Thermodynamic properties of a real gas species can be derived from fundamental thermodynamic relationships. For instance, the variation in the Helmholtz energy of a real gas species with respect to the molar volume at constant pressure is given as

(2–66)

Integrate the equation at constant temperature from the reference volume to the system molar volume

(2–67)

where and are respectively the Helmholtz energy and the molar volume of the same gas species in an ideal state at the same temperature and at a reference pressure such that

(2–68)

The integral in Equation 2–67 can be split into two parts: one for the real-gas regime and the other for the ideal-gas regime

Remove the infinity limit of the ideal-gas part

(2–69)

The integral in Equation 2–69 can be expanded by using an appropriate equation of state to express in terms of and (fixed).

The departure function for , , depends on the value of , that is, the choice of the reference state, and does not vanish even for an ideal gas if . There are two commonly used reference states:

The gas library uses the option number 1 because is needed in the calculation of equilibrium constant (for finding the reverse rate constant) in addition to thermodynamic departure functions.

Other thermodynamic properties of a real gas species can be derived from the Helmholtz energy by using the thermodynamic relationships.

(2–70)

(2–71)

(2–72)

(2–73)

The ideal-gas part of the thermodynamic property is provided by the original Chemkin subroutine, that is, computed from the polynomial with coefficients from the thermodynamic data. For example,

(2–74)

The specific heat can be obtained by differentiating the enthalpy

(2–75)

(2–76)

The departure function of a real gas species, say, the i-th species, depends on pressure, temperature, and parameters of the equation of state (EOS).

(2–77)

And any thermodynamic property (enthalpy, internal energy, ...) of the i-th species can be expressed as

(2–78)

For an ideal gas mixture of composition , any mean thermodynamic property is given as

(2–79)

On the other hand, for a real gas mixture of the same composition , any mean thermodynamic property is given as

(2–80)

In which the EOS parameters of the real gas mixture, , , and , are calculated from EOS parameters of individual species according to specific mixing rule. That is,

Because of the nonlinearity of the mixing rule and the interaction coefficients , the mean thermodynamic property of a real gas mixture cannot be calculated the same way as that of an ideal gas mixture

(2–81)

However, it is desirable to express the mean thermodynamic property as the weighted sum of the thermodynamic property of individual species in the gas mixture such as

(2–82)

And

(2–83)

where is the net molar production rate of the k-th species. The partial molar property of the k-th species is defined as

(2–84)

For an ideal gas mixture, the partial molar property is the same as the species property

(2–85)

For a real gas mixture, partial molar property is different from the species property

(2–86)

These derivatives, , are computed by using a numerical method with fixed (central differencing) [8].