Absolute Internal Energy of the Real Gas

Albrecht Elsner^{*}

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1. Introduction

Investigated here are thermodynamic properties of a real single-component gas of mass in the volume. In accordance with Gibbs [1] , the internal energy is assigned to the gas, composed of a very large number of particles, and the entropy to its distribution in. It is not the total energy of in that is considered, i.e. not the contributions from relativistic or kinetic and potential energies of a mass system moving in external fields, but only the contributions resulting from all time-independent kinetic and potential energies in averaged over all particles. The sum of energies exchanged between the particles can take a positive, negative or else vanishing value, depending on the conditions to which the gas is subjected.

2. Dependence of the Sign of the Internal Energy of the Real Gas on the Aggregate State of Its Mass

Phenomenologically, a distinction is to be made between three thermodynamic conditions under which the gas is in equilibrium. Thermodynamic equilibrium states that the fluid particles are homogeneously distributed in the particular phase volumes and satisfy there the Boltzmann equation, i.e. the equation of kinetic theory. By means of variation of the external parameters, temperature and pressure, the mean distance between fluid particles, which are continually in motion, can be varied; this distance determines the statistical mean value of both the thermal energy and the attractive energy of the particles in each phase volume and is thus related to the aggregate state of the particular fluid mass. In the following, the ratio of the mean distance between fluid particles at given temperature, pressure and volume to that assumed at the critical state, is denoted as.

1) Above the critical point for pressures and volumes

and below the critical point for pressures and volumes, where is the mass-specific volume of vapor, the mass exists as a gas homogeneously distributed in. For values, the averaged kinetic energies of all freely moving particles exceed their mutual bindings, which as potential attractive energy contributions are to be regarded as negative, so that

(2.1)

2) Starting from the condition at the critical point, at temperatures and pressures, the mass disintegrates into a vapor mass with particle volume and a condensed mass with. Under the equilibrium condition one has the relations and, where or

. The internal energy of all freely moving vapor particles is positive, i.e., and that of the more closely bound, not freely moving, condensed particles is negative, i.e.. The internal energy of the gas mass is likewise negative owing to the excess of more closely bound condensed particles. From one obtains

. It then follows that

(2.2)

3) At the critical point the masses and are equally large, i.e.

. The internal energies are likewise equally large, i.e.

. Under the condition the energy contribu- tions of motion and binding cancel out in sum, making it impossible to distinguish between positive and negative values, so that. At the critical point, however, not only do the values of the internal energies of vapor and condensate vanish but, since, the energy value of the total fluid mass does as well. The result is

(2.3)

The sign of the internal energy of the real gas depends on the aggregate state of its mass. The aggregate state, on the other hand, is determined by the mean distances between fluid particles, which are qualitatively described by the value of the relative number.

3. Macroscopic Energy and Density Fluctuations in the Fluid on Approaching the Critical Point

In a fluid, macroscopic density fluctuations are observed in a narrow temperature range before the critical temperature is reached [2] . Andrews discovered these for the first time in 1869 when he irradiated a gas (CO_{2}) with visible light (wavelengths) and found that the previously tran- sparent gas became opaque (within); this phenomenon has since been known as critical opalescence. Macroscopic fluctuations of mass indicate instability of the gas as the consequence of the vanishing internal energy on approaching the critical point.

In the following, the physical processes occuring when the gas approaches the critical point are investigated. In the course of cooling the kinetic energy of all gas particles decreases and at the same time their binding energy increases. This also raises the probability that particles with diameters of the order 10^{−10} [m] aggregate to more voluminous mass conglomerations with diameters of light wavelengths. From their surfaces the light is reflected and scattered proportionally to in different directions. The deflection intensity increases in the course of cooling with the number and size of the scattering centres [2] . The illuminated fluid appears opaque and does not become transparent again till the critical point is reached because there a single aggregate with about 10^{−2} [m] diameter and vapor particles with diameters of less than 10^{−9} [m] exist from which visible light is no longer diffusely scattered.

The reason why there are mass fluctuations in the scattering range between 10^{−6} [m] and 10^{−4} [m] [2] is associated with the macroscopic fluctuations of the internal energy of the real gas at the critical point. The deviation of the fluctuating quantity is considered,; its mean value vanishes owing to equally large positive and negative deviations from, i.e.. As a convenient measure of the amount of scatter of the value is the mean value of the always positive quadratic deviation. Because one has, the mean value of the square of the energy fluctuations occurring is [3]

(3.1)

The square root gives the standard deviation of near the critical point. The fluctuations of energy are thermodynamically associated with fluctuations of mass. These spread out in the fluid on approaching the critical point, becoming stronger and slower because under the condition of decreasing pressure in the fluid there is an increasing volume for fluctuations disposable.

In a Bose gas the phenomenon of critical opalescence is also observed. The shining of the gas cloud illuminated by laser light appears immediately before its condensation at the critical temperature.

Cooling of a Bose gas from room temperature to values of the critical temperature of 10^{−6} [K] within minutes occurs through interaction between laser light and gas atoms. By virtue of the fact that the ensuing aggregating high-energy particles are immediately removed from the available volume there is no coexistent liquid phase. As soon as the low-energy gas particles remaining in have on average the separation at which attractive and repulsive interatomic forces balance one another, the state of vanishing internal energy sets in. The enhanced macroscopic energy and particle density fluctuations show critical opalescence in the laser light. On the assumption that the fluctuations occur above the critical value at a temperature difference of there is spontaneous condensation of the total gas mass at a temperature of just 10^{−9} [K] before the transition temperature is reached.

The values of, , and of both the gas and condensate are needed to describe the real Bose-Einstein gas. These values are thermodynamically defined by the experimental conditions and are given by Equation (7.3) in Section 7.

In 1925 condensation was previously predicted by Einstein for Bose particles: in the lowest quantum mechanical state these assume vanishing entropy and take up equal, lowest possible energy. The first macroscopic quantum phenomenon, which was interpreted in 1938 by H. London as Bose-Einstein-condensate, is suprafluid helium. In 1995 Wieman, Cornell and Ketterle succeeded in the first experimental realisations of heavier alkali atoms. The laboratory conditions for a Bose-Einstein-condensate (BEC) call for temperatures in the range of 10^{−6} [K] and particle densities of 10^{20} [part/m^{3}]. The particles are isolated from vessel walls in a magneto-optical trap (diameter 10^{−2} [m]) and then cooled by means of laser light () to get BEC [4] .

4. Thermodynamic Verification of the Value U = 0 for the Internal Energy at the Critical Point

In accordance with the thermodynamic theory of Willard Gibbs [1] it is possible to calculate every thermodynamic property of a real gas. The quantities uniquely determining the gas are the absolute values of mass, volume, internal energy and entropy, which represent the external or extensive parameters of the gas system. The likewise absolute quantities, temperature, pressure and chemical potential, are referred to as intensive parameters. They are defined by, ,. By measuring the three function values one knows the equilibrium state of the gas in, which is also described by Gibbs’s fundamental equation [3]

(4.1)

Although Gibbs’s theory [1] was already known in the scientific world at the end of the 19th century and the question of absolute values of and had been posed, apparently few scientists were interested in tackling it. It was Nernst, Planck, Boltzmann and physicists in quantum mechanics who solved the problem of the thermodynamic reference point for. The corresponding problem for, on the other hand, remained unsolved [2] [5] . Instead, it sufficed to present determinations of arbitrary zero points, e.g. with

and at the triple point of a liquid without ever taking thermodynamic conditions into account. Consequently, inconsistencies cropped up; in particular, the free energy and the chemical potential were indeterminable functions owing to the missing thermody- namic reference points, so that the fundamental equation could not be applied to every thermodynamic question. Meanwhile the situation has been changed by the proof of found at the critical point [6] .

The theory provides the Gibbs-Duhem equations as the sum of the products of the extensive parameters and the differentials of the corresponding intensive parameters [3] . By enlisting the fundamental equation one then obtains the explicit expressions for the internal energy and entropy [3] :

(4.2)

For the saturated fluid, whose two phases vapor and condensate have the same values and everywhere in, it is immediately possible to give the temperature dependence of the heat capacity:

(4.3)

This relation states that in the case of the saturated fluid there is no difference between a heat capacity at constant volume and one at constant pressure.

In the following, it is to be proved once again that the zero value of is thermodynamically fixed by Gibbs’s theory as at the critical point.

We consider a saturated fluid which according to Gibbs’s phase rule has one thermodynamic degree of freedom fewer than the single phase gas and is thus theoretically simpler to treat. The saturated fluid with the two phases, vapor and condensate, then has the critical density and the mass-specific critical volume. As the fluid values in are locally equal in each of the particular volumes, the two phases can only be distinguished by the densities or mass-specific volumes and. In the temperature range below the critical point one has the relations

(4.4)

(4.5)

The relative mass distributions with the value at the critical point are

(4.6)

In accordance with the masses the values of the internal energy and entropy in the partial volumens and are divided as follows:

(4.7)

where we abbreviate by putting, for example, instead of. The mass-specific quantities are

(4.8)

Whereas is a function of the variables and, the quantities and are pure temperature functions since, for example, it holds that

.

The quantity can be represented by the quantities and as follows:

(4.9)

This representation is possible if, by analogy with Equation (4.2) for, also the functions and are expressed as

(4.10)

The system of Equation (4.10) provides three relations for determining the reference value for the internal energy. Discussion of this is conducted through the mass-specific quantities, and, whose temperature dependences are described as follows:

(4.11)

The difference yields the Clapeyron-Clausius equation:

(4.12)

From and it follows that

and. At the critical point one has, i.e. the value of the evaporation energy is always positive and vanishes at. As, the value of

is smaller than that of and larger than and at takes the value, which is denoted here as. This gives the relations

(4.13)

It is now to be shown that is valid. Because the stated extensive and intensive parameters are absolute quantities, has an absolute value. It is certainly reasonable to assume this to be zero for the time being since otherwise choosing a positive or negative value should be physically justified so as not to appear arbitrary. With the choice it follows from the three Equation (4.11) that

(4.14)

This relation can be enlisted as definition of the critical volume,

. Relations (4.13) then read

(4.15)

If the volume function

(4.16)

is introduced, one can give as upper and lower limits for the temperature

functions and, which at have the value and

with decreasing temperature increase monotonically. This shows that the differential quotient of the chemical potential is a negative temperature function which with increasing values monotonically increases to the negative value at.

The mass-specific energies can thus be put in the simpler form:

(4.17)

It can be shown that the volume function can be explicitly represented by the following thermodynamic expression [7] :

(4.18)

The symmetric occurrence of the variables and in marks the phase independence of the volume function. The internal energies can now be expressed in terms of the measurable quantities and:

(4.19)

Energies as extensive parameters can be added and subtracted. From

it follows in addition to relations (4.13) and (4.19) that

(4.20)

Particularly interesting is the ratio of to, which as a function of the volume ratio assumes universally for every gas values between −1 at the critical point and 0 at absolute zero:

(4.21)

From this it follows that the ratio of the phase-specific energies to the evaporation energy is likewise universal:

(4.22)

It can readily be shown that the internal energy of the saturated fluid

is not positive. Equations (4.4), (4.6), (4.12) and (4.19) yield

. From this

one obtains

(4.23)

The above mentioned relations (4.14)-(4.23) are a consequence of fixing the critical value. If, on the other hand, the assumption were correct, relations (4.13) would read and a unique transformation between thermodynamic parameters would no longer be given; one need only think of the functions, free energy and chemical potential, which would then remain undetermined because of the constant, whereas it is just these functions, which regulate all processes in nature, that should be determinable.

is not compatible with the extensivity of formulated by Gibbs, but it holds that. This then also means that the absolute tempera- ture value can be measured. If a cooling process starts at and ends at and the energy decrease is measured in the form of the heat removed from the system by the amount, is negative and is determined from

(4.24)

can also be calculated from the heat capacity. Both functions and have a discontinuity at the triple point and increase monotonically as:

(4.25)

The value of the internal energy of the saturated fluid is negative for.

5. Thermodynamic Verification of the Value S = 0 for the Entropy at Absolute Zero

In the context of quantum theory the known Planck form of the Nernst postulate, viz.

(5.1)

is interpreted as a macroscopic entropy state of a particle system whose value is proportional to the logarithm of the number of its realized microstates. At the temperature absolute zero the particle system is in the ground state of lowest energy, at which it assumes just a single microstate. In the context of the thermodynamic theory of Willard Gibbs the result (5.1) can likewise be derived.

From and it follows in agreement with Equation (4.2) that

(5.2)

where is an integrable temperature function. Integration between and with the values and yields

(5.3)

because the differential quotients and vanish at absolute zero, and. One therefore gets

(5.4)

The entropy relation analogous to Equation (4.25) is

(5.5)

6. Relation between the Internal Potential and Thermal Energies of the Particle System

By definition, at a given configuration of a particle system the internal energy of the system is the sum of the binding or potential energy and the kinetic or thermal energy of all interacting particles, i.e.

(6.1)

For a certain particle configuration the sum thus expresses the law of internal energy conservation. In turn, with the internal energy value given the particle arrangement in the system is fixed and stable. From relations (6.1) it follows that

(6.2)

According to relations (2.1)-(2.3) the value of depends on the mean inter-particle separation:

(6.3)

Relations (6.2) and (6.3) are represented, for example, in Figure 2 of ref. [8] for the two-phase water fluid. In this special case it holds that

(6.4)

and

(6.5)

Relations (6.3) and (6.5) are plotted in the volume-vs.-pressure chart of water in Figure 3 of ref. [8] . The statement reported there that the isotherm is a line has to be corrected; instead the isochor is the line with; this is shown in the following section.

7. The Conditions U = 0 and S = 0

As the mass in can assume both positive and negative values of the internal energy, the question of the course of the line in the state chart arises. In the case the mass has the entropy value

. These equations yield not only the already known relation (4.14) valid at, but they are also valid along the critical isochor, where we have the following relation between specific volume and the other intensive parameters, temperature, pressure and chemical potential:

(7.1)

This relation can immediately be derived from Equation (4.2). For one has either or. In the case of the saturated fluid it holds that and relations (4.16) and (4.18) are valid.

If, on the other hand, the mass in has the entropy value, then it has the internal-energy value

. This, in turn, yields the Nernst relations (5.4) valid at:

(7.2)

The arrows of in Figure 9.4 and Figure 9.5 of ref. [3] have to be put in the opposite direction to get the correct relation, since for the chemical potential is a negative and concavely curved function of pressure [6] .

The thermodynamically established result states that the statistical mean value of the internal energy of a particle system vanishes under the condition of equal energy contributions of binding and repulsion. The same result must be afforded by quantum mechanical calculation of stationary energy states of a particle system because thermodynamics and quantum mechanics are based on the same suppositions when treating a particle system, viz. the conservation of energy and momentum and application of statistical laws [3] [9] . Cohesion of all system particles is accomplished by interatomic interaction of the particles, which are continually in motion. Quantum mechanical calculation of the stationary and stable energy states provides as result the revealing picture of discrete energy steps arranged above one another. The lowest energy step constitutes the ground state of the particle system and the energy levels above specify the particle system completely. A particle system is then in the ground state of lowest energy when its cohesion by binding energy is maximal and the kinetic energy components are minimal. A particle system may consist of a single atom or a multitude of particles. For example, a Bose-Einstein- condensate. From a statistical point of view, bosons are to be treated as undistinguishable particles; consequently, the sum of the lowest particle energy of N condensed bosons, viz. Nu_{boson} < 0, yields the zero-momentum, ground state energy of the particle system BEC, which is negative and denoted by. If thermal energy is supplied to a structured particle condensate which can absorb it in discrete energy quanta and step by step can then change the arrangement of its particles, with sufficient excitation energy the energy level is reached at which the binding and excitation energies have equal values and particle cohesion is no longer given. This energy level is assigned the internal energy zero. Above this level system particles move freely and have positive internal energy.

The energy zero of a particle system, , is determined by the transition from the bound state particles to freely moving particles. In relation to the number of a particle system, the negative energy value of the condensate is low for a quantum system with, which may be quantum mechanically calculated, and high for a macroscopic system with of order Avogadro numbers, which can only be treated by thermodynamic methods. In other words, the energy reference value is the same for both the microscopic and macroscopic energy scales.

Thermodynamically, the decisive energy transitions in the case BEC can be described as follows: The cold gas mass of critical temperature is contained in the volume and has the critical density. The presence of critical opalescence indicates the state. Under this condition condenses into the volume. The BEC state is denoted by the density, lowest energy and vanishing entropy. is the lowest, experimentally attainable temperature value. At the state of the condensate is more stable than that of the cold gas because in thermodynamic equilibrium the free energy is lower than. From it follows that, i.e. the internal energy of the condensate is negative and, because, gives the lowest energy value of the Bose gas. Conversely, the evaporation energy has to be provided to bring from the condensed to the gaseous state. In thermody- namic terms the experimental conditions at are as follows:

(7.3)

Exact knowledge of all properties of the real Bose-Einstein condensate is a prerequisite to its experimental realisation. This can only be gained on a quantum statistical basis. The value, for example, is quantum mechanically calculated as [4] :

(7.4)

with

Relations (7.3) give the signs of the energy of the condensate (BEC) and of the energy expended on evaporation.

8. Results and Discussion

Gibbs formulated (1873) the Thermodynamic Theory on the properties of matter [1] just a few years after Andrews’s pioneering observations (1869) and prior to the development of quantum physics (as of 1900). Enlisting statistical laws, the theory allows correct description of macroscopically perceived matter, a system composed of a very large number of atomic or molecular particles.

Gibbs investigated the thermodynamic equilibrium of the particle mass in a volume. The thermal properties of the system are characterised by introducing an internal energy and entropy (Equation (4.2)), and the temperature, pressure and chemical potential are defined by partial derivatives of and. The fundamental equation is solved by means of Equation (4.1) in the explicit form. From this follow the negative sign of the free energy,

(because of, ,), and the relation, which can be satisfied by the relations or. For example, in the state chart [-vs.-] of a fluid, the ranges of existence of and are separated from one another, on the one hand, by the critical isochor and, on the other hand, by the vapor volume line at the boundary of the two-phase region; both lines meet at the critical point. In the regions of with, respectively, and with

the mass exists as a gas, while in the adjacent regions of the mass shows more or less distinct structures. Along the line the fluid has the mean energy value and shows near the critical point large fluctuations, which become detectable as macroscopic density fluctu- ations. Crossing the line from to occurs at a continuous change in energy to be removed and crossing the line requires that the energy be extracted from the fluid. The values of the internal energy of vapor and condensate on the saturation lines of the two-phase region can be given in acccordance with Equations (4.6) and (4.19) as functions of the measured values of the vapor pressure and densities of the condensate and vapor, and can be calculated absolutely. One obtains for and for. For, the value of the internal energy of the two-phase fluid is always negative,.

Because the ratio can be given as an analytic function of the argument, its evaluation according to Equation (4.21) at the same is equal for every gas and yields the values −1 at the critical point and 0 at absolute zero. It thus follows that the energy ratios,

and can also be universally represented as functions of for every gas.

The entropy values of the two-phase fluid can likewise be calculated absolutely in acccordance with Equations (4.3), (4.23) and (5.5) from the heat capacity data.

The following four observations are an indication of the thermodynamic zero of a real gas becoming evident: 1) above macroscopic density fluctuations appear within a very narrow temperature range (critical opalescence), 2) at the mass decomposes into a vapor component with density lower than before and a condensate component with higher density, 3) at any heat of evaporation vanishes, and 4) below an interface layer between vapor and condensate appears where on approaching the surface tension of the condensate decreases asymptotically to 0. The relation is thus experimentally proved.

The fundamental Equation (4.1) is valid not only for a gas conforming to the Maxwell-Boltzmann statistics, but also for a gas subject to the Bose statistics. The statement that the internal energy of a condensate is negative therefore also applies to the internal energy of the Bose-Einstein condensate. The quantum mechanical, microscopic and thermodynamical, macroscopic formulations of the internal energy are then only valid if the two energy scales have the common absolute energy reference value 0. The energy values above 0 are positive and those below 0 negative. A system whose particles are homogeneously distributed in the volume available has in the ground state the smallest interparticle distance, or equivalently, the number, the smallest entropy and the lowest energy. By virtue of this most dense particle arrangement the contribution of the binding energy to the internal energy is maximal, and since binding energy is negative energy it constitutes the lowest energy of the system. In an excited state the corresponding discrete energy level is higher. As the energy level increases stepwise, the boundary to the continuum of positive energies is reached; this boundary is assigned the energy value 0, at which the energy contributions by attractive, binding inter-particle forces and repulsive forces are equal. The system is then in the critical state with the critical value. The critical value is the common boundary value 0 of the microscopic and macroscopic energy scales. In the case of a Bose gas, the energy values at the critical transition temperature are zero for the gas and negative for the Bose-Einstein condensate and above they are positive. A macroscopic system has the thermodynamically determined energy zero at the critical point. At the lowest temperature the lowest system energy is equal to the negative value of the heat of vaporization.

Acknowledgements

The author is grateful to A. M. Nicol for the English translation.

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