(181u) Real Pure Component Systems in the View of Discrete Modeling

Discrete modeling is a kind of thermodynamic modeling introducing discrete models for the state functions entropy, internal energy, particle numbers, and system volume. The key item of the method is the usage of Shannon’s formulation of information [1] H = -Σ p_i log p_i, where the p_i are the probabilities of a particle to reside in the system state i. H can be interpreted as a measure of entropy. In combination with Gibbs’ thermodynamics this leads to an explicit expression of the entropy given by internal energy, system volume and particle numbers S=S(U,V,N), that can be used e.g. to deduce thermic and caloric equations of state.

The Case of Ideal Gas

The case of the ideal gas illustrates the most important step of the method: the definition of the system’s state, i.e. what variables define the mechanical state of a particle, and how they have to be used in order to achieve the correct expression for the discrete entropy. With this expression only few steps are necessary to obtain the ideal-gas law. But additionally the analysis of the discrete model of the internal energy yields the Maxwell-Boltzmann distribution of energies and the heat capacity for the ideal gas - an impressive proof of the method.

Treating the Real Gas

With the understanding of discrete modeling of the system’s state in the case of the ideal gas it becomes now clear how the interaction forces between real gas particles have to be incorporated in the discrete model not only for the internal energy, but also for the entropy in form of an expression for the potential energy. It will be shown, how a relatively simple approach of the potential energy yields the Van der Waals equation. Based on this result the modeling of more sophisticated approaches will be discussed.

References

[1] Shannon, C.E., (1948), A mathematical theory of communication, Bell Syst. Tech. J. 27, 379, 623.