Applying the thermodynamic zeros of the entropy ?and internal energy ?of the gas mass ?in the volume ?yields the numerically unique relation between these quantities, thus allowing calculation of the chemical potentia...Applying the thermodynamic zeros of the entropy ?and internal energy ?of the gas mass ?in the volume ?yields the numerically unique relation between these quantities, thus allowing calculation of the chemical potential in the gas fields of temperature ?and pressure , viz. . A difference in chemical potential provides a force for freely moving matter flow. Since ?is intrinsically a negative function, decreasing as the temperature increases, natural flow processes are initiated by high ?values in cold regions directed to low <v:shape id="_x0000_i1034" type="#展开更多
Two-phase fluid properties such as entropy, internal energy, and heat capacity are given by thermodynamically defined fit functions. Each fit function is expressed as a temperature function in terms of a power series ...Two-phase fluid properties such as entropy, internal energy, and heat capacity are given by thermodynamically defined fit functions. Each fit function is expressed as a temperature function in terms of a power series expansion about the critical point. The leading term with the critical exponent dominates the temperature variation between the critical and triple points. With β being introduced as the critical exponent for the difference between liquid and vapor densities, it is shown that the critical exponent of each fit function depends (if at all) on β. In particular, the critical exponent of the reciprocal heat capacity c﹣1 is α=1-2β and those of the entropy s and internal energy u are?2β, while that of the reciprocal isothermal compressibility?κ﹣1T is γ=1. It is thus found that in the case of the two-phase fluid the Rushbrooke equation conjectured α +?2β + γ=2 combines the scaling laws resulting from the two relations c=du/dT and?κT=dlnρ/dp. In the context with c, the second temperature derivatives of the chemical potential μ and vapor pressure p are investigated. As the critical point is approached, ﹣d2μ/dT2 diverges as c, while?d2p/dT2 converges to a finite limit. This is explicitly pointed out for the two-phase fluid, water (with β=0.3155). The positive and almost vanishing internal energy of the one-phase fluid at temperatures above and close to the critical point causes conditions for large long-wavelength density fluctuations, which are observed as critical opalescence. For negative values of the internal energy, i.e. the two-phase fluid below the critical point, there are only microscopic density fluctuations. Similar critical phenomena occur when cooling a dilute gas to its Bose-Einstein condensate.展开更多
The internal energy U of the real, neutral-gas particles of total mass M in the volume V can have positive and negative values, whose regions are identified in the state chart of the gas. Depending on the relations am...The internal energy U of the real, neutral-gas particles of total mass M in the volume V can have positive and negative values, whose regions are identified in the state chart of the gas. Depending on the relations among gas temperature T, pressure p and mass-specific volume v=V/M, the mass exists as a uniform gas of freely-moving particles having positive values U or as more or less structured matter with negative values U. In the regions U>0?above the critical point [Tc , pc , vc] it holds that p(T,v)>pc and v>vc, and below the critical point it holds that p(T,v)c and v>vv , where vv is the mass-specific volume of saturated vapor. In the adjacent regions with negative internal energy values Uc is the line of equal positive and negative energy contributions and thus represents a line of vanishing internal energy ?U=0. At this level along the critical isochor the ever present microscopic fluctuations in energy and density become macroscopic fluctuations as the pressure decreases on approaching the critical point;these are to be observed in experiments on the critical opalescence. Crossing the isochor vc from U>0 to UΔU>0 happens without any discontinuity. The saturation line vv also separates the regions between U>0 and U , but does not represent a line U=0. The internal-energy values of saturated vapor Uv and condensate Ui can be determined absolutely as functions of vapor pressure p and densities (M/V)v and (M/V)i , repectively, yielding the results Uiv, U=Ui+Uvc and U=Ui=Uv=0 at the critical point. Crossing the line Vv from U=Uv>0 to U=Uv+UiΔU=-Ui>0 to be removed from the particle system. The thermodynamic and quantum-mechanical formulations of the internal energy of a particle system only agree if both the macroscopic and microscopic energy scales have the same absolute energy reference value 0. Arguments for the energy reference value in the state of transition from bound to freely- moving particles in macroscopic classical and microscopic quantum particle systems are discussed.展开更多
文摘Applying the thermodynamic zeros of the entropy ?and internal energy ?of the gas mass ?in the volume ?yields the numerically unique relation between these quantities, thus allowing calculation of the chemical potential in the gas fields of temperature ?and pressure , viz. . A difference in chemical potential provides a force for freely moving matter flow. Since ?is intrinsically a negative function, decreasing as the temperature increases, natural flow processes are initiated by high ?values in cold regions directed to low <v:shape id="_x0000_i1034" type="#
文摘Two-phase fluid properties such as entropy, internal energy, and heat capacity are given by thermodynamically defined fit functions. Each fit function is expressed as a temperature function in terms of a power series expansion about the critical point. The leading term with the critical exponent dominates the temperature variation between the critical and triple points. With β being introduced as the critical exponent for the difference between liquid and vapor densities, it is shown that the critical exponent of each fit function depends (if at all) on β. In particular, the critical exponent of the reciprocal heat capacity c﹣1 is α=1-2β and those of the entropy s and internal energy u are?2β, while that of the reciprocal isothermal compressibility?κ﹣1T is γ=1. It is thus found that in the case of the two-phase fluid the Rushbrooke equation conjectured α +?2β + γ=2 combines the scaling laws resulting from the two relations c=du/dT and?κT=dlnρ/dp. In the context with c, the second temperature derivatives of the chemical potential μ and vapor pressure p are investigated. As the critical point is approached, ﹣d2μ/dT2 diverges as c, while?d2p/dT2 converges to a finite limit. This is explicitly pointed out for the two-phase fluid, water (with β=0.3155). The positive and almost vanishing internal energy of the one-phase fluid at temperatures above and close to the critical point causes conditions for large long-wavelength density fluctuations, which are observed as critical opalescence. For negative values of the internal energy, i.e. the two-phase fluid below the critical point, there are only microscopic density fluctuations. Similar critical phenomena occur when cooling a dilute gas to its Bose-Einstein condensate.
文摘The internal energy U of the real, neutral-gas particles of total mass M in the volume V can have positive and negative values, whose regions are identified in the state chart of the gas. Depending on the relations among gas temperature T, pressure p and mass-specific volume v=V/M, the mass exists as a uniform gas of freely-moving particles having positive values U or as more or less structured matter with negative values U. In the regions U>0?above the critical point [Tc , pc , vc] it holds that p(T,v)>pc and v>vc, and below the critical point it holds that p(T,v)c and v>vv , where vv is the mass-specific volume of saturated vapor. In the adjacent regions with negative internal energy values Uc is the line of equal positive and negative energy contributions and thus represents a line of vanishing internal energy ?U=0. At this level along the critical isochor the ever present microscopic fluctuations in energy and density become macroscopic fluctuations as the pressure decreases on approaching the critical point;these are to be observed in experiments on the critical opalescence. Crossing the isochor vc from U>0 to UΔU>0 happens without any discontinuity. The saturation line vv also separates the regions between U>0 and U , but does not represent a line U=0. The internal-energy values of saturated vapor Uv and condensate Ui can be determined absolutely as functions of vapor pressure p and densities (M/V)v and (M/V)i , repectively, yielding the results Uiv, U=Ui+Uvc and U=Ui=Uv=0 at the critical point. Crossing the line Vv from U=Uv>0 to U=Uv+UiΔU=-Ui>0 to be removed from the particle system. The thermodynamic and quantum-mechanical formulations of the internal energy of a particle system only agree if both the macroscopic and microscopic energy scales have the same absolute energy reference value 0. Arguments for the energy reference value in the state of transition from bound to freely- moving particles in macroscopic classical and microscopic quantum particle systems are discussed.
