Math1 Comput. Modelling, Vol. 12, No. 6, pp. 651-657, 1989 Printed in Great Britain. All rights reserved

0895-7177/89 $3.00 + 0.00 Copyright 0 1989 Pergamon Press plc

THE MERE CONCEPT OF AN IDEAL GAS

K. A. MASAVETAS Laboratory of Physical Chemistry, Department of Chemical Engineering,

National Technical University of Athens, 42 Patission Str., 106 82 Athens, Greece

(Received June 1988; accepted for publication July 1988)

Communicated by E. Y. Rodin

Abstract-There exists a tradition to define an ideal gas as one described by the equation of state pV = nRT. The inconsiderate application of this definition meets with serious reservations because it may lead to a host of conceptual inconsistencies. In this paper the view is expounded that the problem of the rigorous definition of the ideal gas presupposes the accurate determination of the scientific region in which it will be applied and requires the existence of an organized formalism for the mathematical foundation of Thermodynamics in whose language it will be laid down. From the attempted analysis it is ascertained that instead of speaking of the ideal gas, in reality we should refer, according to the case, to a different class of ideal gases. For each of them alternative definitions are laid down and the equivalence between them, as well as their difference from the respective alternative definitions of another class of ideal gases, is proved. In conclusion, from the whole investigation it comes out that, as a rule, the traditional definition is simply and solely a necessary, but not sufficient, criterion to define the ideal gas. Depending on the part of Physical Chemistry to which we refer, it is necessary to complement the traditional definition of the ideal gas with other, additional, proper axioms, which are different for each case. Various definitions of the ideal gas are analysed, and changes are proposed.

INTRODUCTION

It is common to introduce an ideal (or perfect) gas as one which obeys the equation of state pV = nRT. This is often represented as a semi-empirical fact: the behaviour of the gas approaches this equation as the pressure p-+0.

However, this manner of defining the ideal gas, even though it does not create problems for a certain class of uses of the concept of the ideal gas, is unacceptably insufficient for a rigorous approach.

In such a case, we find that this definition is simply and solely a necessary condition which by itself is not sufficient to completely define the ideal gas. For such a definition of the ideal gas, this relation will have to be flanked by some other or others, so as to accurately express the concept of the ideal gas. What these additional relations will be, depends on the scientific field from whose point of view the concept of the ideal gas is considered and on the specific formalism of the mathematical foundation of Thermodynamics [lo] we adopt.

Thus, for example, the concepts of the nonrelativistic ideal quantum gas and of the ideal classical gas are not logically equivalent, although both refer to gases each of which, in its own field, is an ideal gas. On the other hand, if our interest is focused, for example, on the concept of the ideal classical gas, then we shall find that the methods of definition employed in Clausius-Kelvin school Thermodynamics are different from those employed in Gibbs school Thermodynamics [l].

Of these two factors determining the nature of the definition of the ideal gas, the former (i.e. the scientific field of reference of the concept) is of an ontological nature, while the latter (i.e. the formalism of the mathematical foundation of Thermodynamics) has a methodological character. Therefore, the first factor applies to matters regarding the content of the definition of the ideal gas, while the second factor is limited to matters related to the form of this definition.

Although the main problems about the definition of the ideal gas proceed from misappreciation of the importance of the contradistinction between specific cases in the ontological status, there is no lesser confusion emanating from an inability to identify the language that is methodologically selected for laying down a definition for some variety of ideal gases.

For example, in the neo-Gibbsian formalisms of the L. Tisza school [2], which follows Gibbs Thermodynamics, an ideal gas of one component is described by one single equation, if this is its

651

652 K. A. MASAVETAS

fundamental equation, or by three equations, if these are its equations of state in a specific representation [3]. It should be pointed out that in this case, the terms fundamental equation, equation of state and representation, have a special, technical content which may possibly differ from that of alternative formalisms of mathematical foundations of Thermodynamics.

Correspondingly, for the C. Carathtodory school [4], which follows the Clausius-Kelvin Thermodynamics, the same ideal gas of one component may be described, by two or more equations called equations of state but, nevertheless the content of the term equation of state is now different from that accepted by the L. Tisza school.

In the following we analyse a series of definitions of the ideal gas, appearing in the literature, and we propose some changes for some of them. All the definitions considered are classified on the basis of the number of axioms on which they are based, into three categories: single-axiom definitions, two-axiom definitions and three-axiom definitions.

SINGLE-AXIOM DEFINITIONS

Definition I

An ideal gas is one for which

pV = nRT, (1)

where p is the pressure, V is the volume, n is the amount of substance, R is the gas constant, and T is the temperature measured on the perfect gas scale (absolute temperature scale). It should be noted that equation (1) is a necessary, but not sufficient, criterion to define an ideal gas. In addition, some other condition(s) is/are needed to complete this definition [5].

The usual name for the class of ideal gases defined by equation (1) is perfect gas or ideal gas. A more apt term would be ideal classical gas. Fairly close to reality would be the term dilute gas, given the semi-empirical fact that the behaviour of the gas approaches this equation as the pressure p-+0.

Dejnition II

An ideal gas is one for which

pV = ngU, (2)

where g is a constant and U is the internal energy. In order to comprehend the physical meaning of g in equation (2), let us evaluate the ratio which

Griineisen studied in connection with equations of state of solids, and which is defined by

l- :=a,v/KTC,,

where aP is the coefficient of volume expansion, v is the specific volume, KT is the isothermal compressibility and C,. is the heat capacity at constant volume. It is found that for systems satisfying equation (2), r has just the value g. Thus, the constant g has a simple interpretation: systems of this type satisfy Griineisens law that r is temperature independent, and this value of r is just g.

But g has also a microscopic interpretation. If one uses quantum statistics for weakly interacting particles, let N(e)de be the number of single particle states in the range (e, e + de). Then, if

N(e) = Qve,

where Q is a constant, it can easily be shown that

g = l/(s + 1).

Thus, g is closely connected with the density of single particle states. One can also show that this class is sufficiently broad to allow for systems which can exist at

arbitrarily low temperatures. For instance, gases of weakly interacting fermions or bosons [6] satisfy equation (2). In the case of nonzero rest masses, we often have s = f and g = 3. For blackbody radiation, s = 2 and g = f .

In quantum statistics the nonrelativistic ideal gases usually satisfy the law (2). Perhaps the most apt term for the class of ideal gases defined by equation (2) is ideal quantum gas [7].

The mere concept of an ideal gas 653

If we now compare from the Logic point of view the concepts corresponding to equations (1) and (2) we find that these, with regard to their connotation are different concepts, while with regard to their denotation, they are partially coincident concepts.

Definition III

TWO-AXIOM DEFINITIONS

An ideal gas is one for which

pV=nRT (3N = (1))

and

(au/a V), = 0. (4)

It is often pointed out that these two properties are not independent: using the so-called thermodynamic equation of state

(g$= T(g),-P, it can be shown thatpV = nRT=s(dU/aV),= 0, and (dU/aV), = O*pVccT(but not definitely that pV = nRT[S]).

In treating nonideal gases, the question naturally arises: to what extent does the property (alJ/dV), determine the equation of state? Or, in other words suppose that the two gases show equal values of (aU/aV), at all pressures, what relation exists between their equations of state? We shall show that gases with different equations of state can have the same values of (dU/dV),.

Suppose that there are two gases, with equations of state p,( V, T) and p2( V, T), such that (dU/dV), is the same for both. By the thermodynamic equation of state:

T@$-P, = T($&-P>,

whence

and, finally,

p, - pz = const . T.

The constant can be any function of V, so that this result can be written in the form: if gases have the property that (all, /a?), = (aU,/d V),, then p, (u, T) - pz(ty, T) = RTf(u), for arbitrary function f(u).

two

any

An immediate consequence of this statement is that, since for an ideal gas p = RT/v, any gas for which the equation of state has the form p = RTf(u) will have the property (aU/dV), = 0. In particular, this applies to gases described by truncated virial expansions of the formpu = RT + Bp, i.e. p(~ -B) = RT.

Definition IV

An ideal gas is one for which

and

pv =cT (5)

where c = C,- C,.

C, is independent of T (6)

With regard to Definition IV it should be noted that equation (5) does not imply that the heat capacities C,, and C, are independent of temperature. But this is just the condition which is always needed in elementary discussions, and the need to stipulate it separately is not very satisfying.

654 K. A. MASAVETAS

Let us make this point clear. If we start with equation (5), or with pu = At and (aU/&), = 0, where A is a constant and t refers to an arbitrary empirical temperature scale (not the absolute temperature scale), it can be shown that any one of the following three conditions implies the other two:

C, = const, C, = const, y := Co/C, = const.

Thus, the basic systems are in fact those which satisfy equation (5) together with one of the above three equivalent conditions.

Dejinition V

An ideal gas is one for which

pu =cT (7X = (5))

and

pv =gu. G3){ = (91

It is proven that Definitions IV and V are logically equivalent, i.e. they define one and the same class of ideal gases.

There are two disadvantages of an excessive use of this class of ideal gases:

(i) It cannot exist near the absolute zero since C, and C, should both tend to zero whereas C, - C, = c for this class, where c = const.

(ii) The heat change dQ has an integrating factor without any appeal to the second law being required. (This illustrates the point that the second law comes in as a new result, in the sense that it yields an integrating factor for 6Q only if the number of independent variables is at least three. For two independent variables an integrating factor always exists).

Logically, the common part of Definitions I and II is Definition V, which, as already mentioned, is logically equivalent to Definition IV.

Definition VZ

An ideal gas is one for which

and

pv = AOG (9)

where A is a constant, Oo is an ideal gas temperature and h is the specific enthalpy. More explicitly, for the two axioms (9) and (10) defining this class of ideal gases, it is pointed

out respectively that:

(i) Boyles law states that, as p +O, pu becomes a function of temperature only, i.e. PV = const, when 0, which can be any empirical temperature, is kept constant. Hence we can define an ideal gas temperature OG by equation (9).

(ii) The Joule-Thomson law states that, as p -0, h becomes a function of tempera- ture only, i.e. h = const when 0, and therefore Oo, is kept constant. Hence, we can write equation (10). Instead of equation (IO), Joules law u = 4(0,) could be used, but it is less securely founded in experiment than the Joule-Thomson law.

From equations (9) and (10) it is shown that Oo _ = T, where T is the temperature measured on the perfect gas scale.

The mere concept of an ideal gas 655

THREE-AXIOM DEFINITIONS

Dejinition VII

An ideal gas is one for which

and

pV =nRT,

(aujav),= 0

(1 l){ = (1))

U2N = (4))

d,, = f[(r.ckT/M3)112p]-2. (13)

In equation (13), we denote by d,, the so-called critical effective collision diameter which has the following meaning: the effective collision diameter d should be greater than zero but less, in order of magnitude, than d,, Of the other symbols in equation (13), k denotes the Boltzmann constant, M denotes the gramme-formula-weight and p denotes the gas density.

It is obvious that the class of ideal gases defined by Definition VII is a special case of the class of ideal gases of Definition III. That is, from the Logic point of view the concept corresponding to Definition VII is with regard to its denotation, a concept subaltern to that defined by Definition III.

The necessity for the existence of an axiom, such as the one described by equation (13), emanates from the fact that the usual defining axioms of the ideal gas-namely, equations (11) and (12)-are, if closely examined, inconsistent with the occurrence of interparticle collisions. The independence of energy on volume at constant temperature excludes long-range forces; therefore, the occurrence of collisions implies that there must be excluded volume. For N, particles an effective hardcore collision diameter d implies an excluded volume per mole of

b = NO(16/3)n(d/2)3,

which in turn implies an equation of state

p( V - nb) = nRT.

Conversely, the equation of state pV = nRT implies that the effective collision diameter is zero: i.e. the particles of a literally ideal gas would not collide with each other, and indeed could support indefinitely any initial velocity distribution, no matter how different from the Maxwell-Boltzmann. For example, if the particles were contained in a rectangular chamber, and all were given initial velocities having nonzero components along the z-axis only, the pressure on the walls perpendicular to the x- and y-axes would be zero!

Thus, inevitably, a complete definition of the ideal gas model for the behaviour of natural gases at low densities should contain an axiom relevant to the collisional properties. Of course, it is an open problem whether the value of d, determined by equation (13) is the appropriate one for a given case of application of the concept of the ideal gas.

Dejinition VIII

An ideal gas is one for which

exp

20 -so> L1 1 3R

and

p = r$X y = (5 - -&)z.40(~)exp[(3;R)], (16) where S is the entropy, 1( is the chemical potential and so, u. and v. are constants representing values of molar entropy, molar internal energy and molar volume in some fiducial state.

656 K. A. MASAVETAS

The first thing that has to be stressed with regard to Definition VIII is that it refers exclusively to monatomic gases. Otherwise, Definition VIII is a typical representative of a whole family of definitions for a class of ideal gases, in energy representation, within the bounds of a neo-Gibbsian formalism of Chemical Thermodynamics. Similar families of definitions exist for enthalpy representation and in general for representation in any thermodynamic potential, as well as for entropy representation and in general for any Massieu function. And, certainly, for each one of these cases, axioms (14)-(16) are differentiated depending on whether the gas is diatomic, triatomic etc.

CONCLUSIONS

The mere concept of the ideal gas proves particularly problematic if a rigorous approach of it is undertaken. Indeed, there are innumerable conceptual inconsistencies that could be encountered in analysing the various definitions used in the literature for the concept of the ideal gas.

A simple example would be sufficient to illustrate the magnitude of problems involved with each attempt to define the concept of the ideal gas: in all definitions of ideal gases it is either explicitly stated (e.g. Definitions I, III etc.) or indirectly implied from their axioms (e.g. Definitions II, IV etc.) that for some class of ideal gases the equation of state is pV = RT. And yet, it is very simple to prove that there are no substances whose equation of state is exactly pV = RT. Indeed, at once S, = pT = R/I/ and so S certainly cannot become independent of V as T+O. The existence of such a substance would therefore be in conflict with the third law! [9].

There is a basic characteristic of the concept of the ideal gas which, if ignored, will inevitably lead to continuous incorrect inferences and improper views about the properties characterizing the ideal gases. This characteristic, which must never escape us, is that the ideal gas is simply and solely a model and no more.

Like every other model, the concept of the ideal gas can be approached by starting from two opposite starting points: we can either accept it by definition and look for the conditions under which the possibility of its relization is secured or, conversely, determine what we expect from its enactment and then investigate to see if it meets our specifications. Anyway, whichever of the two methods we choose, it is certain that we shall come to the same conclusion, i.e. that there is not only one class of the ideal gases but several which sometimes present considerable differences.

The first of the two aforementioned methods for treating the ideal gas model sanctifies each one of the Definitions I-VIII. Indeed, in such a case, for each one of these definitions, we have but to look for the conditions under which it holds true. It is certain that for each one of these definitions we can finally determine some conditions that delimit its range of application, even though, sometimes, these conditions do not correspond with the conventional experimental reality. Also, it is given that the ranges of application of these definitions, as a rule, do not fully coincide.

The second of the two methods mentioned above for treating the ideal gas model is the one which for a given definition can lead to qualifications such as incorrect, inappropriate, insufficient, contradictory definition etc. For example, if the ideal gas model is constructed with the purpose of describing the properties of real gases in the limit of zero pressure, then for example Definition III is at least insufficient.

Indeed, it is very simple to prove that in the sense of the two criteria (3) and (4) set by Definition III, every real gas becomes ideal in the limit of zero pressure. Nevertheless, there are several other properties of the real gases whose value, in the limit of zero pressure, does not coincide with that of the ideal gases determined by either Definition III itself or some variant of it based on the completion of axioms (3) and (4) by some assumptions in the form of tacita conditio.

It is, therefore, not true, as many textbooks state or imply, that the values of all properties of real gases approach the corresponding values calculated on the basis of the ideal gas model (as this is introduced by means of Definition III) at the limit of zero pressure. Thus, statements such as in the limit as the pressure is reduced indefinitely all gases behave in the same way and this is referred to as the ideal gas behaviour or as the pressure approaches zero, the behaviour of any real gas approaches that of the ideal gas as a limit. Thus, all real gases behave ideally at zero pressure . . . are incorrect.

The mere concept of an ideal gas 651

If we choose as an example the Joule-Thompson coefficient, (aT/a~)~, then we shall find that while for the ideal gases (like those defined, for example, by the several variants of Definition III) it is zero, on the contrary for the real gases it is generally finite and nonzero in the limit of zero pressure (with the exception of some extraordinary situations in which it can vanish).

The same situation also holds for other thermodynamic coefficients where there is a second-order effect present. Thus, one can find for 2, the compressibility (which is one for an ideal gas), that for a real gas (82/@),,,=, = 0 and (P2/~3p),,,,, = 0, whereas for an ideal gas of course, these derivatives are all identically zero.

Hence, when one undertakes a critical analysis of the various definitions for ideal gases, one must bear in mind that the concept of the ideal gas is simply and solely a model, and therefore it is necessary to specify each time which of the two above-mentioned approaches one follows. Above all, it should be understood that there is not only one class of ideal gases, but many different kinds of ideal gases.

REFERENCES

1. L. Tisza, Ann. Phys. 13, 1-92 (1961). 2. (a) H. B. Callen, Thermo&zamics. Wiley, New York (1960). (b) A. Miinster, Classical Thermodynamics. Wiley,

New York (1970). 3. M. B. Callen, ibid., pp. 324-326. 4. (a) C. Caratheodory, Math. Annln 67, 355 (1909). (b) M. Born, Physik Z. 22, 218, 249, 282 (1921). (c) A. B. Pippard,

Classical Thermodynamics, Chap. 3. Cambridge Univ. Press, Cambs. (1957). 5. J. J. Martin, Chem. Engng Prog. Symp. Ser. 59(44), 120 (1963). 6. H. Einbinder, Phys. Rev. 74, 805 (1948); 76, 410 (1949). 7. P. T. Landsberg, Am. J. Phys. 29, 695 (1961). 8. J. C. Slater, Infroduction IO Chemical Physics, Sect. H-6. McGraw-Hill, New York (1939). 9. H. A. Buchdahl, Twenty Lectures on Thermodynamics, pp. 30, 94. Pergamon Press, Oxford (1975).

10. K. A. Masavetas, Math1 Comput. Modeiling 10(9), 629-635 (1988).