1. Shroff S.R. Rotary Institute of Chemical Technology Chemical Engineering Chemical Engineering Thermodynamics-II 1

2. Fugacity & Fugacity Coefficient Group Member:- Devanshu Joshi - 130990105010 Jayesh Kikani - 13099015011 Bhavik Mahant - 130990105012 Guided By:- Mrs. Janki Tailor

3. Outline of Presentation 1. Introduction 2. Concepts of Fugacity 3. Effect of Temperature & pressure on Fugacity 4. Important relation of Fugacity Coefficient 5. Vapour Liquid Equilibrium for pure species 6. Fugacity & Fugacity coefficient: Species in solution 7. Reference

4. Introduction  The concept of Fugacity was introduced by Gilbert Newton Lewis.  Fugacity is widely used in solution thermodynamics to represent the behaviour of real gases.

5.  Fugacity is derived from Latin word ‘fleetness’ or the ‘Escaping Tendency’.  Fugacity has been used extensively in the study of phase and chemical reaction equlibria involving gases at high pressures.

6. Fugacity vs. Temperature A liter of water Start: T = 0°C f benzene = 0 (no chemical present) Add: 2100 Joules of heat to raise temperature by 1 °C 0.022 moles benzene to raise fugacity to 10 4.1 Pa. A liter of air Start: T = 0°C f benzene = 0 (no chemical present) Add: 0.001 Joules of heat to raise temperature by 1 °C 0.0051 moles benzene to raise fugacity to 10 4.1 Pa.

7. Concepts of Fugacity  For an infinitesimal reversible change occurring in the system under isothermal condition dG = -SdT + VdP reduces to, dG = VdP  For one mole of an ideal gas V in the above equation may be replaced by RT/P, dG = RT (dP/P) = RT d(ln P)  Above equation is applicable only to ideal gas.

8.  For representing the influence of present on Gibbs free energy of real gases by a similar relationship, then the true pressure in the equation should be replaced by an ‘effective’ pressure, which we call fugacity f of the gas.  Hence, fugacity has the same dimensions as pressure.

9.  The following equation, thus provides the partial definition of fugacity. It is satisfied by gases whether ideal or real. dG = RT d(ln f) Integration of above equation gives, G = RT ln f + C where C is the constant that depends on temperature and nature of the gas.

10. Concepts of Chemical potential:-  The chemical potential μ i provides the fundamental criteria for phase equilibria. This is true as well for chemical reaction equilibria.  The Gibbs energy, and hence μ i, is defined in relation to the internal energy and entropy. Because absolute values of internal energy are unknown and same it is true for μ i.

11.  Moreover, -----(1) shows that approaches negative infinity when either P or y i approches zero.  This is true not just for an ideal gas but for any gas.  Although these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a property that takes the place of μ i but which does not exhibit its less desirable characteristics.

12.  The origin of the fugacity concept resides in equation-1, valid only for pure species i in the ideal-gas state.  For a real fluid, an analogous equation that defines fi, the fugacity of pure species i: --------- (2)  The fugacity of pure species i as an ideal gas is necessarily equal to its pressure. Subtraction of eq. (1) from eq. (2), both written for the same T and P,

13.  we know that, is the Residual Gibb’s energy G i R, where the dimensionless ratio fi/P has been defined as another new property, the fugacity coefficient, given by the symbol ɸ i :  Fugacity Coefficient:- The ratio of fugacity to pressure is referred to as fugacity coefficient and is denoted by ɸ i. It is dimensionless and depends on nature of the gas, the pressure and the temperature.

14. Effect of Temperature & Pressure on Fugacity  By integrating eq. dG = RT d(ln f) between pressure P and P 0.  G 0 and f 0 refer to the molar free energy and fugacity respectively at a very low pressure where the gas behaves ideally. This equation can be rearranged as,

15.  Differentiate this with respect to temperature at constant pressure.  Substituting the Gibbs-Helmholtz equation, into the above result and observing that f 0 is equal to the pressure.  H is the molar enthalpy of the gas at the given pressure and H 0 is the enthalpy at a very low pressure. H 0 – H can be treated as the increase of enthalpy accompanying the expansion of the gas from pressure P to zero pressure at constant temperature.  Above equation indicates the effect of temperature on the fugacity.

16.  The effect of pressure on fugacity is evident from the defining equation for fugacity. dG = V dP = RT d (ln f) which on rearrangement gives,

17. Important relations of Fugacity coefficient  The identification of ln ɸi with G i R / RT by eq. permits its evaluation by the eq.  Compressibility factor is given by, (Const T)

18. Vapour Liquid Equilibrium for Pure Species  Fugacity of pure species i, may be written for species i as a saturated vapour and liquid at the same temperature: --------------(1) --------------(2)  By taking difference,  This equation applies to the change of state from saturated liquid to saturated vapour, both at temperature T and at the vapour pressure P i sat. We know that L.H.S. Became zero,

19.  For a pure species coexisting liquid and vapour phases are in equilibrium when they have same temperature, pressure and fugacity.  An alternative formulation is based on the fugacity coefficients:  This equation expressing equality of fugacity coefficients, is an equally valid criterion of vapour-liquid equilibrium for pure species.

20. Fugacity & Fugacity coefficient: Species in solution  The definition of the fugacity of a species in solution is parallel to the definition of the pure species fugacity. For a species i in a mixture of real gases or in a solution of liquids, -----------(A) where f i ^ is the fugacity of species i in solution, replacing the partial pressure y i P.  This does not make it a partial molar property, therefore identified by circumflex rather than by an over bar.

21.  As we know for the Chemical potential, here also all phases are in equilibrium at the same T, (i = 1,2,...,N)  Multiple phases at the same T and P are in equilibrium when the fugacity of each species is the same in all phases.  for the specific case of vapour/liquid equilibrium above equation becomes: (i = 1,2,...,N)

22.  The residual property is, where M is the molar value of a property and M ig is the value that the property would have for an ideal gas of the same composition at the same T and P. (For n mole)  Differentiation with respect to n i at constant T, P and n j, In the terms of Partial molar property, (Partial Residual Gibbs Energy)

23.  Eq. (A) subtracting from,  The identity gives, where by definition:-  The dimensionless ratio ɸi^ is called the fugacity coefficeint of species i in the solution.

24. Systematic approach to VLE  Please, click on below given link. Click here...

25. References  Introduction to Chemical Engineering Thermodynamics By Van Ness and Smith Mc Graw Hill Publication  Chemical Engineering Thermodynamics K V Narayana, EEE publication URL:- www.sciencedirect.com