Thursday, May 5, 2011

Ideal Gas Law: Limited Density Range

A major result of binary mechanics (BM) [1] is the limited energy density range over which some basic thermodynamic laws apply. This report examines this result presenting BM simulator data pertaining to the BM prediction of absolute maximum pressure [2]. Previous reports found absolute maximum temperature at energy densities far below their absolute maximum [3] [4]. It follows that the energy density range over which the ideal gas law is applicable is limited. Specifically, the ideal gas constant R is far from constant over the full energy density range from zero to maximum. Over a significant portion of this range, work in nuclear physics has quantified this variation in the gas constant with different GAMMA values.

Methods and Results
Fig. 1 plots pressure as a function of energy (bit) density where 0 and 1 represent zero pressure and energy density and one represents maximum possible values.

Fig. 1: Pressure (y-axis) vs Energy Density (x-axis)

The data set defining vacuum thresholds was used in this report [5]. This data was generated with application of bit operations in the VUSF order: vector, unconditional, scalar and strong respectively.

Pressure was operationally defined as number of bits exiting the simulated spatial volume per simulator Tick (OutBits column in the .csv output file) scaled by its absolute maximum value. Since a cubic volume was simulated where the surfaces were orthogonal to spot unit alignment, each bit exiting the volume may be seen as one unit of force applied to the surface, in agreement with the conventional definition of pressure.

As further confirmation of this operational definition of pressure, a subsequent report on blackbody radiation in preparation will show that BM produces a linear relation between pressure and the product of temperature and energy density using the ideal gas law in one of its possible applicable density ranges, where previously defined operational definitions of temperature and energy density were used.

Fig. 1 shows that pressure and energy density each have absolute maximum values and that the relation between pressure and energy density over the entire range from zero to maximum is proportional, but not linear. Instead, visual inspection reveals at least six different slopes. This finding is an early warning, so to speak, that all is not well in traditional thermodynamics in terms of its applicable energy density range or that BM may be fundamentally flawed.

At energy densities below about 0.1, in the perfect vacuum range but above the electron threshold [5], the pressure-energy slope may represent an electron gas. This slope is approximately the same in the 0.1 to 0.2 partial vacuum range where protons and perhaps hydrogen and other low-Z ions and atoms begin to form [5]. In this density range, the results probably represent mostly a hydrogen gas.

Above the Thermal II threshold at about 0.2 bit density [5], at least two different pressure-energy slopes were found, up to about 0.55 density. In the final approach to absolute maximum temperature (approx. 0.55 to 0.6 density), pressure rise per density increment drops to a lower level (slope).

Above absolute maximum temperature at about 0.62 energy density [4], pressure again increased at a faster rate per density increment, which corresponds to the range where both matter and anti-matter baryon counts increase dramatically as available bit loci are filled with 1-state mite and lite bits [5].

Fig. 2: Pressure (y-axis) vs Temperature (x-axis)

Fig. 2 plots pressure as a function of temperature, as defined previously as average kinetic energy due to bit motion caused by electromagnetic (EM) forces (heat) -- the ((S+V)/Mites)^2 column in the simulator output file.

Below the approximate baryon threshold at about 0.1 density, in the electron gas range, a complex, non-linear pressure-temperature relation was observed. Then, over a rather wide range, a much more linear pressure-temperature slope occurred, starting at the baryon threshold up to about 0.55 density close to absolute maximum temperature at 0.6 density.

With the ideal gas law, pressure P is purported to be a function of density (Fig. 1) and temperature (Fig. 2):

P = R(n/V)T

where R is the ideal gas constant, n/V is density and T is temperature.

Fig. 3: Pressure (y-axis) vs Energy Density x Temperature(x-axis)

The slope of the curve shown in Fig. 3 is the ideal gas constant R which obviously is not constant. For example, above 0.9 pressure, R was found to be negative. On the other hand, in the approximate 0.3 to 0.65 pressure range, the slope and hence, the ideal gas constant R, were nearly constant.

In this pressure range, the two independent variables -- energy density and temperature -- are highly redundant. Namely, their product moment correlation is 0.994. That is, either so-called independent variable accounts for some 0.987 (0.994 squared) of the variance of the other.

Discussion
The present results suggest that the ideal gas law was most applicable over a somewhat limited pressure range of about 0.30 to 0.65 where the ideal gas constant R was fairly constant. This range is above the Thermal II threshold [5] where our lives and laboratories most likely are located and well below absolute maximum temperature. In other words, it may be no surprise that thermodynamic laws and concepts were developed based on experience in this pressure range.

This historical development of the work of many great experimentalists has had many consequences. For example, in conventional modern physics, there seems to be essentially no appreciation of any upper limits to key variables such as energy density, temperature and pressure, which, of course, can lead to misleading estimates based on the faulty assumption that the ideal gas constant is in fact constant over any pressure, energy or temperature range. As a result, purely fictional values for one or more of these variables above their absolute maximums may be plugged into equations yielding entirely fictional results.

Indeed, according to BM, a more complete and accurate account of physical events requires better understanding of the range of possible phenomena in the context of absolute maximum energy density, temperature and pressure.

The product-moment correlation between pressure and energy density (Fig. 1) was 0.984 which means that some 96.8 percent of pressure variance (100 x 0.984^2) is accounted for by energy density alone over the entire range of possible densities. Indeed, high energy and nuclear physicists have appreciated this rule of thumb writing equations of state (EOS) such as

P = GAMMA * n/V

where P is pressure, GAMMA is a constant and n/V is total energy density in the ideal gas law [6]. GAMMA represents (1 - gamma) where gamma varies in empirical measurements depending on factors such as the composition of n/V -- the material in a particular volume.

In the present notation, 1-state bit count n in the EOS above is usually written as the total energy U. In short, for decades, nuclear physics has implicitly recognized (1) the BM definition of energy content in a volume, namely the proportion of maximum bit (energy) density [2], without explicit awareness that a maximum value exists and (2) GAMMA, namely dP/dU (using U for energy E) over a more extended energy density range within its zero to maximum limits, varies as a function of both energy density, as reported above in Fig. 1, and the materials used.

In sum, gamma (1 - GAMMA in the EOS) is a composite variable, but might more specifically represent an attribute of a particular atomic element (e.g., uranium) if pressure or energy density is held constant (or factored out).

With the advent of BM, the stage is now set to further elucidate the exact mechanisms which may explain specific gamma values for specific materials, which previously have been based more on empirical measurements than theoretical understanding [6].

Of course, this sort of approximate EOS does not explicitly recognize either (1) substantial variation of the non-constant GAMMA over the entire density range or (2) upper limits to possible density or pressure (Fig. 1). However, the expression does assert that if one variable has an upper limit (absolute maximum) -- pressure or density, the other variable also has an upper limit.

If attention is focused on the limited pressure range described above in which the ideal gas constant R is fairly constant (0.30 - 0.65), the high correlation (0.994) of the two independent variables -- energy density and temperature, implies two pressure estimates using the ideal gas constant R:

P = R(n/V)2 or P = RT2

That is, in this pressure range, from a practical or engineering point of view, pressure might be estimated using the ideal gas constant R with either energy density or temperature, whichever item might be more accurate or available.

Table 1 summarizes and interprets some results in this pilot thermodynamics study by enumerating some pressure ranges defined by clearly different pressure increment per energy density increment (dP/dE).

Table 1: Pressure Ranges by dP/dE
Pressure   Comment
0.00-0.11 Electron gas
0.11-0.21 Baryogenesis; proton and hydrogen gas
0.21-0.47 Hydrogen and low-Z atomic and ionic particles
0.47-0.74 Higher-Z atoms; increased dP/dT (Fig. 2)
0.74-0.78 Decreased dP/dE at absolute maximum temperature range
0.78-0.90 Temperature decreasing while density and pressure increase
0.90-1.00 Ideal gas constant turns negative at 0.9 pressure
How could pressure increase above the 0.78 level even though temperature is decreasing? This apparently paradoxical result arises from the changing incidence of bit motion counts due to the four BM bit operations defining time development of the state of a system.

First, as density increases, fewer 0-state bit loci are available to receive 1-state bits from motion due to electromagnetic potentials. Thus, the resulting kinetic energy from bit motion decreases at higher pressure and energy density levels, and therefore, by definition, temperature decreases. This effect reflects the disassociation of electromagnetic potentials from bit motion at these high density levels, a BM result based on the postulate that a bit locus can contain only one 1-state bit at a time.

Second, at high bit (energy) density and pressures, unconditional bit motion is unaffected and accounts for the final pressure increases toward its absolute maximum.

If higher densities and pressures are not contained, a high energy and extremely unstable, explosive state occurs where unconditional bit motion rapidly disperses energy in all directions. For nuclear physicists working on explosive devices, it may be ironic that the strong (nuclear) force is not responsible for the tremendous energy release upon detonation. Instead, it is the garden-variety unconditional bit operation which causes and initially dominates the event.

Fortunately, BM enables relatively simple calculation to assess the role of each BM bit operation as a function of density. Another article would be appropriate to document the specifics. However, at the higher range of bit densities, the unconditional bit operation dominates with bit motion counts exactly equal to the number of 1-state bits in the volume of interest. Meanwhile, the strong operator approaches a null inter-dimensional bit motion count because destination bits are already occupied. Further, to the extent that empty (0-state) bit loci occur, the strong bit operation would not be considered as explosive since it actually keeps 1-state bits from dispersing by confining them to lepton [7] or baryon [8] bit cycles.

References
[1] Keene, J. J. "Binary mechanics" J. Bin. Mech. July, 2010.
[2] Keene, J. J. "Captives in a binary mechanical universe" J. Bin. Mech. March, 2011.
[3] Keene, J. J. "Maximum temperature below half maximum bit density" J. Bin. Mech. March, 2011.
[4] Keene, J. J. "Absolute maximum temperature" J. Bin. Mech. March, 2011.
[5] Keene, J. J. "Vacuum thresholds" J. Bin. Mech. March, 2011.
[6] Nuclear Weapon Archive. "Section 3: Matter, energy, and radiation hydrodynamics" December, 1997.
[7] Keene, J. J. "Physical interpretation of binary mechanical space" J. Bin. Mech. February, 2011.
[8] Keene, J. J. "The central baryon bit cycle" J. Bin. Mech. March, 2011.
© 2011 James J Keene