Elements and Species Thermodynamics

All phases in Cantera are made up of one or more species, which in turn contain one or more elements.


In Cantera, an element may refer to a chemical element or an isotope. Note that definitions of elements are not often needed, since Cantera has definitions for the standard chemical elements. Explicit element definitions are usually only needed for isotopes.

An element can be defined in the YAML format by adding entries to the elements section of the input file.


For each species, a species definition is required.

A species can be defined in the YAML format by adding an entry to the species section of the input file.

Species Name

The name of a species may contain letters, numbers, or just about anything else that is a printable, non-whitespace character. Some example name specifications:


Elemental Composition

The elemental composition of each species must be specified. For gaseous species, the elemental composition is well-defined, since the species represent distinct molecules. For species in solid or liquid solutions, or on surfaces, there may be several possible ways of defining the species. For example, an aqueous species might be defined with or without including the water molecules in the solvation cage surrounding it.

The special "element" E is used in representing charged species, where it specifies the net number of electrons compared to the number needed to form a neutral species. That is, negatively charged ions will have E > 0, while positively charged ions will have E < 0.

The number of atoms of an element must be non-negative, with the exception of electrons.

For surface species, it is possible to omit the elemental composition, in which case it is composed of nothing, and represents an empty surface site. This can also be done to represent vacancies in solids. A charged vacancy can be defined to be composed solely of electrons.

Thermodynamic Properties

The phase models discussed in the Phases section implement specific models for the thermodynamic properties appropriate for the type of phase or interface they represent. Although each one may use different expressions to compute the properties, they all require thermodynamic property information for the individual species. For the phase types implemented at present, the properties needed are:

  1. the molar heat capacity at constant pressure \(\hat{c}^\circ_p(T)\) for a range of temperatures and a reference pressure \(p^\circ\);

  2. the molar enthalpy \(\hat{h}(T^\circ, p^\circ)\) at \(p^\circ\) and a reference temperature \(T^\circ\);

  3. the absolute molar entropy \(\hat{s}(T^\circ, p^\circ)\) at \((T^\circ, p^\circ)\).

The superscript \(^\circ\) here represents the reference state--a specified state (that is, a set of conditions \(T^\circ\) and \(p^\circ\) and fixed chemical composition) at which thermodynamic properties are known.

Thermodynamic Property Models

The models described in this section can be used to provide thermodynamic data for each species in a phase. Each model implements a different parameterization (functional form) for the heat capacity. Note that there is no requirement that all species in a phase use the same parameterization; each species can use the one most appropriate to represent how the heat capacity depends on temperature.

Currently, several types are implemented that provide species properties appropriate for models of ideal gas mixtures, ideal solutions, and pure compounds.

The NASA 7-Coefficient Polynomial Parameterization

The NASA 7-coefficient polynomial parameterization is used to compute the species reference-state thermodynamic properties \(\hat{c}^\circ_p(T)\), \(\hat{h}^\circ(T)\), and \(\hat{s}^\circ(T)\).

The NASA parameterization represents \(\hat{c}^\circ_p(T)\) with a fourth-order polynomial:

\begin{equation*} \frac{\hat{c}_p^\circ(T)}{\overline{R}} = a_0 + a_1 T + a_2 T^2 + a_3 T^3 + a_4 T^4 \end{equation*}
\begin{equation*} \frac{\hat{h}^\circ (T)}{\overline{R} T} = a_0 + \frac{a_1}{2} T + \frac{a_2}{3} T^2 + \frac{a_3}{4} T^3 + \frac{a_4}{5} T^4 + \frac{a_5}{T} \end{equation*}
\begin{equation*} \frac{\hat{s}^\circ(T)}{\overline{R}} = a_0 \ln T + a_1 T + \frac{a_2}{2} T^2 + \frac{a_3}{3} T^3 + \frac{a_4}{4} T^4 + a_6 \end{equation*}

Note that this is the "old" NASA polynomial form, used in the original NASA equilibrium program and in Chemkin, which uses 7 coefficients in each of two temperature regions. It is not compatible with the form used in the most recent version of the NASA equilibrium program, which uses 9 coefficients for each temperature region.

A NASA-7 parameterization can be defined in the YAML format by specifying NASA7 as the model in the species thermo field.

The NASA 9-Coefficient Polynomial Parameterization

The NASA 9-coefficient polynomial parameterization 1 ("NASA9" for short) is an extension of the NASA 7-coefficient polynomial parameterization which includes two additional terms in each temperature region, as well as supporting an arbitrary number of temperature regions.

The NASA9 parameterization represents the species thermodynamic properties with the following equations:

\begin{equation*} \frac{\hat{c}_p^\circ(T)}{\overline{R}} = a_0 T^{-2} + a_1 T^{-1} + a_2 + a_3 T + a_4 T^2 + a_5 T^3 + a_6 T^4 \end{equation*}
\begin{equation*} \frac{\hat{h}^\circ(T)}{\overline{R} T} = - a_0 T^{-2} + a_1 \frac{\ln T}{T} + a_2 + \frac{a_3}{2} T + \frac{a_4}{3} T^2 + \frac{a_5}{4} T^3 + \frac{a_6}{5} T^4 + \frac{a_7}{T} \end{equation*}
\begin{equation*} \frac{\hat{s}^\circ(T)}{\overline{R}} = - \frac{a_0}{2} T^{-2} - a_1 T^{-1} + a_2 \ln T + a_3 T + \frac{a_4}{2} T^2 + \frac{a_5}{3} T^3 + \frac{a_6}{4} T^4 + a_8 \end{equation*}

A common source for species data in the NASA9 format is the NASA ThermoBuild tool.

A NASA-9 parameterization can be defined in the YAML format by specifying NASA9 as the model in the species thermo field.

The Shomate Parameterization

The Shomate parameterization is:

\begin{equation*} \hat{c}_p^\circ(T) = A + Bt + Ct^2 + Dt^3 + \frac{E}{t^2} \end{equation*}
\begin{equation*} \hat{h}^\circ(T) = At + \frac{Bt^2}{2} + \frac{Ct^3}{3} + \frac{Dt^4}{4} - \frac{E}{t} + F \end{equation*}
\begin{equation*} \hat{s}^\circ(T) = A \ln t + B t + \frac{Ct^2}{2} + \frac{Dt^3}{3} - \frac{E}{2t^2} + G \end{equation*}

where \(t = T / 1000\textrm{ K}\). It requires 7 coefficients \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), and \(G\). This parameterization is used to represent reference-state properties in the NIST Chemistry WebBook. The values of the coefficients \(A\) through \(G\) should be entered precisely as shown there, with no units attached. Unit conversions to SI will be handled internally.

A Shomate parameterization can be defined in the YAML format by specifying Shomate as the model in the species thermo field.

Constant Heat Capacity

In some cases, species properties may only be required at a single temperature or over a narrow temperature range. In such cases, the heat capacity can be approximated as constant, and simple expressions can be used for the thermodynamic properties:

\begin{equation*} \hat{c}_p^\circ(T) = \hat{c}_p^\circ(T^\circ) \end{equation*}
\begin{equation*} \hat{h}^\circ(T) = \hat{h}^\circ\left(T_0\right) + \hat{c}_p^\circ \left(T-T^\circ\right) \end{equation*}
\begin{equation*} \hat{s}^\circ(T) = \hat{s}^\circ(T_0) + \hat{c}_p^\circ \ln{\left(\frac{T}{T^\circ}\right)} \end{equation*}

The parameterization uses four constants: \(T^\circ, \hat{c}_p^\circ(T^\circ), \hat{h}^\circ(T^\circ)\), and \(\hat{s}^\circ(T)\). The default value of \(T^\circ\) is 298.15 K; the default value for the other parameters is 0.0.

A constant heat capacity parameterization can be defined in the YAML format by specifying constant-cp as the model in the species thermo field.



B. J. McBride, M. J. Zehe, S. Gordon. "NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species," NASA/TP-2002-211556, Sept. 2002.