Non-Ideality in 1-component Systems

1. Non-Ideality in 1-component Systems • Purpose of this lecture: • To introduce fugacity (f), a thermodynamic property of non-ideal fluids, and to show how to calculate it • Highlights • Definition of fugacity and fugacity coefficient • Relation between fugacity and compressibility factor • Calculating fugacity of pure non-ideal gases • Reading assignment: Section 11.5 (pp. 394-396) and 11.7 (pp. 407-409) Lecture 9

2. Non-Ideality in 1-component Systems • The ideal gas assumption: • PV = RT • where V = molar volume holds • only for low pressures, where • molecular interactions are • negligible and molecular volume • need not be considered. • At higher pressures, we have used • the compressibility factor, Z, to • characterize gas behaviour. • Z = PV / RT • = 1 for ideal gases Lecture 9

3. Gibbs Energy of Pure Gases • For any pure gas, ideal or non-ideal, the fundamental equation applies: • dG = VdP - SdT • At constant T, changes in the Gibbs energy of a pure gas arise only from changes in pressure, and: • dG = VdP (constant T) • We can integrate between two pressures, Pref and P to obtain: • For an ideal gas, we can substitute for the molar volume, V=RT/P Lecture 9

4. Gibbs Energy of Pure, Ideal Gases • For the ideal gas case, we have • We can simplify the expression: • 11.28 • where i(T) is only a function of temperature (and the reference pressure). • This handy expression provides the Gibbs energy per mole of a pure, ideal gas at a given P and T • We wish that it were true for all gases, but its not. • We used an expression like it to get Raoult’s law Lecture 9

5. Gibbs Energy for Pure, Non-ideal Gases • Let’s define a fudged pressure f called fugacity so that it will be true for all real gases • 11.31 • where • i(T) the same function of temperature • fi is the fugacity of pure i • What units does f have? • What happens to f at low pressures? • We’ll start out with fugacity of pure components and later we will worry about mixtures. Lecture 9

6. Pure Gases: Fugacity and Fugacity Coefficient • In summary, the fugacity of a pure, non-ideal gas is defined as: • with the specification that: • A closely related parameter is the fugacity coefficient, defined by: • What is fi for an ideal gas? Lecture 9

7. Calculating the Fugacity of a Pure Gas • The simplest means of calculating the fugacity of a pure gas is to compare its behaviour to an ideal gas • For the non-ideal gas: • For the ideal gas: • Taking the difference of these equations: Lecture 9

8. Calculating the Fugacity of a Pure Gas • We can simplify this relation by an appropriate choice of Pref. As pressure goes to zero, a real gas approaches ideality. Therefore, • With Pref0, we have: • or • Substituting V = ZRT/P and Vig = RT/P, we arrive at: • 11.35 Lecture 9

9. Calculating the Fugacity of a Pure Gas • Equation 11.35 is commonly written in terms of the fugacity coefficient: • at a given T. • To calculate the fugacity of a pure, non-ideal gas, all we need is information on the relationship of Z as a function of P at T. • Experimental data • Equations of State • Correlations Lecture 9

10. 5. Calculating Fugacity of Pure Gases • We can use several methods to get  for a pure gas, depending on how we choose to calculate Z • In all cases, we can apply the following relation: • Section 11.7 of the text presents a generalized method of calculating fi for pure gases that are non-polar or slightly polar. • Lee-Kesler Correlation: •  = (o)(1) 11.67 • where o and 1 are tabulated functions (Appendix E) of Pr and Tr and  is the acentric factor of the substance Lecture 9

11. Calculating Fugacity of Pure Gases • Virial Equation: • We can also use Virial coefficients • 11.68 • where • 3.65 • 3.66 Lecture 9

12. Applicability of Simple Correlations • It is important to understand under when the simple correlations apply. Lecture 9

13. Example Problem Estimate the fugacity of cyclopentane at 110 C and 2.75 bar. At 110 C the vapor pressure of cyclopentane is 5.267 bar. Lecture 9