The Gibbs Paradox – Towards an axiomatic approach

By Aniket Sudeep Dalvi and Chinmaya Bhargava

ABSTRACT

We discuss the Gibbs Paradox in statistical mechanics from an axiomatic perspective. We begin by introducing the paradox to the reader and discussing some past attempts. We then discuss one attempt that invokes the real gas scenario and uses the van der Waals constants to distinguish between gases. As seen in our example, the simple axiomatic requirements we propose are difficult to fulfill. In closing, we elucidate the relevance of the Gibbs Paradox and the ideas and concepts it motivated in the eld of physics.

INTRODUCTION

In physics, there are two distinct paradoxes which are both known as the Gibbs paradox and are often confused with each other [1]. The one we discuss in this paper falls within the realm of thermodynamics. It addresses the fact that the entropy increase when combining two gases of different kinds is independent of the degree of similarity between the two kinds of gases, and that this entropy increase changes instantaneously at the transition from dissimilar to similar gases. Entropy here is defined as the measurement of disorder in a system. It is typically given by the Boltzmann equation,

where Ω represents the number of states of the system and kB is the Boltzmann constant. State, here, is defined as the condition of a system at a specific time which is specified by state variables like temperature and pressure.

The Gibbs Paradox was originally considered by Josiah Willard Gibbs in his paper On the Equilibrium of Heterogeneous Substances. It involves the change of the entropy of mixing when moving from dissimilar gases to similar ones. This change is paradoxical to the continuous nature of entropy itself with respect to equilibrium and irreversibility in thermodynamic systems. Irreversibility here describes the nature of a process which cannot return to its original state. The entropy of a irreversible process always increases.

Let us consider n1 and n2 moles of two dissimilar ideal gases 1 and 2 at constant temperature and pressure having volumes V1 and V2 at temperature T1 and T2 and pressure P1 and P2 respectively such that T1 = T2, and P1 = P2, present in a container separated by a impermeable membrane. Let these gases be at entropies S1 and S2, respectively. By the ideal gas law PV = nRT, where R is the Universal gas constant, we have,

Dividing these equations and using the given conditions, we get

Now the entropy when these two gases are mixed by removing the impermeable membrane will be equal to S12.

A standard relation of the entropy of gas is described by the equation,

which does not include any variables that point to the nature of the gases [2]. Hence, entropy S1 for gas 1 is given by

and entropy S2 for gas 2 is given by

therefore,

Now, considering mole fraction of gas a; = n1/n = V1/V and n = n1 + n2 we can rearrange equation (5) as,

For the specific case of = 1/2, i.e, taking equal amounts of gases, we get equation (8) as,

which is non-zero.

From equation (10) we can safely conclude that the change in entropy upon mixing two gases is independent of their nature. This, therefore, means that when mixing two similar gases, the change in entropy must be some nite value given by equation (10), which is zero only when = 0 and (1 ) = 0, that is, the mole fraction of either of the gases is zero. However, we know that the change in entropy of mixing two similar gases is zero. This can be deduced by the Boltzmann equation (equation (1)) as similar gases will have the same state variable. We thus have a paradox, as two ways of deduction give two distinct values

In this work, we propose an axiomatic approach towards resolving the Gibbs Paradox. The basis of the resolution is to incorporate variables that point to the nature of the gas into the calculation of the change in entropy in such a way that it vanishes for similar gases, providing continuity. A particular attempt in which the nature of the gases is parametrized using the van der Waals constants is worked as a negative example for implementing our axiomatic approach.

PREVIOUS ATTEMPTS AT RESOLVING THE PARADOX

One historically notable attempt at resolving the paradox was by realizing that if the two gases are composed of indistinguishable particles, they obey different statistics than if they are distinguishable. Since the distinction between the particles is discontinuous, so is the entropy of mixing. The resulting equation for the entropy of a classical ideal gas is known as the Sackur-Tetrode equation. This equation is an expression for the entropy of a monatomic classical ideal gas which incorporates quantum considerations that give a more detailed description of its regime of validity:

where N is the number of particles in the gas, U is the internal energy of the gas, kB is the Boltzmann’s constant, m is the mass of a gas particle, and h is Planck’s constant.

If equation (11) is used, the entropy value will have no difference after mixing two parts of the identical gases. Here, the term:

was introduced to resolve the paradox. This effectively makes the particles of the gas indistinguishable, as all the N! permutations of these N particle gas are identical and should be counted as one due to permutation symmetry. Rewriting this term using the Sterling Approximation for large N, ln(N!) = Nln(N) N, the resulting equation is simplified to equation (11). However, this approach, does not take explore the possibility of resolving the paradox using properties of gases that distinguish them from each other. Our axiomatic resolution described below attempts to do so.

ATTEMPTED RESOLUTION BASED ON PARAMETRIZING NATURE OF GASES UNDER VAN DER WAAL’S CONSTANTS

The mathematical outline of the paradox described earlier was based on the ideal gas equation. One potential attempt to resolve the paradox comes from considering the modified ideal gas equation for real gases, which is,where a and b are the van der Waal’s constants, which are unique for each gas, and describe the correction for molecular forces and the volume of one mole of the atoms or molecules respectively.

Rearranging (13), we get

which on further expansion gives,

On transposing the terms on the right hand side to the left hand side, we get,

which is a cubic equation of V .

Solving for the roots of this cubic equation using Mathematica yields,

where k can take three values:

and

where

As volume V cannot be negative or complex, we get = 1

Now, as V explicitly depends on a and b which are unique for each gas, it implies that change in entropy ( S) also explicitly depends on the nature of the gas (from equation (5)).

Let us now look at another approach of resolving the paradox. Consider a function F defined as follows:

where U is the energy of a purely mechanical system. This function F is known as the Free Energy of the system.

Now, it has been previously established that

and

where W is the external work done, CP and CV are the thermal capacities at constant pressure and volume respectively, and m is the constant of integration [3].

Substituting (23) and (24) in (22), we get

Consider a function G defined as:

where F is the free energy of the system and G is defined as the Gibbs free energy of the system. As we know that CP = CV R [3], substituting this and (25) in the expression for Gibbs free energy we get,

Now, using van der Waal’s corrections we get,

Entropy as defined in the terms of G is,

Applying (29) to (28), we get

where M = (CP log T Rlog P + a + Rlog R + Rlog n).

From equation (29) we have,

where a1; b1 and a2; b2 are the van der Waal’s constants for both the gases respectively.

The mixing rule used here to determine the van der Waal’s constant of the mixture of the gases is a = χa1 + (1 –  χ)a2 and b = χb1 + (1 – χ )b2, where χ and (1 – χ ) are the mole fractions of the gases respectively, which in this case are equal to 1/2 as we take equal moles of each gas [4].

This equation also shows that S depends explicitly on van der Waals’s constants a and b, therefore ΔS, too, depends explicitly on the nature of the gas.

AXIOMATIC CONSIDERATIONS

Now, on considering real gases we know that the change in entropy is dependent on the nature of the gas. Thus, for similar gases such that a and b are the same, all the terms must get cancelled. This leads the entropy to be zero and averts the paradox. However, this does not occur as the above derived equation is nonlinear in n, that is, the highest power of n is not 1, contrary to a linear function which has a highest order of 1 for the variable of that function. This implies that

thereby reinstating the Paradox.

Thus, our first axiom is that in order to resolve the Paradox it is necessary to have linearity in n such that

Furthermore, our second axiom is that it is also necessary for the equation to be non-linear in a and b, such that,

This ensures that they do not get cancelled for dissimilar gases, thereby nullifying their purpose.

IMPLICATIONS AND IMPORTANCE OF STUDYING THE GIBBS PARADOX

Up until now, this paper has discussed what the Gibbs Paradox is, past attempts at resolving the paradox and our axiomatic approach at resolving it. It is however, vital to understand the significance of the paradox and why analyzing it is important.

The Gibbs Paradox played an extremely important role in the emergence of the concept of indistinguishable particles. These are particles that cannot be told apart from each other. If two indistinguishable particles interact in a process and are interchanged to form a new state, then this state is in no way different from the original state and must be considered the same state. It can be seen how the Gibbs Paradox ts this description, and the quantum mechanics approach uses this very principle to resolve the paradox. The theory of blackbody radiation postulated by Max Planck was another development that independently contributed to this concept of indistinguishable particles. Planck postulated the presence of photons that were indistinguishable from each other.

This indistinguishability of particles has some important implications in the field of statistical mechanics as well. The primary one of these is the alteration in the partition function which describes the statistical properties of a system in a state of thermal equilibrium. Moreover, an important similarity that we can draw between the Boltzmann equation and quantum mechanics, is that both of them depend explicitly on states that play an important role in distinguishability in both cases respectively. Furthermore, the indistinguishability of particles had a profound impact on their statistical properties and behavior, playing an important role in the birth of quantum statistics.

CONCLUSION

The Gibbs Paradox tells us that upon mixing two gases, a change in entropy that is independent of the nature of the gas occurs. This implies that even when mixing two same gases, there is a nite value of entropy change that is obtained, which evidently should not happen. This gives rise to the version of paradox considered here. We identify certain axiomatic requirements that are fundamental to solving the paradox. The first one is the requirement of linearity in the number of moles of the gases, and the second is non-linearity in the respective van der Waal constants of the gases. Although these requirements are simple, implementing them may not be straightforward, as is shown by our detailed example using the van der Waals gas constants.

ACKNOLEDGEMENT

This work was carried out as a joint project at the Centre for Fundamental Research and Creative Education (CFRCE), Bangalore, India and the Poornaprajna Institute of Scientic Research (PPISR), Bangalore, India. We thank R. Srikanth for suggesting to us this problem to work on. We thank him, S. Aravinda, B. S. Ramachandra, B. R. Pratiti, Manogna Shastry, Vasudev Shyam, Madhavan Venkatesh and Stefan Westerho for related discussions, suggesting references, and reviewing certain versions of this paper.

REFERENCES

[1] Peters H. Demonstration and resolution of the Gibbs paradox of the first kind. 2013. Available from: arXiv:1306.4638v2

[2] Jaynes ET. The Gibbs Paradox. 1996

[3] Fermi E. Thermodynamics. 1936

[4] Newman A. Comparison of Mixing Rules for a van der Waals Gas Mixture. 2001

This piece was featured in Volume III Issue I of JUST.

2017-12-12T23:56:39+00:00 December 14th, 2017|