Effect of Stagnation Temperature on Supersonic Flow Parameters with Application for Air in Nozzles 439 C F M=2.00 M=3.00 M=4.00 M=5.00 M=6.00 PG (γ=1.402) 1.2078 1.4519 1.5802 1.6523 1.6959 T 0 =298.15 K 1.2078 1.4518 1.5800 1.6521 1.6957 T 0 =500 K 1.2076 1.4519 1.5802 1.6523 1.6958 T 0 =1000 K 1.2072 1.4613 1.5919 1.6646 1.7085 T 0 =1500 K 1.2062 1.4748 1.6123 1.6871 1.7317 T 0 =2000 K 1.2048 1.4832 1.6288 1.7069 1.7527 T 0 =2500 K 1.2042 1.4879 1.6401 1.7221 1.7694 T 0 =3000 K 1.2038 1.4912 1.6479 1.7337 1.7828 T 0 =3500 K 1.2033 1.4936 1.6533 1.7422 1.7932 Table 9. Numerical values of the thrust coefficient at high temperature 123456 Exit Mach number 0.0 0.5 1.0 1.5 2.0 4 3 2 1 Fig. 16. Variation of C F versus exit Mach number. 5.3 Results for the error given by the perfect gas model Figure 17 presents the relative error of the thermodynamic and geometrical parameters between the PG and the HT models for several T 0 values. It can be seen that the error depends on the values of T 0 and M. For example, if T 0 =2000 K and M=3.00, the use of the PG model will give a relative error equal to ε=14.27 % for the temperatures ratio, ε=27.30 % for the density ratio, error ε=15.48 % for the critical sections ratio and ε=2.11 % for the thrust coefficient. For lower values of M and T 0 , the error ε is weak. The curve 3 in the figure 17 is under the error 5% independently of the Mach number, which is interpreted by the use potential of the PG model when T 0 <1000 K. We can deduce for the error given by the thrust coefficient that it is equal to ε=0.0 %, if M E =2.00 approximately independently of T 0 . There is no intersection of the three curves in the same time. When M E =2.00. Thermodynamics – Interaction Studies – Solids, Liquids and Gases 440 123456 Mach number 5 10 15 20 1 2 3 (a) 123456 Mach number 0 20 40 60 80 100 1 2 3 (b) 123456 Mach number 0 10 20 30 40 50 1 2 3 (c) 123456 Exit Mach number 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 3 2 1 1 2 3 (d) Curve 1 Error compared to HT model for (T 0 =3000 K) Curve 2 Error compared to HT model for (T 0 =2000 K) Curve 3 Error compared to HT model for (T 0 =1000 K) (a): Temperature ratio. (b): Density ratio. (c): Critical sections ratio. (d): Thrust coefficient. Fig. 17. Variation of the relative error given by supersonic parameters of PG versus Mach number. 5.4 Results for the supersonic nozzle application Figure 18 presents the variation of the Mach number through the nozzle for T 0 =1000 K, 2000 K and 3000 K, including the case of perfect gas presented by curve 4. The example is selected for M S =3.00 for the PG model. If T 0 is taken into account, we will see a fall in Mach number of the dimensioned nozzle in comparison with the PG model. The more is the temperature T 0 , the more it is this fall. Consequently, the thermodynamics parameters force to design the nozzle with different dimensions than it is predicted by use the PG model. It should be noticed that the difference becomes considerable if the value T 0 exceeds 1000 K. Figure 19 present the correction of the Mach number of nozzle giving exit Mach number M S , dimensioned on the basis of the PG model for various values of T 0 . One can see that the curves confound until Mach number M S =2.0 for the whole range of T 0 . From this value, the difference between the three curves 1, 2 and 3, start to increase. The curves 3 and 4 are almost confounded whatever the Mach number if the value of T 0 is lower than 1000 K. For example, if the nozzle delivers a Mach number M S =3.00 at the exit section, on the assumption of the PG model, the HT model gives Mach number equal to M S =2.93, 2.84 and 2.81 for T 0 =1000 K, 2000 K and 3000 K respectively. The numerical values of the correction of the exit Mach number of the nozzle are presented in the table 10. Effect of Stagnation Temperature on Supersonic Flow Parameters with Application for Air in Nozzles 441 0 2 4 6 8 10121416 0. 0 1.0 2.0 3.0 4.0 (a) 0 2 4 6 8 10 12 14 16 Non-dimensional X-coordinates 1.0 1.5 2.0 2.5 3.0 M 4 3 2 1 (b) (a): Shape of nozzle, dimensioned on the consideration of the PG model for M S =3.00. (b): Variation of the Mach number at high temperature through the nozzle. Fig. 18. Effect of stagnation temperature on the variation of the Mach number through the nozzle. M S (PG γ=1.402) 1.5000 2.0000 3.0000 4.0000 5.0000 6.0000 M S (T 0 =298.15 K) 1.4995 1.9995 2.9995 3.9993 4.9989 5.9985 M S (T 0 =500 K) 1.4977 1.9959 2.9956 3.9955 4.9951 5.9947 M S (T 0 =1000 K) 1.4879 1.9705 2.9398 3.9237 4.9145 5.9040 M S (T 0 =1500 K) 1.4830 1.9534 2.8777 3.8147 4.7727 5.7411 M S (T 0 =2000 K) 1.4807 1.9463 2.8432 3.7293 4.6372 5.5675 M S (T 0 =2500 K) 1.4792 1.9417 2.8245 3.6765 4.5360 5.4209 M S (T 0 =3000 K) 1.4785 1.9388 2.8121 3.6454 4.4676 5.3066 M S (T 0 =3500 K) 1.4778 1.9368 2.8035 3.6241 4.4216 5.2237 Table 10. Correction of the exit Mach number of the nozzle. Figure 20 presents the supersonic nozzles shapes delivering a same variation of the Mach number throughout the nozzle and consequently given the same exit Mach number M S =3.00. The variation of the Mach number through these 4 nozzles is illustrated on curve 4 of figure 18. The three other curves 1, 2, and, 3 of figure 15 are obtained with the HT model use for T 0 =3000 K, 2000 K and 1000 K respectively. The curve 4 of figure 20 is the same as it is in the figure 13a, and it is calculated with the PG model use. The nozzle that is calculated according to the PG model provides less cross-section area in comparison with the HT model. Thermodynamics – Interaction Studies – Solids, Liquids and Gases 442 123456 M ach number f or Per f ect Ga s 0 1 2 3 4 5 6 M (HT) 1 2 3 4 Fig. 19. Correction of the Mach number at High Temperature of a nozzle dimensioned on the perfect gas model. 0246810121416 Non-dimensional X-coordinates 0.0 1.0 2.0 3.0 4.0 5.0 4 3 2 1 Fig. 20. Shapes of nozzles at high temperature corresponding to same Mach number variation througout the nozzle and given M S =3.00 at the exit. 6. Conclusion From this study, we can quote the following points: If we accept an error lower than 5%, we can study a supersonic flow using a perfect gas relations, if the stagnation temperature T 0 is lower than 1000 K for any value of Mach number, or when the Mach number is lower than 2.0 for any value of T 0 up to approximately 3000 K. The PG model is represented by an explicit and simple relations, and do not request a high time to make calculation, unlike the proposed model, which requires the resolution of a nonlinear algebraic equations, and integration of two complex analytical functions. It takes more time for calculation and for data processing. The basic variable for our model is the temperature and for the PG model is the Mach number because of a nonlinear implicit equation connecting the parameters T and M. Effect of Stagnation Temperature on Supersonic Flow Parameters with Application for Air in Nozzles 443 The relations presented in this study are valid for any interpolation chosen for the function C P (T). The essential one is that the selected interpolation gives small error. We can choose another substance instead of the air. The relations remain valid, except that it is necessary to have the table of variation of C P and γ according to the temperature and to make a suitable interpolation. The cross section area ratio presented by the relation (19) can be used as a source of comparison for verification of the dimensions calculation of various supersonic nozzles. It provides a uniform and parallel flow at the exit section by the method of characteristics and the Prandtl Meyer function (Zebbiche & Youbi, 2005a, 2005b, Zebbiche, 2007, Zebbiche, 2010a & Zebbiche, 2010b). The thermodynamic ratios can be used to determine the design parameters of the various shapes of nozzles under the basis of the HT model. We can obtain the relations of a perfect gas starting from the relations of our model by annulling all constants of interpolation except the first. In this case, the PG model becomes a particular case of our model. 7. Acknowledgment The author acknowledges Djamel, Khaoula, Abdelghani Amine, Ritadj Zebbiche and Fettoum Mebrek for granting time to prepare this manuscript. 8. References Anderson J. D. Jr (1982), Modern Compressible Flow. With Historical Perspective, (2 nd edition), Mc Graw-Hill Book Company, ISBN 0-07-001673-9. New York, USA. Anderson J. D. Jr. (1988), Fundamentals of Aerodynamics, (2 nd edition), Mc Graw-Hill Book Company, ISBN 0-07-001656-9, New York, USA. Démidovitch B. et Maron I. (1987), Eléments de calcul numérique, Editions MIR, ISBN 978-2- 7298-9461-0, Moscou, USSR. Fletcher C. A. J. (1988), Computational Techniques for Fluid Dynamics: Specific Techniques for Different Flow Categories, Vol. II, Springer Verlag, ISBN 0-387-18759-6, Berlin, Heidelberg. Moran M. J., (2007). Fundamentals of Engineering Thermodynamics, John Wiley & Sons Inc., 6 th Edition, ISBN 978-8-0471787358, USA Oosthuisen P. H. & Carscallen W. E., (1997), Compressible Fluid Flow. Mc Grw-Hill, ISBN 0- 07-0158752-9, New York, USA. Peterson C.R. & Hill P. G. (1965), Mechanics and Thermodynamics of Propulsion, Addition- Wesley Publishing Company Inc., ISBN 0-201-02838-7, New York, USA. Ralston A. & Rabinowitz P. A. (1985). A First Course in Numerical Analysis. (2 nd Edition), McGraw-Hill Book Company, ISBN 0-07-051158-6, New York, USA. Ryhming I. L. (1984), Dynamique des fluides, Presses Polytechniques Romandes, Lausanne, ISBN 2-88074-224-2, Suisse. Zebbiche T. (2007). Stagnation Temperature Effect on the Prandtl Meyer Function. AIAA Journal, Vol. 45 N° 04, PP. 952-954, April 2007, ISSN 0001-1452, USA Zebbiche T. & Youbi Z. (2005a). Supersonic Flow Parameters at High Temperature. Application for Air in nozzles. German Aerospace Congress 2005, DGLR-2005-0256, 26-29 Sep. 2005, ISBN 978-3-8322-7492-4, Friendrichshafen, Germany. Thermodynamics – Interaction Studies – Solids, Liquids and Gases 444 Zebbiche T. & Youbi Z., (2005b). Supersonic Two-Dimensional Minimum Length Nozzle Conception. Application for Air. German Aerospace Congress 2005, DGLR-2005-0257, 26-29 Sep. 2005, ISBN 978-3-8322-7492-4, Friendrichshafen, Germany. Zebbiche T. & Youbi Z. (2006), Supersonic Plug Nozzle Design at High Temperature. Application for Air, AIAA Paper 2006-0592, 44 th AIAA Aerospace Sciences Meeting and Exhibit, 9-12 Jan. 2006, ISBN 978-1-56347-893-2, Reno Nevada, Hilton, USA. Zebbiche T., (2010a). Supersonic Axisymetric Minimum Length Conception at High Temperature with Application for Air. Journal of British Interplanetary Society (JBIS), Vol. 63, N° 04-05, PP. 171-192, May-June 2010, ISBN 0007-084X, 2010. Zebbiche T., (2010b). Tuyères Supersoniques à Haute Température. Editions Universitaires Européennes. ISBN 978-613-1-50997-1, Dudweiler Landstrabe, Sarrebruck, Germany. Zuker R. D. & Bilbarz O. (2002). Fundamentals of Gas Dynamics, John Wiley & Sons. ISBN 0- 471-05967-6, New York, USA 17 Statistical Mechanics That Takes into Account Angular Momentum Conservation Law - Theory and Application Illia Dubrovskyi Institute for Metal Physics National Academy of Science Ukraine 1. Introduction The fundamental problem of statistical mechanics is obtaining an ensemble average of physical quantities that are described by phase functions (classical physics) or operators (quantum physics). In classical statistical mechanics the ensemble density of distribution is defined in the phase space of the system. In quantum statistical mechanics the space of functions that describe microscopic states of the system play a role similar to the classical phase space. The probability density of the system detection in the phase space must be normalized. It depends on external parameters that determine the macroscopic state of the system. An in-depth study of the statistical mechanics foundations was presented in the works of A.Y. Khinchin (Khinchin, 1949, 1960). For classical statistical mechanics an invariant set was introduced. It would be mapped into itself by transforming with the Hamilton equations. The phase point of the isolated system remains during the process of the motion at the invariant set at all times. If the system is in the stationary equilibrium state, this invariant set has a finite measure. The Ergodic hypothesis asserts that in this case the probability  dP R to detect this system at any point R of the phase space is:     3 d d 2! N P N            R R  (1) where   - the measure (phase volume) of the invariant set  ;     R - the characteristic function of the invariant set, which is equal to one if the point R belongs to this set, and is equal to zero in all other points of the phase space; =1 d= d d N ii i   pr- the phase space volume element. The number of distinguishable states in a phase space volume element d  is  1 3 2! N N      . The system that will be under consideration is a collection of N structureless particles. The averaged value of a phase function   F R is     dFF P    RR. Here the integral goes over all phase space  . This is microcanonical distribution. A characteristic function often would be presented as   f z      R , where   f R is a phase function and z is it’s fixed value. A hypersurface in a hyperspace is a set with zero measure. Therefore the invariant set is determined as a thin layer that nearly envelops the hypersurface in the phase space. The Thermodynamics – Interaction Studies – Solids, Liquids and Gases 446 determining equations of this hypersurface are the equalities that fix the values of controllable motion integrals. A controllable motion integral is a phase function, the value of which does not vary with the motion of the system and can be measured. An isolated system universally has the Hamiltonian that does not depend on the time explicitly, and is the controllable motion integral. A fixed value of the Hamiltonian is the energy of the system. The kinetic energy of majority of systems is a positive definite quadric form of all momenta. It determines a closed hypersurface in the subspace of momenta of the phase space. If motions of all particles are finite, the hypersurface of the fixed energy is closed and the layer that envelops it has the finite measure. Then this hypersurface can determine the invariant set of the system. A finiteness of motions of particles as a rule is provided by enclosing the system in an envelope that reflects particles without changing their energy, if the system is considered as isolated. It is common in statistical mechanics to consider the layer enveloping the energy hypersurface as the invariant set. But A.Y. Khinchin (Khinchin, 1949) shows that other controllable integrals of the system, if they exist, must be taken into account. In the general case an isolated system can have another two vector controllable integrals. That is the total momentum of the system, and the total angular momentum relative to the system’s mass centre. The total momentum is a sum of all momenta of particles. If the volume of the system is bounded by an external field or an envelope, the total momentum does not conserve. In the absence of external fields the total momentum conservation cannot make particle motions finite. Therefore the total momentum cannot be a controllable motion integral that determines the invariant set. The angular momentum is another case. A vector of angular momentum relative to the mass center always is conserved in an isolated system. If this vector is nonzero, a condition should exist that provides a limitation of a gas expansion area. For example, nebulas do not collapse because they rotate, and do not scatter because of the gravitation. In the system of charged particles in a uniform magnetic field the conservation of the angular momentum provides a limitation of a gas expansion area (confinement of plasma). If a gas system is enclosed into envelope, and total system has nonzero angular momentum, the vector of the angular momentum should be conserved. However an envelope can have the non-ideal form and surface. That is the cause of the failure to consider the angular momentum of the gas as a controllable motion integral (Fowler, & Guggenheim, 1939). But if the cylindrical envelope rotates and the gas rotates with the same angular velocity deviations of the angular momentum of the gas from the fixed value as the result of reflections of particles from the envelope should be small and symmetric with respect to a sign. These fluctuations are akin to energy fluctuations for a system that is in equilibrium with a thermostat. Therefore the angular momentum conservation in specific cases can determine the invariant set and the thermodynamical natures of the system together with the energy conservation. Taking into account all controllable motion integrals is the necessary condition of the validity of the Ergodic hypothesis (Khinchin, 1949). There is a contradiction in physics at the present time. Firstly, it has been proven that in the equilibrium state a system spin can exist only if the system is rigid and can rotate as a whole (Landau, & Lifshitz, E.M., 1980a). Therefore a gas, which supposed not be able to rotate as a whole, cannot have any angular momentum and spin. Based on this reasoning R.P. Feynman proves that an electron gas cannot have diamagnetism (the Bohr – van Leeuwen theorem) (Feynman, Leighton, & Sands, 1964). On the other hand, it is well known that density of a gas in a rotating centrifuge is non-uniform. This effect is used for the separation Statistical Mechanics That Takes into Account Angular Momentum Conservation Law - Theory and Application 447 of isotopes (Cohen, 1951). The experiment by R. Tolman, described in the book (Pohl, 1960), is a proof of the existence of the electron gas angular momentum. In this experiment a coil was rotated and then sharply stopped. An electrical potential was observed that generated a moment of force, which decreased to zero the angular momentum of electron gas. The contradiction described above requires creation of statistical mechanics for non-rigid systems taking into account the nonzero angular momentum conservation. This statistical mechanics differs from common one in many respects. If the angular momentum relative to the axis that passes through the mass centre conserves, the system is spatially inhomogeneous. This means that passage to the thermodynamical limit makes no sense, a spatial part of the system is not a subsystem that similar to the total system, specific quantities such as densities or susceptibilities have no physical meaning. The microcanonical distribution is seldom used directly when the computations and the justifications of thermodynamics are done. The more usable Gibbs distribution can be deduced from microcanonical one (Krutkov, 1933; Zubarev, 1974). The Gibbs assembly describes a system that is in equilibrium with environment. These systems do not have motion integrals because they are non-isolated. All elements of the Gibbs assembly must have equal values of parameters that are determined by the equilibrium conditions. In usual thermodynamics this parameters are the temperature and the chemical potential. The physical interpretation of these parameters is getting by statistical mechanics. A rotating system can be in equilibrium only with rotating environment. The equilibrium condition in this case is apparent. That is equality of the both angular velocities of the system and of the environment. The Gibbs assembly density of distribution and thermodynamical functions in the case of a rotating classical system will be obtained in the second section of this work. It was done (Landau, & Lifshitz E.M., 1980a) but an object, to which this distribution is applied, is incomprehensible, because an angular velocity of an equilibrium gas has not been determined. In quantum statistical mechanics the invariant set is the linear manifold of the microscopic states of the system in which the commutative operators that correspond to the controllable motion integrals have fixed eigenvalues. The phase volume of system in this case is the dimension of the manifold, if this dimension is limited. It directly determines the number of distinguishable microstates of the system that are accessible and equiprobable. The role of the angular momentum conservation in quantum statistical mechanics is similar to one in classical statistical mechanics. The method of computing this phase volume will be also proposed in the second section of this work. The Gibbs assembly density of distribution and thermodynamical functions in the case of a rotating quantum system also will be obtained. In the third section of this work statistical mechanics of an electron gas in a magnetic field is considered. This question was investigated by many during the last century. Many hundreds experimental and theoretical works were summarized in the treatises (Lifshits, I.M. et al., 1973; Shoenberg, 1984). However, together with successful theoretical explanations of many experimental effects some paradoxes and discrepancies with observed facts remain unaccounted. “Finally, it is shown that the presence of free electrons, contrary to the generally adopted opinion, will not give rise to any magnetic properties of the metals”. This sentence ends a short report on the presentation “Electron Theory of Metals” by N. Bohr, given at the meeting of the Philosophical Society at Cambridge. It was well-known that a charged particle in a uniform magnetic field moves in a circular orbit with fixed centre in such a way Thermodynamics – Interaction Studies – Solids, Liquids and Gases 448 that the time average value of the magnetic moment, generated by this motion, is directed opposite to the magnetic field and equal to the derivative of the kinetic energy with respect to the magnetic field. N. Bohr computed the magnetic moment of an electron gas by statistical mechanics with the density of distribution that is determined only by a Hamiltonian. Zero result of this theory (Bohr – van Leeuwen theorem) is the first paradox. Many attempts of derivation and explanation of this were summarized in the treatise (van Vleck, 1965). The most widespread explanation was that the magnetization generated by the electrons moving far from the bound is cancelled by the near-boundary electrons that reflect from the bound. But this explanation is not correct because, when formulae are derived in statistical mechanics, any peculiarities of the near-boundary states shall not be taken into account. Another paradox of the common theory went unnoticed. It is well known that a uniform magnetic field restricts an expanse of a charged particles gas in the plane perpendicular to the field. But from common statistical mechanics it follows that the gas uniformly fills all of the bounded area. The diamagnetism of some metals also was left non- explained. L.D. Landau (Landau, 1930) explained the diamagnetism of metals as a quantum effect. He solved the quantum problem of an electron in a uniform magnetic field. The cross-section of the envelope perpendicular to the magnetic field is a rectangle with the sides 2 x L and 2 y L . The solutions are determined by three motion integrals. The first is energy that takes the values   2 12 2 np с n p m   , where с  is the cyclotron frequency с eH m   ,  e is the charge and m is the mass of an electron, H is the magnetic induction, n is a positive integer or zero. The second is the z  component of the momentum p . The third motion integral is the Cartesian coordinate of the centre of the classical orbit. It takes the values  jx y eHL j    , where 0, 1, 2, xy jeHLL      . The thermodynamical potential with this energy spectrum, when the spin degeneracy is taken into account, is: 0 dln1exp 2 np z B n B LeHS kT p kT                      (2) Here B k is the Boltzmann constant, T is the temperature, 4 x y SLL  ,  is the chemical potential. If in this formula the summation over n is changed to the integration, the result 0  does not depend on H , and the magnetic moment   0 0H    M . That agrees to the classical and paradoxical Bohr – van Leeuwen theorem. L.D. Landau uses the Euler – Maclaurin summation formula in the first order and obtains the amendment that depends on the magnetic field. In the limit 0T  the thermodynamical potential has appearance (Abrikosov, 1972): 22 0 2 24 F eHp V m     , (3) where    13 13 2 23 F pm NV    , and  is the Fermi energy, 4 x y z VLLL  is the volume. This result cannot be correct because the magnetic moment does not depend on the Plank constant  and thus it cannot be a quantum effect. This problem is simpler for a two- dimensional gas. In this case the common formula of the thermodynamical potential has the form: [...]... Thermodynamics – Interaction Studies – Solids, Liquids and Gases chemical bodies on the one hand and the aggregative atomic description on the other hand will appear of primary importance at the very beginning of quantum chemistry I propose to study how Linus Pauling and Robert Sanderson Mulliken created the first chemical quantum approaches in the context described before and how they integrated thermodynamics. ..   2 kBT     kBT   5 452 Thermodynamics – Interaction Studies – Solids, Liquids and Gases where hz and R are the dimensions of the envelope, and U 0  NmR 2 2 2 is an appending constant that does the energy positive Going to the thermodynamics (it rather can be entitled by “thermospindynamics”) it is naturally to consider    2 2 as an external parameter and the moment of inertia I  m... Ed Dover, NewYork 468 Thermodynamics – Interaction Studies – Solids, Liquids and Gases Krutkov, Y.A (1933) Zs Phys v 81, p 377, & Supplement by Editor, In translation into Russian of the book by H A Lorentz, Statistical Theory in Thermodynamics, (1935), ONTI, Leningrad – Moscow Landau, L.D (1930) Diamagnetism of Metals, In: Collected papers, Vol 1, p 47, (1969), Nauka, Moscow Landau, L.D., & Lifshitz,... by quantities that are proportional to  , and Tnr  x  is the infinite series that at x    nr , l ; nr  is  l  1  nr r  proportional to x  exp  x  Then the function (35) will have one more null at the large- 460 Thermodynamics – Interaction Studies – Solids, Liquids and Gases scale value x This null X tends to infinity when  tends to zero, and it would be shown that   X 2 nr  l ... relative" 472 Thermodynamics – Interaction Studies – Solids, Liquids and Gases turned to focus on the question of the energy transfer and the direction of collisions between chemical bodies Wilhelm Ostwald succeeded in describing chemical equilibrium without making any reference to atoms (Ostwald, 1919) Two antagonistic approaches of matter were at stake Thermochemistry revolved around energy and denied... ordinary eigenstate ˆ of the Hamiltonian h in absence of the potential energy U  r  is: 462 Thermodynamics – Interaction Studies – Solids, Liquids and Gases  z      n , l  H     e m   n      eH m   H  (41) It is a negative quantity because the positive term that proportional to   H  is small In the paper (Landau, 1930) were taken into account only ordinary states Then... particles If forces of interaction between particles manifest themselves only at distances considerably smaller than the average distance between particles, the interaction energy of particles is essential only in a small fraction of the phase volume Therefore, the interaction of particles can be neglected or be taken into account as a perturbation in calculating the phase volume and the average values... not have any large-scale parameter This multiplier imposes constraints on ensembles that the total angular momentum equal to zero If Hamiltonian and an operator that should be averaged have the 464 Thermodynamics – Interaction Studies – Solids, Liquids and Gases commutative term that is proportional to the total angular momentum operator, this term should be eliminated when the averaging is performed... , X   py eH  p y x , Y x  2 eH 2 (24) Two motion integrals that have the physical importance would be created from it: energy E and squared centre electron orbit distance from the centre of area R 2 : 456 Thermodynamics – Interaction Studies – Solids, Liquids and Gases R 2  p, r   X 2  Y 2  2 2 eHy   1  eHx   p   ;    py  2  x 2   2   e H    2 2  e2 H 2 2 e2 H 2  2... Thermodynamics – Interaction Studies – Solids, Liquids and Gases 2 The integration of thermodynamics into chemical grounds: From a qualitative to a quantitative affinity The new rules of the French Royal Academy of sciences (1699), Wilhelm Homberg’s work on the interchangeability of ‘average’ -now called ‘neutral’- salts, the mechanist philosophy influences at the end of the seventeenth century, and as well . Thermodynamics – Interaction Studies – Solids, Liquids and Gases 440 123456 Mach number 5 10 15 20 1 2 3 (a) 123456 Mach number 0 20 40 60 80 100 1 2 3 (b) 123456 Mach number 0 10 20 30 40 50 1 2 3 (c) .               R (12) Thermodynamics – Interaction Studies – Solids, Liquids and Gases 452 where z h and R are the dimensions of the envelope, and 22 0 2UNm   R is an appending. that 12 12 ,    RR R, and the determining functions possess the values independently, then Thermodynamics – Interaction Studies – Solids, Liquids and Gases 450       