Chemical Thermodynamics II Phase Equilibria Valentim M. B. Nunes ESTT - IPT April 2015

We live in a world of mixtures: the air we breathe, the petrol in the deposits of our automobiles, etc. Many of the chemical technology activities are related with the transfer of substances from a mixture to another. Phase equilibrium: transfer of substances between phases. Many of the operations in the industry, such as extraction, distillation, etc., involve transfer between phases. To scale these units it is necessary to characterize the equilibrium properties of the several phases.

Gibbs Phase Rule Determines the number of degrees of freedom, g, or number of intensive properties necessary to specify a given state of equilibrium, with F phases and C components.

The basic problem:   T p

Solution of the problem: Gibbs! For any component i: To relate  with T and p and x1, x2, etc., it’s necessary to introduce the concepts of fugacity and activity. For instance if  = vapor and  = liquid, then, as we will see later:

Classic thermodynamics of phase equilibria For a homogeneous closed system, the combination of the first and second law of thermodynamics gives, for a reversible process in which the State of equilibrium is maintained: “Work done by the system” “Heat absorbed by the system” Applies to any process, reversible or irreversible, since the initial and final State are equilibrium States. For a finite change:

Prausnitz et al., Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice-Hall, New Jersey, 1986

For an open system, there may be an exchange of matter (and also energy), with the exterior.

For a heterogeneous closed system, we can consider each phase as a homogenous system open. Resuming: Integrating from a state of zero mass (U=S=V=…=0), until a given finite state at constant temperature, pressure and composition we obtain:

Differentiating again: Then: Gibbs-Duhem equation: fundamental equation of the thermodynamics of solutions, restricts the simultaneous variation of T, p and i in a given phase.

For a pure substance, For a ideal gas, Vi = RT/p

To generalize, Lewis introduced the concept of fugacity. Valid for an isothermal process, for any component, whether solid, liquid or gas, pure or in mixtures, ideal or not!

For a ideal gas, f = p. For a component i in an ideal gaseous mixture, the fugacity of i is the partial pressure, fi = pi = yi p. For all systems: Lewis called the relation f/f0 as activity.

An important consequence:

 If we consider: Assuming the standard states of the two phases at the same temperature but not to the same pressure and composition Equilibrium accordingly with Lewis

A simple case, the Raoult's Law Consider the equilibrium between a liquid phase and a vapor phase. For the component 1 the condition of equilibrium implies that:

The Raoult's law in practical calculations its of reduced utility The Raoult's law in practical calculations its of reduced utility. It is necessary to use molecular thermodynamics to relate the fugacity of compounds with its molecular characteristics.