rmgpy.thermo.Wilhoit

class rmgpy.thermo.Wilhoit(Cp0=None, CpInf=None, a0=0.0, a1=0.0, a2=0.0, a3=0.0, H0=None, S0=None, B=None, Tmin=None, Tmax=None, label='', comment='')

A heat capacity model based on the Wilhoit equation. The attributes are:

Attribute Description
a0 The zeroth-order Wilhoit polynomial coefficient
a1 The first-order Wilhoit polynomial coefficient
a2 The second-order Wilhoit polynomial coefficient
a3 The third-order Wilhoit polynomial coefficient
H0 The integration constant for enthalpy (not H at T=0)
S0 The integration constant for entropy (not S at T=0)
E0 The energy at zero Kelvin (including zero point energy)
B The Wilhoit scaled temperature coefficient in K
Tmin The minimum temperature in K at which the model is valid, or zero if unknown or undefined
Tmax The maximum temperature in K at which the model is valid, or zero if unknown or undefined
comment Information about the model (e.g. its source)

The Wilhoit polynomial is an expression for heat capacity that is guaranteed to give the correct limits at zero and infinite temperature, and gives a very reasonable shape to the heat capacity profile in between:

\[C_\mathrm{p}(T) = C_\mathrm{p}(0) + \left[ C_\mathrm{p}(\infty) - C_\mathrm{p}(0) \right] y^2 \left[ 1 + (y - 1) \sum_{i=0}^3 a_i y^i \right]\]

Above, \(y \equiv T/(T + B)\) is a scaled temperature that ranges from zero to one based on the value of the coefficient \(B\), and \(a_0\), \(a_1\), \(a_2\), and \(a_3\) are the Wilhoit polynomial coefficients.

The enthalpy is given by

\[\begin{split}H(T) & = H_0 + C_\mathrm{p}(0) T + \left[ C_\mathrm{p}(\infty) - C_\mathrm{p}(0) \right] T \\ & \left\{ \left[ 2 + \sum_{i=0}^3 a_i \right] \left[ \frac{1}{2}y - 1 + \left( \frac{1}{y} - 1 \right) \ln \frac{T}{y} \right] + y^2 \sum_{i=0}^3 \frac{y^i}{(i+2)(i+3)} \sum_{j=0}^3 f_{ij} a_j \right\}\end{split}\]

where \(f_{ij} = 3 + j\) if \(i = j\), \(f_{ij} = 1\) if \(i > j\), and \(f_{ij} = 0\) if \(i < j\).

The entropy is given by

\[S(T) = S_0 + C_\mathrm{p}(\infty) \ln T - \left[ C_\mathrm{p}(\infty) - C_\mathrm{p}(0) \right] \left[ \ln y + \left( 1 + y \sum_{i=0}^3 \frac{a_i y^i}{2+i} \right) y \right]\]

The low-temperature limit \(C_\mathrm{p}(0)\) is \(3.5R\) for linear molecules and \(4R\) for nonlinear molecules. The high-temperature limit \(C_\mathrm{p}(\infty)\) is taken to be \(\left[ 3 N_\mathrm{atoms} - 1.5 \right] R\) for linear molecules and \(\left[ 3 N_\mathrm{atoms} - (2 + 0.5 N_\mathrm{rotors}) \right] R\) for nonlinear molecules, for a molecule composed of \(N_\mathrm{atoms}\) atoms and \(N_\mathrm{rotors}\) internal rotors.

B

The Wilhoit scaled temperature coefficient.

Cp0

The heat capacity at zero temperature.

CpInf

The heat capacity at infinite temperature.

E0

The ground state energy (J/mol) at zero Kelvin, including zero point energy.

For the Wilhoit class, this is calculated as the Enthalpy at 0.001 Kelvin.

H0

The integration constant for enthalpy.

NB. this is not equal to the enthlapy at 0 Kelvin, which you can access via E0

S0

The integration constant for entropy.

Tmax

The maximum temperature at which the model is valid, or None if not defined.

Tmin

The minimum temperature at which the model is valid, or None if not defined.

a0

a0: ‘double’

a1

a1: ‘double’

a2

a2: ‘double’

a3

a3: ‘double’

comment

comment: str

copy(self) → Wilhoit

Return a copy of the Wilhoit object.

discrepancy(self, HeatCapacityModel other) → double

Return some measure of how dissimilar self is from other.

The measure is arbitrary, but hopefully useful for sorting purposes. Discrepancy of 0 means they are identical

fitToData(self, ndarray Tdata, ndarray Cpdata, double Cp0, double CpInf, double H298, double S298, double B0=500.0)

Fit a Wilhoit model to the data points provided, allowing the characteristic temperature B to vary so as to improve the fit. This procedure requires an optimization, using the fminbound function in the scipy.optimize module. The data consists of a set of heat capacity points Cpdata in J/mol*K at a given set of temperatures Tdata in K, along with the enthalpy H298 in kJ/mol and entropy S298 in J/mol*K at 298 K. The linearity of the molecule, number of vibrational frequencies, and number of internal rotors (linear, Nfreq, and Nrotors, respectively) is used to set the limits at zero and infinite temperature.

fitToDataForConstantB(self, ndarray Tdata, ndarray Cpdata, double Cp0, double CpInf, double H298, double S298, double B)

Fit a Wilhoit model to the data points provided using a specified value of the characteristic temperature B. The data consists of a set of dimensionless heat capacity points Cpdata at a given set of temperatures Tdata in K, along with the dimensionless heat capacity at zero and infinite temperature, the dimensionless enthalpy H298 at 298 K, and the dimensionless entropy S298 at 298 K.

getEnthalpy(self, double T) → double

Return the enthalpy in J/mol at the specified temperature T in K.

getEntropy(self, double T) → double

Return the entropy in J/mol*K at the specified temperature T in K.

getFreeEnergy(self, double T) → double

Return the Gibbs free energy in J/mol at the specified temperature T in K.

getHeatCapacity(self, double T) → double

Return the constant-pressure heat capacity in J/mol*K at the specified temperature T in K.

isIdenticalTo(self, HeatCapacityModel other) → bool

Returns True if self and other report very similar thermo values for heat capacity, enthalpy, entropy, and free energy over a wide range of temperatures, or False otherwise.

isSimilarTo(self, HeatCapacityModel other) → bool

Returns True if self and other report similar thermo values for heat capacity, enthalpy, entropy, and free energy over a wide range of temperatures, or False otherwise.

isTemperatureValid(self, double T) → bool

Return True if the temperature T in K is within the valid temperature range of the thermodynamic data, or False if not. If the minimum and maximum temperature are not defined, True is returned.

label

label: str

toNASA(self, double Tmin, double Tmax, double Tint, bool fixedTint=False, bool weighting=True, int continuity=3) → NASA

Convert the Wilhoit object to a NASA object. You must specify the minimum and maximum temperatures of the fit Tmin and Tmax in K, as well as the intermediate temperature Tint in K to use as the bridge between the two fitted polynomials. The remaining parameters can be used to modify the fitting algorithm used:

  • fixedTint - False to allow Tint to vary in order to improve the fit, or True to keep it fixed

  • weighting - True to weight the fit by \(T^{-1}\) to emphasize good fit at lower temperatures, or False to not use weighting

  • continuity - The number of continuity constraints to enforce at Tint:

    • 0: no constraints on continuity of \(C_\mathrm{p}(T)\) at Tint
    • 1: constrain \(C_\mathrm{p}(T)\) to be continous at Tint
    • 2: constrain \(C_\mathrm{p}(T)\) and \(\frac{d C_\mathrm{p}}{dT}\) to be continuous at Tint
    • 3: constrain \(C_\mathrm{p}(T)\), \(\frac{d C_\mathrm{p}}{dT}\), and \(\frac{d^2 C_\mathrm{p}}{dT^2}\) to be continuous at Tint
    • 4: constrain \(C_\mathrm{p}(T)\), \(\frac{d C_\mathrm{p}}{dT}\), \(\frac{d^2 C_\mathrm{p}}{dT^2}\), and \(\frac{d^3 C_\mathrm{p}}{dT^3}\) to be continuous at Tint
    • 5: constrain \(C_\mathrm{p}(T)\), \(\frac{d C_\mathrm{p}}{dT}\), \(\frac{d^2 C_\mathrm{p}}{dT^2}\), \(\frac{d^3 C_\mathrm{p}}{dT^3}\), and \(\frac{d^4 C_\mathrm{p}}{dT^4}\) to be continuous at Tint

Note that values of continuity of 5 or higher effectively constrain all the coefficients to be equal and should be equivalent to fitting only one polynomial (rather than two).

Returns the fitted NASA object containing the two fitted NASAPolynomial objects.

toThermoData(self) → ThermoData

Convert the Wilhoit model to a ThermoData object.