Thermodynamic flows

Ti = Temperature of the gas at the point of origin, °F W = Thermodynamic flow work term... 

In Sections IVA, VA, and VI the nonequilibrium probability distribution is given in phase space for steady-state thermodynamic flows, mechanical work, and quantum systems, respectively. (The second entropy derived in Section II gives the probability of fluctuations in macrostates, and as such it represents the nonequilibrium analogue of thermodynamic fluctuation theory.) The present phase space distribution differs from the Yamada-Kawasaki distribution in that... 

A thermodynamic flow system may be fully described in n-dimensions of flow and w-dimensions of conjugate force. According to Tellegen s theorem, we have... 

Jolicoeur, Carmel, Thermodynamic Flow Methods in Biochemistry ... 

Thermodynamic Flow Methods in Biochemistry Calorimetry, Densimetry and Dilatometry... 

Onsager (1931) used a set of equations that expresses in an explicit manner the linear dependence of the thermodynamic flows on the thermodynamic forces. These equations, known as the phenomenological equations, can be expressed as... 

Our task now is to obtain explicit expressions for d S and dyS in terms of experimentally measurable quantities. Irreversible processes can be described in terms of thermodynamic forces and thermodynamic flows. The thermodynamic flows are a consequence of the thermodynamic forces. Figure 3.8 shows how a difference in temperature between adjacent parts of a system (or temperature gradient) is the thermodynamic force that causes the irreversible flow of heat. Similarly, a concentration difference between two adjacent parts of a system is the thermodynamic force that causes the flow of matter. In general, the irreversible change d S is associated with a flow dXofa quantity such as heat or matter that has occurred in a time dt. For the flow of heat, we have dX = dQ, the amount of heat that flowed in time dv, for the case of matter, we have dX — dN, the number of moles of the substance that flowed in time dt. In each case the change in entropy can be written in the form... 

F = (1/rcoid — 1/ hot)- For the flow of matter, the corresponding thermodynamic force is expressed in terms of affinity, a concept developed in Chapter 4. All irreversible processes can be described in terms of thermodynamic forces and thermodynamic flows. The entropy change is the sum of all the changes due to irreversible flows dXk. We then have the general expression... 

De Donder also defined the affinity of a chemical reaction that enables us to write expression (4.1.10) in an elegant form, as the product of a thermodynamic force and a thermodynamic flow. The concept of affinity can be understood through the following simple example. 

At equilibrium the thermodynamic flows and hence the entropy production must vanish. This implies that in the state of equilibrium the affinity of a chemical reaction A = 0. Thus we arrive at the conclusion that, at thermodynamic equilibrium, the chemical potentials of the compounds X, Y and Z will reach values such that... 

The condition that the thermodynamic force, affinity A equals zero implies that the corresponding thermodynamic flow, i.e. the reaction velocity d /dt, also equals zero. The condition A = 0 means that at equilibrium the stoichiometric sums for the chemical potentials of the reactants and products are equal, as in (9.3.3). It is easy to generalize this result to an arbitrary chemical reaction of the form... 

A gradient in this chemical potential will result in a thermodynamic flow... 

Let us go back to our book. Entropy production plays a central role in our presentation. As we show in Chapter 15, the entropy production can be expressed in terms of thermodynamic flows /, and thermodynamic forces Z,. An example is heat conduction where 7,- is the flow of heat and Xi the gradient of temperature. We can now distinguish three stages. At equilibrium both the flows and the forces vanish. This is the domain of traditional thermodynamics. It is covered in Chapters 5-11. The reader will find many results familiar from all textbooks on thermodynamics. 

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