W10 Solution Thermodynamics (II) v2 PDF

KMJ22003/3 Chemical Engineering Thermodynamics Solution Thermodynamics (II) Dr. Norzilah binti Abdul Halif [email protected] MBM 5, Materials Department Taman Muhibbah...

KMJ22003/3 Chemical Engineering Thermodynamics Solution Thermodynamics (II) Dr. Norzilah binti Abdul Halif [email protected] MBM 5, Materials Department Taman Muhibbah KMJ22003|NAH|2024/2025 1  CONTENT 1. Ideal-Gas Mixture Model 2. Fugacity & Fugacity Coefficient 3. The Fundamental Residual-Property Relation 4. Fugacity Coefficients from the Virial Equation of State 5. Generalized Correlations for the Fugacity Coefficient 6. The Ideal Solution Model 7. Excess Properties KMJ22003|NAH|2024/2025 2  1. Ideal-Gas Mixture Model Mathematical model to evaluate/describe Ideal-Gas Mixture thermodynamically An ideal gas mixture: a collection of multiple ideal gases that occupy the same volume and maintain uniform temperature and pressure conditions. Each gas in the mixture behaves independently, and the overall properties of the mixture can be determined by considering the properties of each individual gas. Ideal Gas Behavior No Intermolecular Forces Objective: To design and analyze processes involving gas mixtures Negligible Molecular Volume KMJ22003|NAH|2024/2025 3  Ideal-Gas Mixture Model: Partial Pressure For the ideal-gas state at given T and P the partial molar volume, the pure- species molar volume, and the mixture molar volume are identical. Partial pressure of species i in the ideal-gas- state mixture (pi) = the pressure that species i would exert if it alone occupied the molar volume of the mixture. yi = mole fraction of species i. The partial pressures obviously sum to the total pressure. KMJ22003|NAH|2024/2025 4  Ideal-Gas Mixture Model: Gibb’s Theorem o Engineers use Gibbs' theorem to optimize the design of chemical reactors and separation processes. o By understanding the thermodynamic properties of gas mixtures, they can predict reaction yields, energy requirements, and product purity. A partial molar property (other than volume) of a constituent species in an ideal-gas-state mixture = corresponding molar property of the species in the pure ideal-gas state at the mixture temperature but at a pressure equal to its partial pressure in the mixture. Enthalpy (independent of Entropy (dependent of Gibbs energy Chemical Potential pressure) pressure) Hiig is the pure-species value at the mixture T KMJ22003|NAH|2024/2025 5  Ideal-Gas Mixture Model: Gibb’s Theorem Summability relation: For the ideal-gas state, this enthalpy change of mixing is zero. Entropy change of mixing for the ideal-gas state. KMJ22003|NAH|2024/2025 6  2. Fugacity, fi & Fugacity Coefficient, Øi A way to quantify how much a substance "wants" to move from one phase to another or from one mixture to another. It is a useful concept in thermodynamics because it allows us to predict the behavior of real gases and mixtures, which often deviate from ideal behavior. Often thought of as a "corrected pressure" that accounts for the non-ideal behavior of real gases. Key Points about Fugacity For Ideal Gases Fugacity is equal to pressure Real Gases For Real Gases Fugacity is often lower than pressure, especially are NOT Ideal at high pressures and low temperatures. Fugacity Coefficient The ratio of fugacity to pressure, quantifying the Fugacity as a deviation from ideal behavior. Correction Equilibrium At equilibrium, the fugacity of a substance is the same in all phases. Application: Designing and optimizing chemical processes. 7 KMJ22003|NAH|2024/2025  Fugacity & Fugacity Coefficient: Pure Species For a real fluid: Apply to pure species i in any phase at any Residual Gibbs energy, GiR condition. The difference between the actual Gibbs energy of a For the ideal-gas state, for which G iR = 0, ϕi = 1 real gas and that of an ideal gas at the same Zi can be determined from temperature and pressure. Equation of State Methods KMJ22003|NAH|2024/2025 8 Vapor/Liquid Equilibrium (VLE) for Pure Species For a pure species, coexisting liquid and vapor phases are in equilibrium when they have the same T, P, and f KMJ22003|NAH|2024/2025 9  Fugacity of Pure Liquid The exponential is known as a Poynting factor. Values of Ziv for calculation of ϕisat. These may come The liquid-phase molar from an equation of state, volume Vil , usually the A value for P is at. from experiment, or from a value for saturated liquid. generalized correlation. KMJ22003|NAH|2024/2025 10  Fugacity & Fugacity Coefficient: Species in Solution For species i in a mixture of real gases or in a solution of liquids: fugacity of species i in solution Multiple phases at the same T and P are in equilibrium when the fugacity of each constituent species is the same in all phases. For the specific case of multicomponent vapor/liquid equilibrium KMJ22003|NAH|2024/2025 11  Fugacity & Fugacity Coefficient: Species in Solution ideal-gas state: n mol of mixture Differentiation with respect to ni at constant T, P, and nj Fugacity of species i in an Partial residual Gibbs ideal-gas-state mixture is energy equal to its partial pressure - Fugacity coefficient 12 of species i in solution KMJ22003|NAH|2024/2025 3. The Fundamental Residual-Property Relation G = H -TS All terms have the units of moles A general relation expressing nG/RT as a function of all of its canonical variables, T, P, and the mole numbers. Special case of the ideal-gas state: KMJ22003|NAH|2024/2025 13 3. The Fundamental Residual-Property Relation Fundamental residual-property relation: Fugacity coefficient The logarithm of the fugacity coefficient of a species in solution is a partial property with respect to GR/RT KMJ22003|NAH|2024/2025 14 4. Fugacity Coefficients from the Virial Equation of State Start with: characterizes bimolecular interactions between The mixture second virial coefficient, B is a function of molecules of species i and species j temperature and composition. For a binary mixture: cross coefficient. Mixing rules: KMJ22003|NAH|2024/2025 15  4. Fugacity Coefficients from the Virial Equation of State For n mol of gas mixture: Differentiation with respect to n1: Substitute in: Thermodynamics handbooks like Perry's Chemical Engineers' Handbook or CRC Handbook of Chemistry and Physics. Online databases such as NIST Chemistry WebBook, which has virial coefficients for a wide range of gases. Earlier editions of textbooks Multicomponent gas mixtures KMJ22003|NAH|2024/2025 16 4. Fugacity Coefficients from the Virial Equation of State The general equation to determine the fugacity coefficient in multicomponent gas mixture: KMJ22003|NAH|2024/2025 17 5. Generalized Correlations for the Fugacity Coefficient: Pure Species Fugacity Coefficients for Pure Species: General equation to calculate ln ϕˆi values Integration at constant Tr. from compressibility-factor data. Substitution for Zi by Refer to Volumetric properties topic * Evaluated numerically or graphically for various values of Tr and Pr from the data for Z0 and Z1 given in Tables D.1 through D.4 (App. D). KMJ22003|NAH|2024/2025 18  5. Generalized Correlations for the Fugacity Coefficient: Gas Mixtures Previously… From: Prausnitz et al.: KMJ22003|NAH|2024/2025 19  6. The Ideal Solution Model Refer to liquid solutions where the solute and solvent mix perfectly without any change in enthalpy or volume. KMJ22003|NAH|2024/2025 20 Fugacity & Fugacity Coefficients of Ideal Solution From: Subtraction: Special case of an ideal solution: Compare to: It shows that the fugacity of each species in an Lewis/Randall rule: ideal solution is proportional to its mole Applies to each species in an ideal solution at all conditions of T, P, and xi. fraction; the proportionality constant is the fugacity of pure species i in the same physical state as the solution and at the same T and P. Alternative form: KMJ22003|NAH|2024/2025 21  7. Excess Properties o Thermodynamic quantities that measure the deviation of a real mixture from ideal behavior. o The difference between the actual property of the mixture and the property it would have if it were an ideal solution at the same T, P, and composition. o The mathematical formalism of excess properties is analogous to that of the residual properties. Understanding Non- Design and Thermodynamic Material Properties Excess properties are Ideal Behavior Optimization Models key to accurately Most real-world Helps in designing Used to develop & Provide insights mixtures do not & optimizing refine into the modeling and behave ideally. processes such as distillation, thermodynamic models. interactions between different understanding the Understanding these deviations extraction, & Predicting the components in a behavior of real allows for more mixing. behavior of mixture. accurate Lead to more mixtures under Valuable for mixtures, which is predictions & efficient & cost- different developing new essential for both models effective conditions. materials & processes understanding scientific research and their properties practical applications in KMJ22003|NAH|2024/2025 industry. 22  7. Excess Properties Excess Properties Residual Properties Quantify the deviation of a real mixture Describe the deviation of a real gas from from an ideal solution. ideal gas behavior. The difference between the actual The difference between the actual property of the mixture and the property it property of the gas and the property it would have if it were an ideal solution at would have if it were an ideal gas at the the same temperature, pressure, and same temperature and density. composition. KMJ22003|NAH|2024/2025 23  7. Excess Properties Excess properties have a simple relation to residual properties: Partial-property relation: Fundamental excess-property relation: KMJ22003|NAH|2024/2025 24 7. Excess Properties KMJ22003|NAH|2024/2025 25