Chapter 8: The Thermodynamics of Multicomponent Mixtures

Important Notation

Learning objectives Be able to: Understand the difference between partial molar properties and pure component properties; Use the mass, energy, and entropy balance for mixtures; Compute partial molar properties from experimental data; Derive the criteria for phase and chemical equilibria in multicomponent systems.

Thermodynamic description of mixtures In previous chapters (which dealt with pure components), it was identified that properties such as volume, internal energy, enthalpy, Helmholtz and Gibbs energies are extensive, i.e., they depend on the amount of substance present.

Thermodynamic description of mixtures For mixtures, it is reasonable to expect that these properties will depend on the amount of each of the C components present, i.e.:

Thermodynamic description of mixtures For mixtures, it is reasonable to expect that the corresponding molar properties will depend on the mole fractions of the C components present. But, there is a subtle point to note: Consider an open system. Depending on the type of interaction with the surroundings, the number of moles of each component may vary independently: can each be treated as an independent variable. The same does not happen with mole fractions. Why?

Thermodynamic description of mixtures The summation of the mole fractions of all C components is always equal to 1. Only (C-1) mole fractions can be treated as independent. One of them, say that of component C, can be calculated as function of the other mole fractions

Thermodynamic description of mixtures For the molar properties:

Thermodynamic description of mixtures Would a relationship such as this one be true?

Thermodynamic description of mixtures Would a relationship such as this one be true? Let us watch a movie to help answer this question. http://www.youtube.com/watch?v=84k206qaVRU

Thermodynamic description of mixtures We see that:

Thermodynamic description of mixtures We see that: Can we quantify this difference, perhaps give it a name?

Thermodynamic description of mixtures Volume change of mixing: Enthalpy change of mixing:

Thermodynamic description of mixtures Volume change of mixing: Enthalpy change of mixing:

Thermodynamic description of mixtures Methyl formate+ethanol 298.15 K Methyl formate+methanol Benzene+C6F5Y at 298.15 K

Example 1 What is the density of an equimolar liquid solution of methyl formate and ethanol at 298.15 K, 1 atm? Data Density of pure ethanol at 298.15 K: 0.789 g/cm3 Density of pure methyl formate at 298.15 K: 0.977 g/cm3 Molar mass of ethanol: 46.07 g/mol Molar mass of methyl formate : 60.1 g/mol

Example 2 A continuous mixer operating at steady-state blends has two input streams (streams 1 and 2) to produce one output stream (stream 3), all of them liquid binary mixtures of water and sulfuric acid. Stream 1 has a temperature of 21.1oC and a sulfuric acid mass fraction of 0.3 and flow rate of 1 kg/s. Stream 2 is at the same temperature and a water mass fraction of 0.2 and flow rate of 2 kg/s. What is the molar enthalpy change of mixing of stream 1? What is the mass fraction of sulfuric acid in stream 3? What is the heat transfer rate to/from the mixer if the temperature of output stream is 21.1oC? What is the temperature of the output stream if the mixer operation is adiabatic?

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Thermodynamic description of mixtures Partial molar properties The discussion that follows is applicable to V, U, H, S, A, G, but it will be developed based on volume. Consider a mixture:

Thermodynamic description of mixtures Partial molar properties If the amounts of all components double:

Thermodynamic description of mixtures Partial molar properties If the amounts of all components are multiplied by a positive constant : The total volume is a homogeneous function of degree 1.

Thermodynamic description of mixtures Partial molar properties Now differentiate this expression with respect to :

Thermodynamic description of mixtures Partial molar properties

Thermodynamic description of mixtures The partial molar volume is defined as: The molar volume of the mixture is the average weighted by the mole fractions of the partial molar volumes (and not of the pure component molar volumes).

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Gibbs-Duhem equation The Gibbs-Duhem equation can be developed in generalized form (see the textbook). Here, it is developed based on the Gibbs energy:

Gibbs-Duhem equation Side note: Chemical potential of component i.

Gibbs-Duhem equation From the discussion of partial molar properties, we also have that:

Gibbs-Duhem equation The previous slides have two expressions for changes in Gibbs energy: Subtracting them:

Gibbs-Duhem equation This is the Gibbs-Duhem equation: Its interpretation is that the changes in temperature, pressure, and chemical potentials are interrelated. In other words, they cannot all change independently.

Generalized Gibbs-Duhem equation For a molar thermodynamic property (other than temperature, pressure, and mole numbers): Please refer to the proof in the textbook.

Experimental determination of partial molar volume and enthalpy The next slide shows a table of experimental density data for liquid mixtures of water and methanol at 298.15 K.

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Experimental determination of partial molar volume and enthalpy How to use the information available in this table to determine the partial molar volumes?

Binary solutions

Obtain dM/dx1 from (a)

Example 11.3 We need 2,000 cm3 of antifreeze solution: 30 mol% methanol in water. What volumes of methanol and water (at 25oC) need to be mixed to obtain 2,000 cm3 of antifreeze solution at 25oC Data:

solution Calculate total molar volume of the 30% mixture We know the total volume, calculate the number of moles required, n Calculate n1 and n2 Calculate the total volume of each pure species needed to make that mixture

Note curves for partial molar volumes

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Experimental determination of partial molar volume and enthalpy How to use the information available in this table to determine the partial molar volumes?

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Experimental determination of partial molar volume and enthalpy The molar volume change of mixing is: For a binary mixture:

Experimental determination of partial molar volume and enthalpy Let’s evaluate the slope of the curve in the binary diagram (at constant temperature and pressure):

Experimental determination of partial molar volume and enthalpy 0: Why?

Experimental determination of partial molar volume and enthalpy Generalized Gibbs-Duhem equation

Experimental determination of partial molar volume and enthalpy

Experimental determination of partial molar volume and enthalpy Note that:

Experimental determination of partial molar volume and enthalpy Note that:

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Experimental determination of partial molar volume and enthalpy The experimental data are often fitted to a polynomial form, known as Redlich-Kister expansion: For water(1) + methanol(2) at 298.15K with volume in m3/mol:

Equilibrium in multicomponent systems The discussion extends that of Chapter 7, which dealt with pure components. All the steps are very, very similar.

Equilibrium in an isolated system No mass in or out; neglect changes in kinetic and potential energies. The energy and entropy balances are: with because of the 2nd law of thermodynamics.

Equilibrium in an isolated system 0, no heat transfer to an isolated system 0, no change in volume in an isolated system 0, no heat transfer to an isolated system

Equilibrium in an isolated system for systems in which M, U, V are constant. Away from equilibrium, the system conditions change, increasing its entropy. After a long enough wait (from a fraction of a second to many years, depending on the system), the system attains a condition in which its state properties no longer change, including the entropy. At this state, the equilibrium state, the entropy has a maximum value.

Equilibrium in multicomponent systems Chemical reactions may occur in a multicomponent system. Let us begin with a situation in which chemical reactions do not occur.

Equilibrium in multicomponent systems Consider an isolated system composed of two subsystems containing a non-reactive multicomponent mixture:

Equilibrium in multicomponent systems Consider an isolated system composed of two subsystems containing a non-reactive multicomponent mixture: Find the equilibrium condition if the internal wall is diathermal, rigid, and impermeable. Find the equilibrium condition if the internal wall is diathermal, moveable, and impermeable.

Equilibrium in multicomponent systems Consider an isolated system composed of two subsystems containing a non-reactive multicomponent mixture: Find the equilibrium condition if the internal wall is diathermal, moveable, and permeable.

Equilibrium in multicomponent systems At constant U, V, Ni

Equilibrium in multicomponent systems At equilibrium:

Equilibrium in multicomponent systems In summary, at equilibrium:

Equilibrium in multicomponent systems Find the two-phase equilibrium conditions of a non-reactive multicomponent system with specified temperature, pressure, and amounts of each component.

Closed system, constant T and P At constant T and P 

Recommendation Read chapter 8 and review the corresponding examples.