Detailed Balance - Dissipation in Systems With Detailed Balance

Dissipation in Systems With Detailed Balance

To describe dynamics of the systems that obey the generalized mass action law, one has to represent the activities as functions of the concentrations c_j and temperature. For this purpose, let us the representation of the activity through the chemical potential:

where μ_i is the chemical potential of the species under the conditions of interest, μo_i is the chemical potential of that species in the chosen standard state, R is the gas constant and T is the thermodynamic temperature. The chemical potential can be represented as a function of c and T, where c is the vector of concentrations with components c_j. For the ideal systems, and : the activity is the concentration and the generalized mass action law is the usual law of mass action.

Let us consider a system in isothermal (T=const) isochoric (the volume V=const) condition. For these conditions, the Helmholtz free energy F(T,V,N) measures the “useful” work obtainable from a system. It is a functions of the temperature T, the volume V and the amounts of chemical components N_j (usually measured in moles), N is the vector with components N_j. For the ideal systems,

The chemical potential is a partial derivative: .

The chemical kinetic equations are

If the principle of detailed balance is valid then for any value of T there exists a positive point of detailed balance ceq:

Elementary algebra gives

where

For the dissipation we obtain from these formulas:

The inequality holds because ln is a monotone function and, hence, the expressions and have always the same sign.

Similar inequalities are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for isothermal isobaric conditions the Gibbs free energy decreases, for the isochoric systems with the constant internal energy (isolated systems) the entropy increases as well as for isobaric systems with the constant enthalpy.