V}{\partial T\partial P}\right) =
            - \left[\left(\frac{\partial^2 P}{\partial T \partial V}\right)
            \left(\frac{\partial P}{\partial V}\right)_T
            - \left(\frac{\partial P}{\partial T}\right)_V
            \left(\frac{\partial^2 P}{\partial V^2}\right)_T
            \right]\left(\frac{\partial P}{\partial V}\right)_T^{-3}

        Excess properties

        .. math::
            H_{dep} = \int_{\infty}^V \left[T\frac{\partial P}{\partial T}_V
            - P\right]dV + PV - RT= P V - R T + \frac{2}{\sqrt{
            \delta^{2} - 4 \epsilon}} \left(T a \frac{d \alpha{\left (T \right
            )}}{d T}  - a \alpha{\left (T \right )}\right) \operatorname{atanh}
            {\left (\frac{2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}}
            \right)}

            S_{dep} = \int_{\infty}^V\left[\frac{\partial P}{\partial T}
            - \frac{R}{V}\right] dV + R\log\frac{PV}{RT} = - R \log{\left (V
            \right )} + R \log{\left (\frac{P V}{R T} \right )} + R \log{\left
            (V - b \right )} + \frac{2 a \frac{d\alpha{\left (T \right )}}{d T}
            }{\sqrt{\delta^{2} - 4 \epsilon}} \operatorname{atanh}{\left (\frac
            {2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}} \right )}

            V_{dep} = V - \frac{RT}{P}

            U_{dep} = H_{dep} - P V_{dep}

            G_{dep} = H_{dep} - T S_{dep}

            A_{dep} = U_{dep} - T S_{dep}

            \text{fugacity} = P\exp\left(\frac{G_{dep}}{RT}\right)

            \phi = \frac{\text{fugacity}}{P}

            C_{v, dep} = T\int_\infty^V \left(\frac{\partial^2 P}{\partial
            T^2}\right) dV = - T a \left(\sqrt{\frac{1}{\delta^{2} - 4
            \epsilon}} \log{\left (V - \frac{\delta^{2}}{2} \sqrt{\frac{1}{
            \delta^{2} - 4 \epsilon}} + \frac{\delta}{2} + 2 \epsilon \sqrt{
            \frac{1}{\delta^{2} - 4 \epsilon}} \right )} - \sqrt{\frac{1}{
            \delta^{2} - 4 \epsilon}} \log{\left (V + \frac{\delta^{2}}{2}
            \sqrt{\frac{1}{\delta^{2} - 4 \epsilon}} + \frac{\delta}{2}
            - 2 \epsilon \sqrt{\frac{1}{\delta^{2} - 4 \epsilon}} \right )}
            \right) \frac{d^{2} \alpha{\left (T \right )} }{d T^{2}}

            C_{p, dep} = (C_p-C_v)_{\text{from EOS}} + C_{v, dep} - R

        References
        ----------
        .. [1] Thorade, Matthis, and Ali Saadat. "Partial Derivatives of
           Thermodynamic State Properties for Dynamic Simulation."
           Environmental Earth Sciences 70, no. 8 (April 10, 2013): 3497-3503.
           doi:10.1007/s12665-013-2394-z.
        .. [2] Poling, Bruce E. The Properties of Gases and Liquids. 5th
           edition. New York: McGraw-Hill Professional, 2000.
        .. [3] Walas, Stanley M. Phase Equilibria in Chemical Engineering.
           Butterworth-Heinemann, 1985.
        '''
        ([dP_dT, dP_dV, dV_dT, dV_dP, dT_dV, dT_dP],
            [d2P_dT2, d2P_dV2, d2V_dT2, d2V_dP2, d2T_dV2, d2T_dP2],
            [d2V_dPdT, d2P_dTdV, d2T_dPdV],
            [H_dep, S_dep, Cv_dep]) = self.derivatives_and_departures(T, P, V, b, delta, epsilon, a_alpha, da_alpha_dT, d2a_alpha_dT2, quick=quick)

        beta = dV_dT/V # isobaric_expansion(V, dV_dT)
        kappa = -dV_dP/V # isothermal_compressibility(V, dV_dP)
        Cp_m_Cv = -T*dP_dT*dP_dT/dP_dV # Cp_minus_Cv(T, dP_dT, dP_dV)

        Cp_dep = Cp_m_Cv + Cv_dep - R

        V_dep = (V - R*T/P)
        U_dep = H_dep - P*V_dep
        G_dep = H_dep - T*S_dep
        A_dep = U_dep - T*S_dep
        fugacity = P*exp(G_dep/(R*T))
        phi = fugacity/P

        PIP = V*(d2P_dTdV/dP_dT - d2P_dV2/dP_dV) # phase_identification_parameter(V, dP_dT, dP_dV, d2P_dV2, d2P_dTdV)
        phase = 'l' if PIP > 1 else 'g' # phase_identification_parameter_phase(PIP)

        if phase == 'l':
            self.Z_l = self.P*V/(R*self.T)
            self.beta_l, self.kappa_l = beta, kappa
            self.PIP_l, self.Cp_minus_Cv_l = PIP, Cp_m_Cv

            self.dP_dT_l, self.dP_dV_l, self.dV_dT_l = dP_dT, dP_dV, dV_dT
            self.dV_dP_l, self.dT_dV_l, self.dT_dP_l = dV_dP, dT_dV, dT_dP

            self.d2P_dT2_l, self.d2P_dV2_l = d2P_dT2, d2P_dV2
            self.d2V_dT2_l, self.d2V_dP2_l = d2V_dT2, d2V_dP2
            self.d2T_dV2_l, self.d2T_dP2_l = d2T_dV2, d2T_dP2

            self.d2V_dPdT_l, self.d2P_dTdV_l, self.d2T_dPdV_l = d2V_dPdT, d2P_dTdV, d2T_dPdV

            self.H_dep_l, self.S_dep_l, self.V_dep_l = H_dep, S_dep, V_dep,
            self.U_dep_l, self.G_dep_l, self.A_dep_l = U_dep, G_dep, A_dep,
            self.fugacity_l, self.phi_l = fugacity, phi
            self.Cp_dep_l, self.Cv_dep_l = Cp_dep, Cv_dep
        else:
            self.Z_g = self.P*V/(R*self.T)
            self.beta_g, self.kappa_g = beta, kappa
            self.PIP_g, self.Cp_minus_Cv_g = PIP, Cp_m_Cv

            self.dP_dT_g, self.dP_dV_g, self.dV_dT_g = dP_dT, dP_dV, dV_dT
            self.dV_dP_g, self.dT_dV_g, self.dT_dP_g = dV_dP, dT_dV, dT_dP

            self.d2P_dT2_g, self.d2P_dV2_g = d2P_dT2, d2P_dV2
            self.d2V_dT2_g, self.d2V_dP2_g = d2V_dT2, d2V_dP2
            self.d2T_dV2_g, self.d2T_dP2_g = d2T_dV2, d2T_dP2

            self.d2V_dPdT_g, self.d2P_dTdV_g, self.d2T_dPdV_g = d2V_dPdT, d2P_dTdV, d2T_dPdV

            self.H_dep_g, self.S_dep_g, self.V_dep_g = H_dep, S_dep, V_dep,
            self.U_dep_g, self.G_dep_g, self.A_dep_g = U_dep, G_dep, A_dep,
            self.fugacity_g, self.phi_g = fugacity, phi
            self.Cp_dep_g, self.Cv_dep_g = Cp_dep, Cv_dep
        return phase

Response: r Sets all interesting properties of a single entry in the system from the solution.


Instruction:
Explain what the following Python 3 code does
def dPsat_dT(self, T):
        r'''Generic method to calculate the temperature derivative of vapor
        pressure for a specified `T`. Implements the analytical derivative
        of the two polynomials described in `Psat`.

        As with `Psat`, results above the critical temperature are meaningless.
        The first-order polynomial which is used to calculate it under 0.32 Tc
        may not be physicall meaningful, due to there normally not being a
        volume solution to the EOS which can produce that low of a pressure.

        Parameters
        ----------
        T : float
            Temperature, [K]

        Returns
        -------
        dPsat_dT : float
            Derivative of vapor pressure with respect to temperature, [Pa/K]

        Notes
        -----
        There is a small step change at 0.32 Tc for all EOS due to the two
        switch between polynomials at that point.

        Useful for calculating enthalpy of vaporization with the Clausius
        Clapeyron Equation. Derived with SymPy's diff and cse.
        '''
        a_alphas = self.a_alpha_and_derivatives(T)
        alpha, d_alpha_dT = a_alphas[0]/self.a, a_alphas[1]/self.a
        Tr = T/self.Tc
        if Tr >= 0.32:
            c = self.Psat_coeffs
            x0 = alpha/T
            x1 = -self.Tc*x0 + 1
            x2 = c[0]*x1
            x3 = c[2] - x1*(c[1] - x2)
            x4 = c[3] - x1*x3
            x5 = c[4] - x1*x4
            x6 = c[5] - x1*x5
            x7 = c[6] - x1*x6
            x8 = c[7] - x1*x7
            x9 = c[8] - x1*x8
            return self.Pc*(-(d_alpha_dT - x0)*(-c[9] + x1*x9 + x1*(-x1*(-x1*(-x1*(-x1*(-x1*(-x1*(-x1*(c[1] - 2*x2) + x3) + x4) + x5) + x6) + x7) + x8) + x9)) + 1./self.Tc)*exp(c[10] - x1*(c[9] - x1*(c[8] - x1*(c[7] - x1*(c[6] - x1*(c[5] - x1*(c[4] - x1*(c[3] - x1*(c[2] + x1*(-c[1] + x2))))))))))
        else:
            c = self.Psat_coeffs_limiting
            return self.Pc*T*c[0]*(self.Tc*d_alpha_dT/T - self.Tc*alpha/(T*T))*exp(c[0]*(-1. + self.Tc*alpha/T) + c[1])/self.Tc + self.Pc*exp(c[0]*(-1. + self.Tc*alpha/T) + c[1])/self.Tc

Response: Generic method to calculate the temperature derivative of vapor edition for a specified temperature.


Instruction:
Can you tell what is the following Python 3 function doing
def V_g_sat(self, T):
        r'''Method to calculate molar volume of the vapor phase along the
        saturation line.

        Parameters
        ----------
        T : float
            Temperature, [K]

        Returns
        -------
        V_g_sat : float
            Gas molar volume along the saturation line, [m^3/mol]

        Notes
        -----
        Computers `Psat`, and then uses `volume_solutions` to obtain the three
        possible molar volumes. The highest value is returned.
        '''
        Psat = self.Psat(T)
        a_alpha = self.a_alpha_and_derivatives(T, full=False)
        Vs = self.volume_solutions(T, Psat, self.b, self.delta, self.epsilon, a_alpha)
        # Assume we can safely take the Vmax as gas, Vmin as l on the saturation line
        return max([i.real for i in Vs])

Response: r Method to calculate the gas molar volume of the vapor phase along the saturation line.


Instruction:
Make a summary of the following Python 3 code
def Heyen(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Heyen (1980) [1]_. Returns `a_alpha`, `da_alpha_dT`,
        and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives` for more
        documentation. Two coefficients needed.

        .. math::
            \alpha = e^{c_{1} \left(- \left(\frac{T}{Tc}\right)^{c_{2}}
            + 1\right)}

        References
        ----------
        .. [1] Heyen, G. Liquid and Vapor Properties from a Cubic Equation of
           State. In "Proceedings of the 2nd International Conference on Phase
           Equilibria and Fluid Properties in the Chemical Industry". DECHEMA:
           Frankfurt, 1980; p 9-13.
        '''
        c1, c2 = self.alpha_function_coeffs
        T, Tc, a = self.T, self.Tc, self.a
        a_alpha = a*exp(c1*(1 -(T/Tc)**c2))
        if not full:
            return a_alpha
        else:
            da_alpha_dT = -a*c1*c2*(T/Tc)**c2*exp(c1*(-(T/Tc)**c2 + 1))/T
            d2a_alpha_dT2 = a*c1*c2*(T/Tc)**c2*(c1*c2*(T/Tc)**c2 - c2 + 1)*exp(-c1*((T/Tc)**c2 - 1))/T**2
            return a_alpha, da_alpha_dT, d2a_alpha_dT2

Response: Method to calculate a_alpha da_alpha_dT d2a_alpha_dT2 based on Heyen method.


Instruction:
Given the following Python 3 function, write the documentation
def Yu_Lu(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Yu and Lu (1987) [1]_. Returns `a_alpha`,
        `da_alpha_dT`, and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives`
        for more documentation. Four coefficients needed.

        .. math::
            \alpha = 10^{c_{4} \left(- \frac{T}{Tc} + 1\right) \left(
            \frac{T^{2} c_{3}}{Tc^{2}} + \frac{T c_{2}}{Tc} + c_{1}\right)}

        References
        ----------
        .. [1] Yu, Jin-Min, and Benjamin C. -Y. Lu. "A Three-Parameter Cubic
           Equation of State for Asymmetric Mixture Density Calculations."
           Fluid Phase Equilibria 34, no. 1 (January 1, 1987): 1-19.
           doi:10.1016/0378-3812(87)85047-1.
        '''
        c1, c2, c3, c4 = self.alpha_function_coeffs
        T, Tc, a = self.T, self.Tc, self.a
        a_alpha = a*10**(c4*(-T/Tc + 1)*(T**2*c3/Tc**2 + T*c2/Tc + c1))
        if not full:
            return a_alpha
        else:
            da_alpha_dT = a*(10**(c4*(-T/Tc + 1)*(T**2*c3/Tc**2 + T*c2/Tc + c1))*(c4*(-T/Tc + 1)*(2*T*c3/Tc**2 + c2/Tc) - c4*(T**2*c3/Tc**2 + T*c2/Tc + c1)/Tc)*log(10))
            d2a_alpha_dT2 = a*(10**(-c4*(T/Tc - 1)*(T**2*c3/Tc**2 + T*c2/Tc + c1))*c4*(-4*T*c3/Tc - 2*c2 - 2*c3*(T/Tc - 1) + c4*(T**2*c3/Tc**2 + T*c2/Tc + c1 + (T/Tc - 1)*(2*T*c3/Tc + c2))**2*log(10))*log(10)/Tc**2)
            return a_alpha, da_alpha_dT, d2a_alpha_dT2

Response: Method to calculate a_alpha da_alpha_dT and d2a_alpha_dT2 according to Yu and Lu.


Instruction:
Can you generate a brief explanation for the following Python 3 code
def Trebble_Bishnoi(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Trebble and Bishnoi (1987) [1]_. Returns `a_alpha`,
        `da_alpha_dT`, and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives`
        for more documentation. One coefficient needed.

        .. math::
            \alpha = e^{c_{1} \left(- \frac{T}{Tc} + 1\right)}

        References
        ----------
        .. [1] Trebble, M. A., and P. R. Bishnoi. "Development of a New Four-
           Parameter Cubic Equation of State." Fluid Phase Equilibria 35, no. 1
           (September 1, 1987): 1-18. doi:10.1016/0378-3812(87)80001-8.
        '''
        c1 = self.alpha_function_coeffs
        T, Tc, a = self.T, self.Tc, self.a
        a_alpha = a*exp(c1*(-T/Tc + 1))
        if not full:
            return a_alpha
        else:
            da_alpha_dT = a*-c1*exp(c1*(-T/Tc + 1))/Tc
            d2a_alpha_dT2 = a*c1**2*exp(-c1*(T/Tc - 1))/Tc**2
            return a_alpha, da_alpha_dT, d2a_alpha_dT2

Response: Method to calculate a_alpha da_alpha_dT and d2a_alpha_dT2 according to Trebble and Bishnoi.


Instruction:
Can you generate the documentation for the following Python 3 function
def Almeida(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives according to Almeida et al. (1991) [1]_. Returns `a_alpha`,
        `da_alpha_dT`, and `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives`
        for more documentation. Three coefficients needed.

        .. math::
            \alpha = e^{c_{1} \left(- \frac{T}{Tc} + 1\right) \left|{
            \frac{T}{Tc} - 1}\right|^{c_{2} - 1} + c_{3} \left(-1
            + \frac{Tc}{T}\right)}

        References
        ----------
        .. [1] Almeida, G. S., M. Aznar, and A. S. Telles. "Uma Nova Forma de
           Dependência Com a Temperatura Do Termo Atrativo de Equações de
           Estado Cúbicas." RBE, Rev. Bras. Eng., Cad. Eng. Quim 8 (1991): 95.
        '''
        # Note: For the second derivative, requires the use a CAS which can
        # handle the assumption that Tr-1 != 0.
        c1, c2, c3 = self.alpha_function_coeffs
        T, Tc, a = self.T, self.Tc, self.a
        a_alpha = a*exp(c1*(-T/Tc + 1)*abs(T/Tc - 1)**(c2 - 1) + c3*(-1 + Tc/T))
        if not full:
            return a_alpha
        else:
            da_alpha_dT = a*((c1*(c2 - 1)*(-T/Tc + 1)*abs(T/Tc - 1)**(c2 - 1)*copysign(1, T/Tc - 1)/(Tc*Abs(T/Tc - 1)) - c1*abs(T/Tc - 1)**(c2 - 1)/Tc - Tc*c3/T**2)*exp(c1*(-T/Tc + 1)*abs(T/Tc - 1)**(c2 - 1) + c3*(-1 + Tc/T)))
            d2a_alpha_dT2 = a*exp(c3*(Tc/T - 1) - c1*abs(T/Tc - 1)**(c2 - 1)*(T/Tc - 1))*((c1*abs(T/Tc - 1)**(c2 - 1))/Tc + (Tc*c3)/T**2 + (c1*abs(T/Tc - 1)**(c2 - 2)*copysign(1, T/Tc - 1)*(c2 - 1)*(T/Tc - 1))/Tc)**2 - exp(c3*(Tc/T - 1) - c1*abs(T/Tc - 1)**(c2 - 1)*(T/Tc - 1))*((2*c1*abs(T/Tc - 1)**(c2 - 2)*copysign(1, T/Tc - 1)*(c2 - 1))/Tc**2 - (2*Tc*c3)/T**3 + (c1*abs(T/Tc - 1)**(c2 - 3)*copysign(1, T/Tc - 1)**2*(c2 - 1)*(c2 - 2)*(T/Tc - 1))/Tc**2)
            return a_alpha, da_alpha_dT, d2a_alpha_dT2

Response: Method to calculate a_alpha da_alpha_dT and d2a_alpha_dT2 according to Almeida et al.


Instruction:
How would you explain what the following Python 3 function does
def a_alpha_and_derivatives(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives for this EOS. Returns `a_alpha`, `da_alpha_dT`, and
        `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives` for more
        documentation. Uses the set values of `Tc`, `kappa`, and `a`.

        For use in `solve_T`, returns only `a_alpha` if full is False.

        .. math::
            a\alpha = a \left(\kappa \left(- \frac{T^{0.5}}{Tc^{0.5}}
            + 1\right) + 1\right)^{2}

            \frac{d a\alpha}{dT} = - \frac{1.0 a \kappa}{T^{0.5} Tc^{0.5}}
            \left(\kappa \left(- \frac{T^{0.5}}{Tc^{0.5}} + 1\right) + 1\right)

            \frac{d^2 a\alpha}{dT^2} = 0.5 a \kappa \left(- \frac{1}{T^{1.5}
            Tc^{0.5}} \left(\kappa \left(\frac{T^{0.5}}{Tc^{0.5}} - 1\right)
            - 1\right) + \frac{\kappa}{T^{1.0} Tc^{1.0}}\right)
        '''
        if not full:
            return self.a*(1 + self.kappa*(1-(T/self.Tc)**0.5))**2
        else:
            if quick:
                Tc, kappa = self.Tc, self.kappa
                x0 = T**0.5
                x1 = Tc**-0.5
                x2 = kappa*(x0*x1 - 1.) - 1.
                x3 = self.a*kappa

                a_alpha = self.a*x2*x2
                da_alpha_dT = x1*x2*x3/x0
                d2a_alpha_dT2 = x3*(-0.5*T**-1.5*x1*x2 + 0.5/(T*Tc)*kappa)
            else:
                a_alpha = self.a*(1 + self.kappa*(1-(T/self.Tc)**0.5))**2
                da_alpha_dT = -self.a*self.kappa*sqrt(T/self.Tc)*(self.kappa*(-sqrt(T/self.Tc) + 1.) + 1.)/T
                d2a_alpha_dT2 = self.a*self.kappa*(self.kappa/self.Tc - sqrt(T/self.Tc)*(self.kappa*(sqrt(T/self.Tc) - 1.) - 1.)/T)/(2.*T)
            return a_alpha, da_alpha_dT, d2a_alpha_dT2

Response: Method to calculate a_alpha and a_alpha_dT for a given T.


Instruction:
Here you have a function in Python 3, explain what it does
def solve_T(self, P, V, quick=True):
        r'''Method to calculate `T` from a specified `P` and `V` for the PRSV
        EOS. Uses `Tc`, `a`, `b`, `kappa0`  and `kappa` as well, obtained from
        the class's namespace.

        Parameters
        ----------
        P : float
            Pressure, [Pa]
        V : float
            Molar volume, [m^3/mol]
        quick : bool, optional
            Whether to use a SymPy cse-derived expression (somewhat faster) or
            individual formulas.

        Returns
        -------
        T : float
            Temperature, [K]

        Notes
        -----
        Not guaranteed to produce a solution. There are actually two solution,
        one much higher than normally desired; it is possible the solver could
        converge on this.
        '''
        Tc, a, b, kappa0, kappa1 = self.Tc, self.a, self.b, self.kappa0, self.kappa1
        if quick:
            x0 = V - b
            R_x0 = R/x0
            x3 = (100.*(V*(V + b) + b*x0))
            x4 = 10.*kappa0
            kappa110 = kappa1*10.
            kappa17 = kappa1*7.
            def to_solve(T):
                x1 = T/Tc
                x2 = x1**0.5
                return (T*R_x0 - a*((x4 - (kappa110*x1 - kappa17)*(x2 + 1.))*(x2 - 1.) - 10.)**2/x3) - P
        else:
            def to_solve(T):
                P_calc = R*T/(V - b) - a*((kappa0 + kappa1*(sqrt(T/Tc) + 1)*(-T/Tc + 7/10))*(-sqrt(T/Tc) + 1) + 1)**2/(V*(V + b) + b*(V - b))
                return P_calc - P
        return newton(to_solve, Tc*0.5)

Response: r Method to calculate T from a specified P and V.


Instruction:
Can you generate a brief explanation for the following Python 3 code
def a_alpha_and_derivatives(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives for this EOS. Returns `a_alpha`, `da_alpha_dT`, and
        `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives` for more
        documentation. Uses the set values of `Tc`, `kappa0`, `kappa1`, and
        `a`.

        For use in root-finding, returns only `a_alpha` if full is False.

        The `a_alpha` function is shown below; its first and second derivatives
        are long available through the SymPy expression under it.

        .. math::
            a\alpha = a \left(\left(\kappa_{0} + \kappa_{1} \left(\sqrt{\frac{
            T}{Tc}} + 1\right) \left(- \frac{T}{Tc} + \frac{7}{10}\right)
            \right) \left(- \sqrt{\frac{T}{Tc}} + 1\right) + 1\right)^{2}

        >>> from sympy import *
        >>> P, T, V = symbols('P, T, V')
        >>> Tc, Pc, omega = symbols('Tc, Pc, omega')
        >>> R, a, b, kappa0, kappa1 = symbols('R, a, b, kappa0, kappa1')
        >>> kappa = kappa0 + kappa1*(1 + sqrt(T/Tc))*(Rational(7, 10)-T/Tc)
        >>> a_alpha = a*(1 + kappa*(1-sqrt(T/Tc)))**2
        >>> # diff(a_alpha, T)
        >>> # diff(a_alpha, T, 2)
        '''
        Tc, a, kappa0, kappa1 = self.Tc, self.a, self.kappa0, self.kappa1
        if not full:
            return a*((kappa0 + kappa1*(sqrt(T/Tc) + 1)*(-T/Tc + 0.7))*(-sqrt(T/Tc) + 1) + 1)**2
        else:
            if quick:
                x1 = T/Tc
                x2 = x1**0.5
                x3 = x2 - 1.
                x4 = 10.*x1 - 7.
                x5 = x2 + 1.
                x6 = 10.*kappa0 - kappa1*x4*x5
                x7 = x3*x6
                x8 = x7*0.1 - 1.
                x10 = x6/T
                x11 = kappa1*x3
                x12 = x4/T
                x13 = 20./Tc*x5 + x12*x2
                x14 = -x10*x2 + x11*x13
                a_alpha = a*x8*x8
                da_alpha_dT = -a*x14*x8*0.1
                d2a_alpha_dT2 = a*(x14*x14 - x2/T*(x7 - 10.)*(2.*kappa1*x13 + x10 + x11*(40./Tc - x12)))/200.
            else:
                a_alpha = a*((kappa0 + kappa1*(sqrt(T/Tc) + 1)*(-T/Tc + 0.7))*(-sqrt(T/Tc) + 1) + 1)**2
                da_alpha_dT = a*((kappa0 + kappa1*(sqrt(T/Tc) + 1)*(-T/Tc + 0.7))*(-sqrt(T/Tc) + 1) + 1)*(2*(-sqrt(T/Tc) + 1)*(-kappa1*(sqrt(T/Tc) + 1)/Tc + kappa1*sqrt(T/Tc)*(-T/Tc + 0.7)/(2*T)) - sqrt(T/Tc)*(kappa0 + kappa1*(sqrt(T/Tc) + 1)*(-T/Tc + 0.7))/T)
                d2a_alpha_dT2 = a*((kappa1*(sqrt(T/Tc) - 1)*(20*(sqrt(T/Tc) + 1)/Tc + sqrt(T/Tc)*(10*T/Tc - 7)/T) - sqrt(T/Tc)*(10*kappa0 - kappa1*(sqrt(T/Tc) + 1)*(10*T/Tc - 7))/T)**2 - sqrt(T/Tc)*((10*kappa0 - kappa1*(sqrt(T/Tc) + 1)*(10*T/Tc - 7))*(sqrt(T/Tc) - 1) - 10)*(kappa1*(40/Tc - (10*T/Tc - 7)/T)*(sqrt(T/Tc) - 1) + 2*kappa1*(20*(sqrt(T/Tc) + 1)/Tc + sqrt(T/Tc)*(10*T/Tc - 7)/T) + (10*kappa0 - kappa1*(sqrt(T/Tc) + 1)*(10*T/Tc - 7))/T)/T)/200
            return a_alpha, da_alpha_dT, d2a_alpha_dT2

Response: r Method to calculate a_alpha and a_alpha_dT for a specific set of entry points.


Instruction:
Explain what the following Python 3 code does
def a_alpha_and_derivatives(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives for this EOS. Returns `a_alpha`, `da_alpha_dT`, and
        `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives` for more
        documentation. Uses the set values of `a`.

        .. math::
            a\alpha = a

            \frac{d a\alpha}{dT} = 0

            \frac{d^2 a\alpha}{dT^2} = 0
        '''
        if not full:
            return self.a
        else:
            a_alpha = self.a
            da_alpha_dT = 0.0
            d2a_alpha_dT2 = 0.0
            return a_alpha, da_alpha_dT, d2a_alpha_dT2

Response: r Method to calculate a_alpha da_alpha_dT and d2a_alpha_dT2 for this EOS.


Instruction:
Can you tell what is the following Python 3 function doing
def main_derivatives_and_departures(T, P, V, b, delta, epsilon, a_alpha,
                                        da_alpha_dT, d2a_alpha_dT2, quick=True):
        '''Re-implementation of derivatives and excess property calculations,
        as ZeroDivisionError errors occur with the general solution. The
        following derivation is the source of these formulas.

        >>> from sympy import *
        >>> P, T, V, R, b, a = symbols('P, T, V, R, b, a')
        >>> P_vdw = R*T/(V-b) - a/(V*V)
        >>> vdw = P_vdw - P
        >>>
        >>> dP_dT = diff(vdw, T)
        >>> dP_dV = diff(vdw, V)
        >>> d2P_dT2 = diff(vdw, T, 2)
        >>> d2P_dV2 = diff(vdw, V, 2)
        >>> d2P_dTdV = diff(vdw, T, V)
        >>> H_dep = integrate(T*dP_dT - P_vdw, (V, oo, V))
        >>> H_dep += P*V - R*T
        >>> S_dep = integrate(dP_dT - R/V, (V,oo,V))
        >>> S_dep += R*log(P*V/(R*T))
        >>> Cv_dep = T*integrate(d2P_dT2, (V,oo,V))
        >>>
        >>> dP_dT, dP_dV, d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, Cv_dep
        (R/(V - b), -R*T/(V - b)**2 + 2*a/V**3, 0, 2*(R*T/(V - b)**3 - 3*a/V**4), -R/(V - b)**2, P*V - R*T - a/V, R*(-log(V) + log(V - b)) + R*log(P*V/(R*T)), 0)
        '''
        dP_dT = R/(V - b)
        dP_dV = -R*T/(V - b)**2 + 2*a_alpha/V**3
        d2P_dT2 = 0
        d2P_dV2 = 2*(R*T/(V - b)**3 - 3*a_alpha/V**4)
        d2P_dTdV = -R/(V - b)**2
        H_dep = P*V - R*T - a_alpha/V
        S_dep = R*(-log(V) + log(V - b)) + R*log(P*V/(R*T))
        Cv_dep = 0
        return [dP_dT, dP_dV, d2P_dT2, d2P_dV2, d2P_dTdV, H_dep, S_dep, Cv_dep]

Response: Re - implementation of derivatives and excess property calculations and ZeroDivisionError errors occur with the general solution.


Instruction:
Make a summary of the following Python 3 code
def a_alpha_and_derivatives(self, T, full=True, quick=True):
        r'''Method to calculate `a_alpha` and its first and second
        derivatives for this EOS. Returns `a_alpha`, `da_alpha_dT`, and
        `d2a_alpha_dT2`. See `GCEOS.a_alpha_and_derivatives` for more
        documentation. Uses the set values of `Tc`, `m`, and `a`.

        .. math::
            a\alpha = a \left(m \left(- \sqrt{\frac{T}{Tc}} + 1\right)
            + 1\right)^{2}

            \frac{d a\alpha}{dT} = \frac{a m}{T} \sqrt{\frac{T}{Tc}} \left(m
            \left(\sqrt{\frac{T}{Tc}} - 1\right) - 1\right)

            \frac{d^2 a\alpha}{dT^2} = \frac{a m \sqrt{\frac{T}{Tc}}}{2 T^{2}}
            \left(m + 1\right)
        '''
        a, Tc, m = self.a, self.Tc, self.m
        sqTr = (T/Tc)**0.5
        a_alpha = a*(m*(1. - sqTr) + 1.)**2
        if not full:
            return a_alpha
        else:
            da_alpha_dT = -a*m*sqTr*(m*(-sqTr + 1.) + 1.)/T
            d2a_alpha_dT2 =  a*m*sqTr*(m + 1.)/(2.*T*T)
            return a_alpha, da_alpha_dT, d2a_alpha_dT2

Response: Method to calculate a_alpha da_alpha_dT and d2a_alpha_dT2 for this EOS.


Instruction:
Explain what the following Python 3 code does
def Tb(CASRN, AvailableMethods=False, Method=None, IgnoreMethods=[PSAT_DEFINITION]):
    r'''This function handles the retrieval of a chemical's boiling
    point. Lookup is based on CASRNs. Will automatically select a data
    source to use if no Method is provided; returns None if the data is not
    available.

    Prefered sources are 'CRC Physical Constants, organic' for organic
    chemicals, and 'CRC Physical Constants, inorganic' for inorganic
    chemicals. Function has data for approximately 13000 chemicals.

    Parameters
    ----------
    CASRN : string
        CASRN [-]

    Returns
    -------
    Tb : float
        Boiling temperature, [K]
    methods : list, only returned if AvailableMethods == True
        List of methods which can be used to obtain Tb with the given inputs

    Other Parameters
    ----------------
    Method : string, optional
        A string for the method name to use, as defined by constants in
        Tb_methods
    AvailableMethods : bool, optional
        If True, function will determine which methods can be used to obtain
        Tb for the desired chemical, and will return methods instead of Tb
    IgnoreMethods : list, optional
        A list of methods to ignore in obtaining the full list of methods,
        useful for for performance reasons and ignoring inaccurate methods

    Notes
    -----
    A total of four methods are available for this function. They are:

        * 'CRC_ORG', a compillation of data on organics
          as published in [1]_.
        * 'CRC_INORG', a compillation of data on
          inorganic as published in [1]_.
        * 'YAWS', a large compillation of data from a
          variety of sources; no data points are sourced in the work of [2]_.
        * 'PSAT_DEFINITION', calculation of boiling point from a
          vapor pressure calculation. This is normally off by a fraction of a
          degree even in the best cases. Listed in IgnoreMethods by default
          for performance reasons.

    Examples
    --------
    >>> Tb('7732-18-5')
    373.124

    References
    ----------
    .. [1] Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of
       Chemistry and Physics, 95E. Boca Raton, FL: CRC press, 2014.
    .. [2] Yaws, Carl L. Thermophysical Properties of Chemicals and
       Hydrocarbons, Second Edition. Amsterdam Boston: Gulf Professional
       Publishing, 2014.
    '''
    def list_methods():
        methods = []
        if CASRN in CRC_inorganic_data.index and not np.isnan(CRC_inorganic_data.at[CASRN, 'Tb']):
            methods.append(CRC_INORG)
        if CASRN in CRC_organic_data.index and not np.isnan(CRC_organic_data.at[CASRN, 'Tb']):
            methods.append(CRC_ORG)
        if CASRN in Yaws_data.index:
            methods.append(YAWS)
        if PSAT_DEFINITION not in IgnoreMethods:
            try:
                # For some chemicals, vapor pressure range will exclude Tb
                VaporPressure(CASRN=CASRN).solve_prop(101325.)
                methods.append(PSAT_DEFINITION)
            except:  # pragma: no cover
                pass
        if IgnoreMethods:
            for Method in IgnoreMethods:
                if Method in methods:
                    methods.remove(Method)
        methods.append(NONE)
        return methods
    if AvailableMethods:
        return list_methods()
    if not Method:
        Method = list_methods()[0]

    if Method == CRC_INORG:
        return float(CRC_inorganic_data.at[CASRN, 'Tb'])
    elif Method == CRC_ORG:
        return float(CRC_organic_data.at[CASRN, 'Tb'])
    elif Method == YAWS:
        return float(Yaws_data.at[CASRN, 'Tb'])
    elif Method == PSAT_DEFINITION:
        return VaporPressure(CASRN=CASRN).solve_prop(101325.)
    elif Method == NONE:
        return None
    else:
        raise Exception('Failure in in function')

Response: r Returns a new object with the boiling temperature for the given CASRN and methods.


Instruction:
Explain what the following Python 3 code does
def Clapeyron(T, Tc, Pc, dZ=1, Psat=101325):
    r'''Calculates enthalpy of vaporization at arbitrary temperatures using the
    Clapeyron equation.

    The enthalpy of vaporization is given by:

    .. math::
        \Delta H_{vap} = RT \Delta Z \frac{\ln (P_c/Psat)}{(1-T_{r})}

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Pc : float
        Critical pressure of fluid [Pa]
    dZ : float
        Change in compressibility factor between liquid and gas, []
    Psat : float
        Saturation pressure of fluid [Pa], optional

    Returns
    -------
    Hvap : float
        Enthalpy of vaporization, [J/mol]

    Notes
    -----
    No original source is available for this equation.
    [1]_ claims this equation overpredicts enthalpy by several percent.
    Under Tr = 0.8, dZ = 1 is a reasonable assumption.
    This equation is most accurate at the normal boiling point.

    Internal units are bar.

    WARNING: I believe it possible that the adjustment for pressure may be incorrect

    Examples
    --------
    Problem from Perry's examples.

    >>> Clapeyron(T=294.0, Tc=466.0, Pc=5.55E6)
    26512.354585061985

    References
    ----------
    .. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
       New York: McGraw-Hill Professional, 2000.
    '''
    Tr = T/Tc
    return R*T*dZ*log(Pc/Psat)/(1. - Tr)

Response: r Returns an enthalpy of vaporization at arbitrary temperatures using the Clapeyron equation.


Instruction:
Can you tell what is the following Python 3 function doing
def SMK(T, Tc, omega):
    r'''Calculates enthalpy of vaporization at arbitrary temperatures using a
    the work of [1]_; requires a chemical's critical temperature and
    acentric factor.

    The enthalpy of vaporization is given by:

    .. math::
         \frac{\Delta H_{vap}} {RT_c} =
         \left( \frac{\Delta H_{vap}} {RT_c} \right)^{(R1)} + \left(
         \frac{\omega - \omega^{(R1)}} {\omega^{(R2)} - \omega^{(R1)}} \right)
         \left[\left( \frac{\Delta H_{vap}} {RT_c} \right)^{(R2)} - \left(
         \frac{\Delta H_{vap}} {RT_c} \right)^{(R1)} \right]

        \left( \frac{\Delta H_{vap}} {RT_c} \right)^{(R1)}
        = 6.537 \tau^{1/3} - 2.467 \tau^{5/6} - 77.251 \tau^{1.208} +
        59.634 \tau + 36.009 \tau^2 - 14.606 \tau^3

        \left( \frac{\Delta H_{vap}} {RT_c} \right)^{(R2)} - \left(
        \frac{\Delta H_{vap}} {RT_c} \right)^{(R1)}=-0.133 \tau^{1/3} - 28.215
        \tau^{5/6} - 82.958 \tau^{1.208} + 99.00 \tau  + 19.105 \tau^2 -2.796 \tau^3

        \tau = 1-T/T_c

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    omega : float
        Acentric factor [-]

    Returns
    -------
    Hvap : float
        Enthalpy of vaporization, [J/mol]

    Notes
    -----
    The original article has been reviewed and found to have coefficients with
    slightly more precision. Additionally, the form of the equation is slightly
    different, but numerically equivalent.

    The refence fluids are:

    :math:`\omega_0` = benzene = 0.212

    :math:`\omega_1` = carbazole = 0.461

    A sample problem in the article has been verified. The numerical result
    presented by the author requires high numerical accuracy to obtain.

    Examples
    --------
    Problem in [1]_:

    >>> SMK(553.15, 751.35, 0.302)
    39866.17647797959

    References
    ----------
    .. [1] Sivaraman, Alwarappa, Joe W. Magee, and Riki Kobayashi. "Generalized
       Correlation of Latent Heats of Vaporization of Coal-Liquid Model Compounds
       between Their Freezing Points and Critical Points." Industrial &
       Engineering Chemistry Fundamentals 23, no. 1 (February 1, 1984): 97-100.
       doi:10.1021/i100013a017.
    '''
    omegaR1, omegaR2 = 0.212, 0.461
    A10 = 6.536924
    A20 = -2.466698
    A30 = -77.52141
    B10 = 59.63435
    B20 = 36.09887
    B30 = -14.60567

    A11 = -0.132584
    A21 = -28.21525
    A31 = -82.95820
    B11 = 99.00008
    B21 = 19.10458
    B31 = -2.795660

    tau = 1. - T/Tc
    L0 = A10*tau**(1/3.) + A20*tau**(5/6.) + A30*tau**(1-1/8. + 1/3.) + \
        B10*tau + B20*tau**2 + B30*tau**3

    L1 = A11*tau**(1/3.) + A21*tau**(5/6.0) + A31*tau**(1-1/8. + 1/3.) + \
        B11*tau + B21*tau**2 + B31*tau**3

    domega = (omega - omegaR1)/(omegaR2 - omegaR1)
    return R*Tc*(L0 + domega*L1)

Response: r Returns a string that represents the smack of the chemical s critical temperature and acentric factor of the chemical s vaporization at arbitrary temperatures.


Instruction:
How would you explain what the following Python 3 function does
def Velasco(T, Tc, omega):
    r'''Calculates enthalpy of vaporization at arbitrary temperatures using a
    the work of [1]_; requires a chemical's critical temperature and
    acentric factor.

    The enthalpy of vaporization is given by:

    .. math::
        \Delta_{vap} H = RT_c(7.2729 + 10.4962\omega + 0.6061\omega^2)(1-T_r)^{0.38}

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    omega : float
        Acentric factor [-]

    Returns
    -------
    Hvap : float
        Enthalpy of vaporization, [J/mol]

    Notes
    -----
    The original article has been reviewed. It is regressed from enthalpy of
    vaporization values at 0.7Tr, from 121 fluids in REFPROP 9.1.
    A value in the article was read to be similar, but slightly too low from
    that calculated here.

    Examples
    --------
    From graph, in [1]_ for perfluoro-n-heptane.

    >>> Velasco(333.2, 476.0, 0.5559)
    33299.41734936356

    References
    ----------
    .. [1] Velasco, S., M. J. Santos, and J. A. White. "Extended Corresponding
       States Expressions for the Changes in Enthalpy, Compressibility Factor
       and Constant-Volume Heat Capacity at Vaporization." The Journal of
       Chemical Thermodynamics 85 (June 2015): 68-76.
       doi:10.1016/j.jct.2015.01.011.
    '''
    return (7.2729 + 10.4962*omega + 0.6061*omega**2)*(1-T/Tc)**0.38*R*Tc

Response:
r Calculates the enthalpy of vaporization at arbitrary temperatures using a
    the work of the given chemical.


Instruction:
Here you have a function in Python 3, explain what it does
def Riedel(Tb, Tc, Pc):
    r'''Calculates enthalpy of vaporization at the boiling point, using the
    Ridel [1]_ CSP method. Required information are critical temperature
    and pressure, and boiling point. Equation taken from [2]_ and [3]_.

    The enthalpy of vaporization is given by:

    .. math::
        \Delta_{vap} H=1.093 T_b R\frac{\ln P_c-1.013}{0.930-T_{br}}

    Parameters
    ----------
    Tb : float
        Boiling temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Pc : float
        Critical pressure of fluid [Pa]

    Returns
    -------
    Hvap : float
        Enthalpy of vaporization at the normal boiling point, [J/mol]

    Notes
    -----
    This equation has no example calculation in any source. The source has not
    been verified. It is equation 4-144 in Perry's. Perry's also claims that
    errors seldom surpass 5%.

    [2]_ is the source of example work here, showing a calculation at 0.0%
    error.

    Internal units of pressure are bar.

    Examples
    --------
    Pyridine, 0.0% err vs. exp: 35090 J/mol; from Poling [2]_.

    >>> Riedel(388.4, 620.0, 56.3E5)
    35089.78989646058

    References
    ----------
    .. [1] Riedel, L. "Eine Neue Universelle Dampfdruckformel Untersuchungen
       Uber Eine Erweiterung Des Theorems Der Ubereinstimmenden Zustande. Teil
       I." Chemie Ingenieur Technik 26, no. 2 (February 1, 1954): 83-89.
       doi:10.1002/cite.330260206.
    .. [2] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
       New York: McGraw-Hill Professional, 2000.
    .. [3] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    '''
    Pc = Pc/1E5  # Pa to bar
    Tbr = Tb/Tc
    return 1.093*Tb*R*(log(Pc) - 1.013)/(0.93 - Tbr)

Response: r References of the enthalpy of vaporization at the boiling point.


Instruction:
Can you generate the documentation for the following Python 3 function
def Vetere(Tb, Tc, Pc, F=1):
    r'''Calculates enthalpy of vaporization at the boiling point, using the
    Vetere [1]_ CSP method. Required information are critical temperature
    and pressure, and boiling point. Equation taken from [2]_.

    The enthalpy of vaporization is given by:

    .. math::
        \frac {\Delta H_{vap}}{RT_b} = \frac{\tau_b^{0.38}
        \left[ \ln P_c - 0.513 + \frac{0.5066}{P_cT_{br}^2}\right]}
        {\tau_b + F(1-\tau_b^{0.38})\ln T_{br}}

    Parameters
    ----------
    Tb : float
        Boiling temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Pc : float
        Critical pressure of fluid [Pa]
    F : float, optional
        Constant for a fluid, [-]

    Returns
    -------
    Hvap : float
        Enthalpy of vaporization at the boiling point, [J/mol]

    Notes
    -----
    The equation cannot be found in the original source. It is believed that a
    second article is its source, or that DIPPR staff have altered the formulation.

    Internal units of pressure are bar.

    Examples
    --------
    Example as in [2]_, p2-487; exp: 25.73

    >>> Vetere(294.0, 466.0, 5.55E6)
    26363.430021286465

    References
    ----------
    .. [1] Vetere, Alessandro. "Methods to Predict the Vaporization Enthalpies
       at the Normal Boiling Temperature of Pure Compounds Revisited."
       Fluid Phase Equilibria 106, no. 1-2 (May 1, 1995): 1–10.
       doi:10.1016/0378-3812(94)02627-D.
    .. [2] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    '''
    Tbr = Tb/Tc
    taub = 1-Tb/Tc
    Pc = Pc/1E5
    term = taub**0.38*(log(Pc)-0.513 + 0.5066/Pc/Tbr**2) / (taub + F*(1-taub**0.38)*log(Tbr))
    return R*Tb*term

Response: r Returns an enthalpy of vaporization at the boiling point.


Instruction:
Can you create a Python 3 function that
calculates Watson vaporization of enthalpy for one temperature.

Response:
def Watson(T, Hvap_ref, T_Ref, Tc, exponent=0.38):
    '''
    Adjusts enthalpy of vaporization of enthalpy for another temperature, for one temperature.
    '''
    Tr = T/Tc
    Trefr = T_Ref/Tc
    H2 = Hvap_ref*((1-Tr)/(1-Trefr))**exponent
    return H2


Instruction:
How would you explain what the following Python 3 function does
def Hfus(T=298.15, P=101325, MW=None, AvailableMethods=False, Method=None, CASRN=''):  # pragma: no cover
    '''This function handles the calculation of a chemical's enthalpy of fusion.
    Generally this, is used by the chemical class, as all parameters are passed.
    Calling the function directly works okay.

    Enthalpy of fusion is a weak function of pressure, and its effects are
    neglected.

    This API is considered experimental, and is expected to be removed in a
    future release in favor of a more complete object-oriented interface.

    '''
    def list_methods():
        methods = []
        if CASRN in CRCHfus_data.index:
            methods.append('CRC, at melting point')
        methods.append('None')
        return methods
    if AvailableMethods:
        return list_methods()
    if not Method:
        Method = list_methods()[0]
    # This is the calculate, given the method section
    if Method == 'CRC, at melting point':
        _Hfus = CRCHfus_data.at[CASRN, 'Hfus']
    elif Method == 'None' or not MW:
        _Hfus = None
    else:
        raise Exception('Failure in in function')
    _Hfus = property_molar_to_mass(_Hfus, MW)
    return _Hfus

Response: This function calculates the Hfus of a chemical s enthalpy of fusion.


Instruction:
Can you tell what is the following Python 3 function doing
def load_all_methods(self):
        r'''Method which picks out coefficients for the specified chemical
        from the various dictionaries and DataFrames storing it. All data is
        stored as attributes. This method also sets :obj:`Tmin`, :obj:`Tmax`,
        and :obj:`all_methods` as a set of methods for which the data exists for.

        Called on initialization only. See the source code for the variables at
        which the coefficients are stored. The coefficients can safely be
        altered once the class is initialized. This method can be called again
        to reset the parameters.
        '''
        methods = []
        Tmins, Tmaxs = [], []
        if has_CoolProp and self.CASRN in coolprop_dict:
            methods.append(COOLPROP)
            self.CP_f = coolprop_fluids[self.CASRN]
            Tmins.append(self.CP_f.Tt); Tmaxs.append(self.CP_f.Tc)
        if self.CASRN in _VDISaturationDict:
            methods.append(VDI_TABULAR)
            Ts, props = VDI_tabular_data(self.CASRN, 'Hvap')
            self.VDI_Tmin = Ts[0]
            self.VDI_Tmax = Ts[-1]
            self.tabular_data[VDI_TABULAR] = (Ts, props)
            Tmins.append(self.VDI_Tmin); Tmaxs.append(self.VDI_Tmax)
        if self.CASRN in Alibakhshi_Cs.index and self.Tc:
            methods.append(ALIBAKHSHI)
            self.Alibakhshi_C = float(Alibakhshi_Cs.at[self.CASRN, 'C'])
            Tmaxs.append( max(self.Tc-100., 0) )
        if self.CASRN in CRCHvap_data.index and not np.isnan(CRCHvap_data.at[self.CASRN, 'HvapTb']):
            methods.append(CRC_HVAP_TB)
            self.CRC_HVAP_TB_Tb = float(CRCHvap_data.at[self.CASRN, 'Tb'])
            self.CRC_HVAP_TB_Hvap = float(CRCHvap_data.at[self.CASRN, 'HvapTb'])
        if self.CASRN in CRCHvap_data.index and not np.isnan(CRCHvap_data.at[self.CASRN, 'Hvap298']):
            methods.append(CRC_HVAP_298)
            self.CRC_HVAP_298 = float(CRCHvap_data.at[self.CASRN, 'Hvap298'])
        if self.CASRN in GharagheiziHvap_data.index:
            methods.append(GHARAGHEIZI_HVAP_298)
            self.GHARAGHEIZI_HVAP_298_Hvap = float(GharagheiziHvap_data.at[self.CASRN, 'Hvap298'])
        if all((self.Tc, self.omega)):
            methods.extend(self.CSP_methods)
            Tmaxs.append(self.Tc); Tmins.append(0)
        if all((self.Tc, self.Pc)):
            methods.append(CLAPEYRON)
            Tmaxs.append(self.Tc); Tmins.append(0)
        if all((self.Tb, self.Tc, self.Pc)):
            methods.extend(self.boiling_methods)
            Tmaxs.append(self.Tc); Tmins.append(0)
        if self.CASRN in Perrys2_150.index:
            methods.append(DIPPR_PERRY_8E)
            _, Tc, C1, C2, C3, C4, self.Perrys2_150_Tmin, self.Perrys2_150_Tmax = _Perrys2_150_values[Perrys2_150.index.get_loc(self.CASRN)].tolist()
            self.Perrys2_150_coeffs = [Tc, C1, C2, C3, C4]
            Tmins.append(self.Perrys2_150_Tmin); Tmaxs.append(self.Perrys2_150_Tmax)
        if self.CASRN in VDI_PPDS_4.index:
            _,  MW, Tc, A, B, C, D, E = _VDI_PPDS_4_values[VDI_PPDS_4.index.get_loc(self.CASRN)].tolist()
            self.VDI_PPDS_coeffs = [A, B, C, D, E]
            self.VDI_PPDS_Tc = Tc
            self.VDI_PPDS_MW = MW
            methods.append(VDI_PPDS)
            Tmaxs.append(self.VDI_PPDS_Tc);
        self.all_methods = set(methods)
        if Tmins and Tmaxs:
            self.Tmin, self.Tmax = min(Tmins), max(Tmaxs)

Response: r This method loads all methods for the specified chemicalical class and stores them as attributes.


Instruction:
Can you generate the documentation for the following Python 3 function
def solubility_eutectic(T, Tm, Hm, Cpl=0, Cps=0, gamma=1):
    r'''Returns the maximum solubility of a solute in a solvent.

    .. math::
        \ln x_i^L \gamma_i^L = \frac{\Delta H_{m,i}}{RT}\left(
        1 - \frac{T}{T_{m,i}}\right) - \frac{\Delta C_{p,i}(T_{m,i}-T)}{RT}
        + \frac{\Delta C_{p,i}}{R}\ln\frac{T_m}{T}

        \Delta C_{p,i} = C_{p,i}^L - C_{p,i}^S

    Parameters
    ----------
    T : float
        Temperature of the system [K]
    Tm : float
        Melting temperature of the solute [K]
    Hm : float
        Heat of melting at the melting temperature of the solute [J/mol]
    Cpl : float, optional
        Molar heat capacity of the solute as a liquid [J/mol/K]
    Cpls: float, optional
        Molar heat capacity of the solute as a solid [J/mol/K]
    gamma : float, optional
        Activity coefficient of the solute as a liquid [-]

    Returns
    -------
    x : float
        Mole fraction of solute at maximum solubility [-]

    Notes
    -----
    gamma is of the solute in liquid phase

    Examples
    --------
    From [1]_, matching example

    >>> solubility_eutectic(T=260., Tm=278.68, Hm=9952., Cpl=0, Cps=0, gamma=3.0176)
    0.24340068761677464

    References
    ----------
    .. [1] Gmehling, Jurgen. Chemical Thermodynamics: For Process Simulation.
       Weinheim, Germany: Wiley-VCH, 2012.
    '''
    dCp = Cpl-Cps
    x = exp(- Hm/R/T*(1-T/Tm) + dCp*(Tm-T)/R/T - dCp/R*log(Tm/T))/gamma
    return x

Response: r Returns the maximum solubility of a system in a liquid phase.


Instruction:
Can you tell what is the following Python 3 function doing
def Tm_depression_eutectic(Tm, Hm, x=None, M=None, MW=None):
    r'''Returns the freezing point depression caused by a solute in a solvent.
    Can use either the mole fraction of the solute or its molality and the
    molecular weight of the solvent. Assumes ideal system behavior.

    .. math::
        \Delta T_m = \frac{R T_m^2 x}{\Delta H_m}

        \Delta T_m = \frac{R T_m^2 (MW) M}{1000 \Delta H_m}

    Parameters
    ----------
    Tm : float
        Melting temperature of the solute [K]
    Hm : float
        Heat of melting at the melting temperature of the solute [J/mol]
    x : float, optional
        Mole fraction of the solute [-]
    M : float, optional
        Molality [mol/kg]
    MW: float, optional
        Molecular weight of the solvent [g/mol]

    Returns
    -------
    dTm : float
        Freezing point depression [K]

    Notes
    -----
    MW is the molecular weight of the solvent. M is the molality of the solute.

    Examples
    --------
    From [1]_, matching example.

    >>> Tm_depression_eutectic(353.35, 19110, .02)
    1.0864594900639515

    References
    ----------
    .. [1] Gmehling, Jurgen. Chemical Thermodynamics: For Process Simulation.
       Weinheim, Germany: Wiley-VCH, 2012.
    '''
    if x:
        dTm = R*Tm**2*x/Hm
    elif M and MW:
        MW = MW/1000. #g/mol to kg/mol
        dTm = R*Tm**2*MW*M/Hm
    else:
        raise Exception('Either molality or mole fraction of the solute must be specified; MW of the solvent is required also if molality is provided')
    return dTm

Response: r Returns the freezing point depression caused by a solute in a solvent.


Instruction:
Can you tell what is the following Python 3 function doing
def Yen_Woods_saturation(T, Tc, Vc, Zc):
    r'''Calculates saturation liquid volume, using the Yen and Woods [1]_ CSP
    method and a chemical's critical properties.

    The molar volume of a liquid is given by:

    .. math::
        Vc/Vs = 1 + A(1-T_r)^{1/3} + B(1-T_r)^{2/3} + D(1-T_r)^{4/3}

        D = 0.93-B

        A = 17.4425 - 214.578Z_c + 989.625Z_c^2 - 1522.06Z_c^3

        B = -3.28257 + 13.6377Z_c + 107.4844Z_c^2-384.211Z_c^3
        \text{ if } Zc \le 0.26

        B = 60.2091 - 402.063Z_c + 501.0 Z_c^2 + 641.0 Z_c^3
        \text{ if } Zc \ge 0.26

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Vc : float
        Critical volume of fluid [m^3/mol]
    Zc : float
        Critical compressibility of fluid, [-]

    Returns
    -------
    Vs : float
        Saturation liquid volume, [m^3/mol]

    Notes
    -----
    Original equation was in terms of density, but it is converted here.

    No example has been found, nor are there points in the article. However,
    it is believed correct. For compressed liquids with the Yen-Woods method,
    see the `YenWoods_compressed` function.

    Examples
    --------
    >>> Yen_Woods_saturation(300, 647.14, 55.45E-6, 0.245)
    1.7695330765295693e-05

    References
    ----------
    .. [1] Yen, Lewis C., and S. S. Woods. "A Generalized Equation for Computer
       Calculation of Liquid Densities." AIChE Journal 12, no. 1 (1966):
       95-99. doi:10.1002/aic.690120119
    '''
    Tr = T/Tc
    A = 17.4425 - 214.578*Zc + 989.625*Zc**2 - 1522.06*Zc**3
    if Zc <= 0.26:
        B = -3.28257 + 13.6377*Zc + 107.4844*Zc**2 - 384.211*Zc**3
    else:
        B = 60.2091 - 402.063*Zc + 501.0*Zc**2 + 641.0*Zc**3
    D = 0.93 - B
    Vm = Vc/(1 + A*(1-Tr)**(1/3.) + B*(1-Tr)**(2/3.) + D*(1-Tr)**(4/3.))
    return Vm

Response: r A function that calculates the saturation liquid volume using the Yen - Woods method.


Instruction:
Given the following Python 3 function, write the documentation
def Yamada_Gunn(T, Tc, Pc, omega):
    r'''Calculates saturation liquid volume, using Yamada and Gunn CSP method
    and a chemical's critical properties and acentric factor.

    The molar volume of a liquid is given by:

    .. math::
        V_s = \frac{RT_c}{P_c}{(0.29056-0.08775\omega)}^{[1+(1-{T/T_c})^{2/7}]}

    Units are in m^3/mol.

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Pc : float
        Critical pressure of fluid [Pa]
    omega : float
        Acentric factor for fluid, [-]

    Returns
    -------
    Vs : float
        saturation liquid volume, [m^3/mol]

    Notes
    -----
    This equation is an improvement on the Rackett equation.
    This is often presented as the Rackett equation.
    The acentric factor is used here, instead of the critical compressibility
    A variant using a reference fluid also exists

    Examples
    --------
    >>> Yamada_Gunn(300, 647.14, 22048320.0, 0.245)
    2.1882836429895796e-05

    References
    ----------
    .. [1] Gunn, R. D., and Tomoyoshi Yamada. "A Corresponding States
        Correlation of Saturated Liquid Volumes." AIChE Journal 17, no. 6
        (1971): 1341-45. doi:10.1002/aic.690170613
    .. [2] Yamada, Tomoyoshi, and Robert D. Gunn. "Saturated Liquid Molar
        Volumes. Rackett Equation." Journal of Chemical & Engineering Data 18,
        no. 2 (1973): 234-36. doi:10.1021/je60057a006
    '''
    return R*Tc/Pc*(0.29056 - 0.08775*omega)**(1 + (1 - T/Tc)**(2/7.))

Response: r Returns a base class for a given temperature Tc and pressure omega.


Instruction:
Make a summary of the following Python 3 code
def Townsend_Hales(T, Tc, Vc, omega):
    r'''Calculates saturation liquid density, using the Townsend and Hales
    CSP method as modified from the original Riedel equation. Uses
    chemical critical volume and temperature, as well as acentric factor

    The density of a liquid is given by:

    .. math::
        Vs = V_c/\left(1+0.85(1-T_r)+(1.692+0.986\omega)(1-T_r)^{1/3}\right)

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Vc : float
        Critical volume of fluid [m^3/mol]
    omega : float
        Acentric factor for fluid, [-]

    Returns
    -------
    Vs : float
        Saturation liquid volume, [m^3/mol]

    Notes
    -----
    The requirement for critical volume and acentric factor requires all data.

    Examples
    --------
    >>> Townsend_Hales(300, 647.14, 55.95E-6, 0.3449)
    1.8007361992619923e-05

    References
    ----------
    .. [1] Hales, J. L, and R Townsend. "Liquid Densities from 293 to 490 K of
       Nine Aromatic Hydrocarbons." The Journal of Chemical Thermodynamics
       4, no. 5 (1972): 763-72. doi:10.1016/0021-9614(72)90050-X
    '''
    Tr = T/Tc
    return Vc/(1 + 0.85*(1-Tr) + (1.692 + 0.986*omega)*(1-Tr)**(1/3.))

Response: r Calculates the saturation liquid density using the Townsend and Hales ArcGIS method as modified from the original Riedel equation.


Instruction:
Can you generate the documentation for the following Python 3 function
def Bhirud_normal(T, Tc, Pc, omega):
    r'''Calculates saturation liquid density using the Bhirud [1]_ CSP method.
    Uses Critical temperature and pressure and acentric factor.

    The density of a liquid is given by:

    .. math::
        &\ln \frac{P_c}{\rho RT} = \ln U^{(0)} + \omega\ln U^{(1)}

        &\ln U^{(0)} = 1.396 44 - 24.076T_r+ 102.615T_r^2
        -255.719T_r^3+355.805T_r^4-256.671T_r^5 + 75.1088T_r^6

        &\ln U^{(1)} = 13.4412 - 135.7437 T_r + 533.380T_r^2-
        1091.453T_r^3+1231.43T_r^4 - 728.227T_r^5 + 176.737T_r^6

    Parameters
    ----------
    T : float
        Temperature of fluid [K]
    Tc : float
        Critical temperature of fluid [K]
    Pc : float
        Critical pressure of fluid [Pa]
    omega : float
        Acentric factor for fluid, [-]

    Returns
    -------
    Vm : float
        Saturated liquid molar volume, [mol/m^3]

    Notes
    -----
    Claimed inadequate by others.

    An interpolation table for ln U values are used from Tr = 0.98 - 1.000.
    Has terrible behavior at low reduced temperatures.

    Examples
    --------
    Pentane

    >>> Bhirud_normal(280.0, 469.7, 33.7E5, 0.252)
    0.00011249654029488583

    References
    ----------
    .. [1] Bhirud, Vasant L. "Saturated Liquid Densities of Normal Fluids."
       AIChE Journal 24, no. 6 (November 1, 1978): 1127-31.
       doi:10.1002/aic.690240630
    '''
    Tr = T/Tc
    if Tr <= 0.98:
        lnU0 = 1.39644 - 24.076*Tr + 102.615*Tr**2 - 255.719*Tr**3 \
            + 355.805*Tr**4 - 256.671*Tr**5 + 75.1088*Tr**6
        lnU1 = 13.4412 - 135.7437*Tr + 533.380*Tr**2-1091.453*Tr**3 \
            + 1231.43*Tr**4 - 728.227*Tr**5 + 176.737*Tr**6
    elif Tr > 1:
        raise Exception('Critical phase, correlation does not apply')
    else:
        lnU0 = Bhirud_normal_lnU0_interp(Tr)
        lnU1 = Bhirud_normal_lnU1_interp(Tr)

    Unonpolar = exp(lnU0 + omega*lnU1)
    Vm = Unonpolar*R*T/Pc
    return Vm

Response: r Calculates the saturation liquid density using the Bhirud method.