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 cj and temperature. For this purpose, let us the representation of the activity through the chemical potential:
where μi is the chemical potential of the species under the conditions of interest, μ
oi 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 cj. 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 Nj (usually measured in moles), N is the vector with components Nj. 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
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.
Read more about this topic: Detailed Balance
Famous quotes containing the words balance, detailed, dissipation and/or systems:
“... idleness is an evil. I dont think man can maintain his balance or sanity in idleness. Human beings must work to create some coherence. You do it only through work and through love. And you can only count on work.”
—Barbara Terwilliger (b. c. 1940)
“[The Republicans] offer ... a detailed agenda for national renewal.... [On] reducing illegitimacy ... the state will use ... funds for programs to reduce out-of-wedlock pregnancies, to promote adoption, to establish and operate childrens group homes, to establish and operate residential group homes for unwed mothers, or for any purpose the state deems appropriate. None of the taxpayer funds may be used for abortion services or abortion counseling.”
—Newt Gingrich (b. 1943)
“Crowds without company, and dissipation without pleasure.”
—Edward Gibbon (17371794)
“No civilization ... would ever have been possible without a framework of stability, to provide the wherein for the flux of change. Foremost among the stabilizing factors, more enduring than customs, manners and traditions, are the legal systems that regulate our life in the world and our daily affairs with each other.”
—Hannah Arendt (19061975)