Non-ideal Gases Fugacity

The fugacity is a pressure like quantity that is used to treat the free energy of nonideal gases.

Now we begin the steps that allow us to relate free energy changes to the equilibrium constant of real, nonideal gases. The thermodynamic reaction (∂G/∂P) t = V is used with the ideal gas relation PV = RT, or V = RT/P, to obtain G = G° = R in P. it was this equation that led to the familiar equilibrium constant expression. If the ideal gas relation PV = RT is not satisfactory, some other quality equations, that of van der Waals, for example, could be used to express the pressure dependence of V. if that were done, the integration of (∂G/∂P)T = V would produce an awkward expression for the equilibrium constant. Thus a route that preserves the simple form of the equilibrium constant expression is preferable.

A satisfactory procedure is the introduction of a function called the fugacy ƒ.  This procedure insists on the free energy equation having the convenient form of the nonideal complications are hidden in the fugacy term. A number of manipulations are necessary; we begin with the thermodynamic equation for mol 1 of gas at constant temperature.

G2 - G1 = V dP

The quantity RT/P can be added to and subtracted from the integrand to give

G2 - G1 = [RT/P + (V - RT/P0] dP

= RT/P dP = (V - RT/P dP

= RTY in P2/P1 + (V - RT/P) dP

Thus the ratio f/P can be calculated at any temperature for which viral coefficient data are available and for any pressure in the range in which these data are applicable. If the real gas behavior is expressed by any other equation of state, the integration can be carried out graphically or with the help of a computer.

Fugacity and the law of corresponding states: for gases for which molar volume measurements have not been made and an equation of state is not available, the law of corresponding states can be used to estimate the fugacities at various reduced variables PR, VR and TR all gases follow the same imperfection and therefore the same nonideality. Furthermore, the variation of the compressibility factor Z with the reduced pressure has been represented for various values TR. These data are all that is necessary for the integration values of:

Z = PV/RT

From which we obtain:

V = RT/P × Z

With this relation eq. can be written as:

RT In ƒ/P = ∫PO (RT/P × Z - RT/P) dP = RT  ∫PO (Z - 1) dP/P

Or, In ƒ/P =
∫PO (Z - 1) dP/P =  ∫PO (Z - 1) d PR/PR

The data of Z as a function of PR for a given value of TR then allow graphical integrations to be performed to give curves.

Example: estimate the fugacity of methane at 200 bar and 25°C, but use the correlation that is based on the law of corresponding states. The critical data give = 46.3 bar and T = 190.6 K for methane.

Solution: at 200 bar the reduced pressure is 200 bar/46.3 bar = 4.32. At 25°C the reduced temperature is 298.15/190.6 K = 1.56. From the value of ƒ/P is estimated at about 0.8, given ƒ = 160 bar.

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