Phase Modeling Tools: Applications to Gases

By Michel Soustelle

This book is part of a set of books which offers advanced students successive characterization tool phases, the study of all types of phase (liquid, gas and solid, pure or multi-component), process engineering, chemical and electrochemical equilibria, and the properties of surfaces and phases of small sizes. Macroscopic and microscopic models are in turn covered with a constant correlation between the two scales. Particular attention has been given to the rigor of mathematical developments.

• Language: English
• Publisher: Wiley
• Release date: Aug 5, 2015
• ISBN: 9781119178477

Book preview

1

Thermodynamic Functions and Variables

In thermodynamics, each state of a system is characterized by the value given by a certain characteristic function, which depends on a certain number of variables: the state variables. The most commonly cited variables include the internal energy U, enthalpy H, Helmholtz energy F and Gibbs energy G. Therefore, the complete properties of a system are known if one of the characteristic functions is known according to the variables chosen to define the problem.

These characteristic functions also possess two properties, according to the two first principles of thermodynamics:

– these are the state functions, that is to say, their variations during a transformation only depend on the initial and final state and their differentials are exact total differentials (see Appendix 3, section A3.2);

– they are each a potential function in a set of given variables, i.e. they have a minimal value for a system in equilibrium and vice versa.

If the system is composed of several phases, the value of the characteristic function of the overall system is the sum of the values of this function for each phase.

NOTE.– We know that if a phase contains several components, the value of the characteristic function of the phase is not the sum of the values of these same functions for each of the components.

1.1. State variables and characteristic functions of a phase

The number and nature of variables, used to study the state or thermodynamic modifications of a phase, are chosen by the operator. This limits the number of variables in order to not complicate the problem.

1.1.1. Intensive and extensive conjugate variables

Variables that can be chosen are classed in couples. The product of the variables of the same couple is homogeneous at a given energy, that is to say, the equation with dimensions: ML²T-2. One of the variables in the couple is an intensive variable, i.e. homogeneous to zero degrees with regards to quantities of matter, and the other is an extensive variable, i.e. homogeneous at degree one with regards to quantities of matter (see Appendix 3, section A3.1).

Table 1.1 shows the main couples, where pressure–volume and temperature–entropy are the two couples most frequently encountered in chemical thermodynamics, with, for each component, the couple quantities of matter–chemical potentials. The couples electric charge–electric potential and area–surface tension are found in electrochemistry and surface chemistry, respectively.

The third column in the table provides an expression of the corresponding work in the forms YdX, X and Y, the extensive variable and intensive variable of the couple, respectively.

Note that with volume being the intensive variable, opposed pressure must respect the form of the pressure work forces.

When dealing with the thermodynamic modeling of a transformation, it is best to first choose the physical phenomena involved. From Table 1.1, the conjugate variable couples used are deduced.

For example, in the case of a steam engine machine in which the phenomena involved are heating and changes in the state of water, the following thermomechanical couples are used: temperature–entropy (T, S) and opposed pressure–volume (-P, V). To study a chemical transformation in condensed phases at atmospheric pressure, incapable of varying or even influencing the phenomenon, the couple temperature–entropy is chosen and, for each component, the couple quantity of matter and chemical potential.

Table 1.1. Couples of conjugate variables and corresponding reversible work