Thermodynamic similarity

Imagine a gas consisting of a single component in a big vessel. There should not be any external forces acting on the gas, such as gravity. It should be sufficient to describe the gas in the thermodynamic sense completely by three variables, i.e., the entropy, the volume, and the number of particles. 

To find out the state of the gas, we probe a small, but still macroscopic, region within the vessel with the volume V. We will find a certain mol number of particles n and we will find a certain content of entropy S in this particular probed volume. If we probe in another region a volume of the same size, we will find the same values of entropy and mol number. Thus, from the thermodynamic data afterward we cannot find out from which region we had probed. 

We address the two regions from where we had probed as two subsystems of the gas in the vessel. If we combine the two subsystems into a single one, it is intuitively obvious that in the combined subsystem the volume, the number of particles, and further the entropy will be additive. Also, the energy will be additive. This is the condition of a homogeneous system, which is also addressed as a phase. 

If the vessel contains both liquid and gas, then the additivity will no longer hold in general, but is, if at all, restricted to carefully selected regions. In particular, it must be probed along an interphase border, in a certain position to catch equal amounts of the phases and surface. 

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Borishansky, V. M., 1961, Allowing for the Influence of Pressure on the Heat Transfer and Critical Thermal Loads during Boiling in Accordance with the Theory of Thermodynamic Similarity, Voprosy Teplootdachi i gidravliki Dvukhfaznykh Sred, Gosenergoizdet, Moscow. (2)... 

Borishansky, V. M., I. I. Novikov, and S. S. Kutateladze, 1961, Use of Thermodynamic Similarity in Generalizing Experimental Data of Heat Transfer, Int. Dev. Heat Transfer, Part 2, pp. 475-482, ASME, New York. (5)... 

Leiner, W., 1994, Heat Transfer by Nucleate Pool Boiling—General Correlation Based on Thermodynamic Similarity, Int. J. Heat Mass Transfer 37(5) 763-769. (2)... 

GEOMETRICAL REPRESENTATION OF EQUILIBRIUM THERMODYNAMICS Similarly, we can state the general Schwarz inequality... 

Solovev, A. N. Thermodynamic similarity and viscosity of molten metals. Atomnaya Energiya, Moskau 3, 550 (1957). 

If the excess thermodynamic properties of the three binary subsystems of the A-B-C ternary system are similar to each other, the ternary system is symmetric. If the deviation of the binary system A-B and A-C from the ideal behavior are similar, but differ markedly from that of the binary system B-C, then the A-B-C ternary system is asymmetric. In the asymmetric system the component A in two binary subsystems with thermodynamic similarity should be chosen as the thermodynamic asymmetric component. ... 

Skripov, V.P., and Faizullin, M.Z. (2006) Crystal-Liquid-Gas Phase Transitions and Thermodynamic Similarity, Wiley-VCH... 

Figure 2. The elliptical temperature-pressure stability phase diagram characteristic for proteins. After Suzuki [2] and Hawley [4], Note the thermodynamic similarities between the cold, c, and pressure, p, unfolding and the contrast with heat, h, unfolding.

W. Leiner and D. Gorenflo, Methods of Predicting the Boiling Curve and a New Equation Based on Thermodynamic Similarity, in Pool and External Flow Boiling, V. K. Dhir and A. E. Bergles eds., pp. 99-103, ASME, New York, 1992. 

A theoretical study has been performed in order to estimate the ability of 1,2,4-triazines to undergo N-protonation reactions. The ab initio calculations, obtained using HF/6-31G, have shown that the N2-H triazinium cation is the most stable thermodynamically similar characteristics have been obtained for Nj- and N2-H 1,2,4-triazinium salts derived from protonation of 6-phenyl-l,2,4-triazine <2003CHE616>. 

Of practical importance are electrochemical systems with electrodes. Electrical and gravitational systems may be of a pseudo first order with respect to charge and mass additions, when only the contribution of the external field to the energy is considered. However, this does not mean that these electrical and gravitational contributions should be included into the Gibbs -Duhem equation [16]. Of course, it is a matter of definition of the thermodynamic variables, whether the thermodynamic similarity exists or not. 

We can use this fact to check whether the variable set and the functional dependence of the variable set obey the demand that the complete Legendre transformation is zero. Only then, we have a function of the energy that obeys thermodynamic similarity. 

Because of the thermodynamic similarity, i.e., the volume is a homogeneous function of first order... 

We will address the homogeneity according to Eq. (2.35) as thermodynamic similarity. This similarity outs itself that an enlargement of the whole system is effective, when two systems of the same kind are put into one system. For instance, with regard to kinetic energy, if we consider two particles with the same velocity, we may integrate these two particles into one system to double the momentum, but having the same velocity. 

Question 6.4. Is the form U S, V, n, h) thermodynamically similar, or in other words, is it Eulerian homogeneous of first order ... 

Try to reformulate it with the gravity term Mgq with q = nh and set C " = 0. For comparison, recall the momentum as extensive variable p = mv. The derivation of Eq. (6.57) works because we do not make there explicit use of the thermodynamic similarity. 

The equation (5.146) contains only the reduced parameters, therefore if two different substances are described by the Van der Waals equation and have identical values of pr and T, their reduced molar volumes Or must also be the same. This law is known as the principle of homologous states. It lies at the basis of the theory of thermodynamic similarity. 

Thus, Sadi Carnot s analysis of Carnot cycle provided the theory for the formulation of the first and the second law of thermodynamics. His concept is that for a system undergoing a cycle, the net heat transfer is equal to the net work done, which led to the first law of thermodynamics. Similarly, the concept that a heat engine cannot convert all the heat absorbed from a heat source at a single temperature into work even under ideal condition led to the second law of thermodynamics. Carnot cycle efficiency gives the idea about the maximmn theoretical efficiency of an engine. Sadi Carnot was rightly honored with the title Father of Thermodynamics for his invaluable contribution to thermodynamics. 

Blends are classified as either thermodynamically miscible or immiscible, with the latter dominating. However, imposition of a flow affects the thermodynamic equilibrium and it may enhance the miscibiUty of immiscible blends or vice-versa - there is an interrelation between rheology and thermodynamics. Similarly, flow affects the degree of deformation of the dispersed phase, thus in immiscible blends there are other interrelations between rheology and morphology, which affect the blend performance. To the complexity of polymer alloys and blends (PAB) behavior one must add the incorporation of soUds, either in the form of filler and nanofiller particles or by simple factofblendingtwocomponents with widely differenttransitiontemperatures. 

Crystal-Liquid-Gas Phase Transitions and Thermodynamic Similarity... 

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