Equation of State

Real Gas Relationships

• Relates PVT properties of a pure substance/mixture via Theoratical and empirical relations

• Source of information: computer databases• Effective Eq. of state: experimental data with

good precision

Van-der Waals Equation• Simple, explains computational

problems.• Also illustrates Theoretical

developments

• Fortan computer Programs for non linear equations

• One Code: Newton Method, Other Minimization

Critical Temperature and Pressure

• The highest temperature at which a species can coexist in two phases (liquid vapor) is the critical temperature of that species, Tc, and the corresponding pressure is the critical pressure, Pc.

• A substance at Tc and Pc is said to be at its critical state.• A vapor is a gaseous species below its critical temperature, and a gas

is a species above its critical temperature at a pressure low enough for the species to be more like a vapor than a liquid (i.e., a density closer to 1 g/L than 1000 g/L).

• Substances at temperatures above Tc and pressures above Pc are referred to as supercritical fluids.

Cubic Equation of State

• Fit the experimental data with as few constants in equation as possible

• Concise summary of a large of experimental data• Accurate interpolation between experimental data

points• Provide a continuous function to facilitate

calculation of physical properties involving differentiation and integration.

• Point of departure for the treatment of thermodynamic properties of mixtures.

Virial Equation of state• A virial equation of state expresses the quantity PV/

RT as a power series in the inverse of specific volume:

• B, C, and D are functions of temperature …known as the second, third, and fourth virial coefficients,

• Theoretical basis in statistical mechanics • procedures for estimating the coefficients are not well

developed, those beyond B.• The ideal gas equation of state is obtained if B=C=D…

• For polar compounds (asymmetrical compounds with a nonzero dipole moment, such as water).

• Can estimate V or P for a given T for a nonpolar species (one with a dipole moment close to zero, such as hydrogen and oxygen and all other molecularly symmetrical compounds).

• Solution for P is straightforward. • For V, the equation can be rearranged into a quadratic and

solved using the quadratic formula. • One of the two solutions is reasonable and the other is not

and should be discarded; • if there is any doubt, estimate V from the ideal gas equation

of state and accept the virial equation solution that comes closest to V ideal'

Steps Involved

• Look up the critical temperature and pressure (Tc and Pc) for the species of interest

• Pitzer acentric factor, w. a parameter that reflects the geometry and polarity of a molecule.

• Calculate the reduced temperature, Tr = TITc.• Estimate B using the following equations

• Substitute into Equation the values of B and whichever of the variables P and V is known and solve for the other variable

Cubic equations ofstate (SRK)

• Empirical but have proved remarkably robust in describing a 'Wide variety of systems.

• SRK equation of state is• Empirical functions of critical temperature and

Pressure

Correlations

Calculating volume

By Spreadsheet Software data

Considerations

• SRK equation of state (and every other equation of state) is itself an

• approximation.• All equations of state have parameters obtained by fitting

empirical expressions to experimental PVT data. • The fit may be excellent in the temperature and pressure ranges

where the data were obtained but may be terrible elsewhere. • Always try to ascertain the region of validity of any equation of

state intend to be used. • No assurance of the accuracy of the equation for condition far

from acceptable region.