Pure Component VLE in Terms of Fugacity Purpose of this lecture: To derive an expression for the calculation of fugacity of pure liquids Highlights Phase equilibrium expressed in terms of fugacities The fugacity of a pure liquid at a given temperature can be calculated through its vapour-phase fugacity coefficient and saturation pressure The effect of pressure on liquid-phase fugacity is captured by the Poynting factor Reading assignment: Section 11.5 (pp. 396-401) CHEE 311 Lecture 10

Pure Component VLE in Terms of Fugacity Consider a pure component at its vapour pressure: Phase rule tells us, F=2-2+1 = 1 degree of freedom Therefore, at a given T, there can only be a single pressure, Psat for which a vapour and a liquid are in equilibrium Along the phase boundary, the chemical potentials are equal How do the fugacities of the liquid and gas relate? P liquid gas T CHEE 311 Lecture 10

Pure Component VLE in Terms of Fugacity For the non-ideal, pure gas we can write: 11.38a For a non-ideal liquid, we can define an analogous expression: 11.38b At equilibrium 11.39 In terms of fugacity: 11.41 CHEE 311 Lecture 10

Review of Chemical Equilibrium Criteria We have several different criteria for phase equilibrium. While they stem from the same theory, they differ in practical applicability. A system at equilibrium has the following properties: the total Gibbs energy of the system is minimized, meaning that no change in the number of phases or their composition could lower the Gibbs energy further the chemical potential of each component, i, is the same in every phase within the system in p phases the fugacity of each component, i, is equal in every phase of the system CHEE 311 Lecture 10

Calculating the Fugacity of Pure Liquids The derivation of the fugacity of a pure liquid at a given T, P is comprised of four steps: Step 1. Calculate the fugacity of a vapour at Pisat Step 2. Calculate the change in Gibbs energy between Pisat and the given pressure P using the fundamental equation: dG = VdP - SdT (constant T) which after integration yields: Given that liquids are nearly incompressible (Viliq is not a strong function of P) the integral is approximated as: (A) CHEE 311 Lecture 10

Calculating the Fugacity of Pure Liquids 3. Using the definitions of fugacity: we can take the difference: (B) 4. Substituting A into B: or 11.44 CHEE 311 Lecture 10

Calculating the Fugacity of Pure Liquids We can now calculate the fugacity of any pure liquid using two equations: 11.44 and 11.35 The exponential within Equation 11.44 accounts for the change in Gibbs energy as we compress the liquid from Pisat to the specified pressure, P. This is known as the Poynting factor. This contribution to fugacity is slight at all pressures near Pisat, and is often assumed to be unity. CHEE 311 Lecture 10

Example Problem Calculate the fugacity of n-pentane at T= 25 oC and P=101 kPa. The saturation pressure of n-pentane at 25 oC is Pisat =67.5 kPa. CHEE 311 Lecture 10