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*s = solid, 1 = liquid, g = gas. Several species are listed twice, as a liquid and as a gas; the difference is due to evaporation or condensation. The molar mass can be in g/g-mol or kg/kg-mol and Cp can be in J/g-mol-K or kJ/kg-mol-K. Source: Refs. 5-8 and 5-9.

*s = solid, 1 = liquid, g = gas. Several species are listed twice, as a liquid and as a gas; the difference is due to evaporation or condensation. The molar mass can be in g/g-mol or kg/kg-mol and Cp can be in J/g-mol-K or kJ/kg-mol-K. Source: Refs. 5-8 and 5-9.

Various thermodynamic criteria that represent the necessary and sufficient conditions for an equilibrium to be stable were first advanced by J. W. Gibbs early in the 20th century; they are based on minimizing the free energy. The Gibbs free energy G (often called the chemical potential) is a convenient derived function or property of the state of a chemical material describing its thermodynamic potential and is directly related to the internal energy U, the pressure p, molar volume V, enthalpy h, temperature T, and entropy S. For a single species j the free energy is defined as Gf, it can be determined for specific thermodynamic conditions, for mixtures of gas as well as an individual gas species.

For most materials used as rocket propellant the free energy has been determined and tabulated as a function of temperature. It can be corrected for pressure. Its units are J/kg-mol. For a series of different species the mixture free energy G is n

The free energy is a function of temperature and pressure. It is another property of a material, just like enthalpy or density; only two such independent parameters are required to characterize a gas condition. The free energy may be thought of as the tendency or driving force for a chemical material to enter into a chemical (or physical) change. Although it cannot be measured directly, differences in chemical potential can be measured. When the chemical potential of the reactants is higher than that of the likely products, a chemical reaction can occur and the chemical composition can change. The change in free energy AG for reactions at constant temperature and pressure is the chemical potential of the products less that of the reactants.

Here the superscript m gives the number of gas species in the combustion products, the superscript n gives the number of gas species in the reactants, and the AG represents the maximum energy that can be "freed" to do work on an "open" system where mass enters and leaves the system. At equilibrium the free energy is a minimum; at its minimum a small change in mixture fractions causes almost no change in AG and the free energies of the products and the reactants are essentially equal. Then d a G/dn = 0

and a curve of molar concentration n versus AG would have a minimum.

If reacting propellants are liquid or solid materials, energy will be needed to change phase, vaporize them, or break them down into other gaseous species. This energy has to be subtracted from the heat or the energy available to heat the gases from the reference temperature to the combustion temperature. Therefore, the values of AH0 and AG0 for liquid and solid species are different from those of the same species in a gaseous state. The standard free energy of formation AfG° is the increment in free energy associated with the reaction of forming a given compound or species from its elements at their reference state. Table 5-2 gives values of AfH° and AfG° and other properties of carbon monoxide as a function of temperature. Similar data for other species can be obtained from Refs. 5-7 and 5-13. The entropy is another thermodynamic property of matter that is relative, which means that it is determined as a change in entropy. In the analysis of isentropic nozzle flow, it is assumed that the entropy remains constant. It is defined as and the corresponding integral is

Po where the zero applies to the reference state. In an isentropic process, entropy is constant. For a mixture the entropy is

TABLE 5-2. Variation of Thermochemical Data with Temperature for Carbon Monoxide (CO) as an Ideal Gas

Temp.

C° ^p

H° - H°(T)

AfH°

A fG°

log Kf

(K)

(J/mol-K)

0 0

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