Absolute Internal Energy of the Real Gas

Abstract

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 [ T c , p c , v c ] it holds that p(T,v)>p c and v>v c , and below the critical point it holds that p(T,v ) c and v>v v , where v v is the mass-specific volume of saturated vapor. In the adjacent regions with negative internal energy values U<0 the mean distances between particles are short enough to yield negative energy contributions to U due to interparticle attraction that exceeds the thermal, positive energy contributions due to particle motion. The critical isochor v c 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 v c from U>0 to U<0, the change in energy ΔU>0 happens without any discontinuity. The saturation line v v also separates the regions between U>0 and U<0 , but does not represent a line U=0 . The internal-energy values of saturated vapor U v and condensate U i can be determined absolutely as functions of vapor pressure p and densities (M/V) v and (M/V ) i , repectively, yielding the results U i <0 v, U=U i +U v <0, for T c and U=U i =U v =0 at the critical point. Crossing the line V v from U=U v >0 to U=U v +U i <0 requires the energy ΔU=-U i >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.