Gibbs Paradox: Mixing and Non Mixing Potentials T

Physics Education 1 Jul - Sep 2016

Gibbs paradox: Mixing and non mixing potentials T. P. Suresh 1∗, Lisha Damodaran 1 and K. M. Udayanandan 2 1 School of Pure and Applied Physics Kannur University, Kerala- 673 635, INDIA *[email protected] 2 Nehru Arts and Science College, Kanhangad Kerala- 671 314, INDIA (Submitted 12-03-2016, Revised 13-05-2016)

Abstract

Entropy of mixing leading to Gibbs paradox is studied for diﬀerent physical systems with and without potentials. The eﬀect of potentials on the mixing and non mixing character of physical systems is discussed. We hope this article will encourage students to search for new problems which will help them understand statistical mechanics better.

1 Introduction namics of any system (which is the main aim of statistical mechanics) we need to calculate Statistical mechanics is the study of macro- the canonical partition function, and then ob- scopic properties of a system from its mi- tain macro properties like internal energy, en- croscopic description. In the ensemble for- tropy, chemical potential, speciﬁc heat, etc. malism introduced by Gibbs[1] there are of the system from the partition function. In three ensembles-micro canonical, canonical this article we make use of the canonical en- and grand canonical ensemble. Since we are semble formalism to study the extensive char- not interested in discussing quantum statis- acter of entropy and then to calculate the tics we will use the canonical ensemble for in- entropy of mixing. It is then used to ex- troducing our ideas. To study the thermody- plain the Gibbs paradox. Most text books

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[2, 3, 4, 5, 6, 7, 8, 9] discuss Gibbs paradox in the denominator of the partition function. considering only the case of the classical ideal The corrected N particle partition function gas, but here we analyze diﬀerent physical is QN systems with and without potentials. The Q = 1 , study of Gibbs paradox is very fundamental N N! as pointed out by Erwin Schrdinger- “ It has where Q1 is the single particle partition func- always been believed that Gibbs paradox em- tion. bodied profound thought. That it was inti- With this correction the entropy become mately linked up with something so impor- extensive which resolves the paradox. tant and entirely new could hardly have been foreseen.” V 5 S = N k ln + . N λ3 2

2 Gibbs paradox The entropy of mixing then turns out to be When entropy of an ideal gas was calculated V1 + V2 ∆S =k (N1 + N2) ln using ensemble theory, it was found to be not N1 + N2 extensive. V1 V2 − k N1 ln + N2 ln . V 3 N1 N2 S = N k ln + . λ3 2 If the initial particle densities of two similar The entropy of mixing for two ideal gases is mixing systems are equal, this equation gives the difference between the total entropy after ∆S = 0. The recipe of Gibbs corrects the mixing and that of individual systems before enumeration of micro states as necessitated mixing. For the same particle density, the by the indistinguishability of identical parti- entropy of mixing is given by cles [3]. N1 + N2 N1 + N2 ∆S = k N1 ln + N2 ln . N1 N2 3 Thermodynamics in If we find ∆S for two different ideal gases it is the canonical ensemble found that there is a finite entropy of mixing, but a paradoxical situation is there for sim- In the canonical ensemble the bridging equa- ilar ones - instead of getting ∆S = 0 there tion to find the thermodynamics is is a finite entropy of mixing. Thus the non extensive character of the entropy equation A = −kT ln QN , causes the Gibbs paradox when there is mixing. This paradox is resolved in an ad hoc where A is the Helmholtz free energy, k is way by Gibbs. He put a correction factor N! Boltzmann’s constant and T is the absolute

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temperature. To circumvent Gibbs paradox 1. Non-relativistic free particles the above equation is modified as 2. Massive particles with a relativistic for- QN mulation A = −kT ln . N! 3. Ultra relativistic Using Stirling’s approximation, ln N! = N ln N − N, 4.1 Non-relativistic A = −kT (ln Q − N ln N + N) . N For a single non-relativistic particle, the en- With the inclusion of Gibbs’ correction factor ergy is p2 A contains new terms which are T and N E = dependent. So the entropy (S) given by the 2m equation and the N particle partition function is ∂A S = − V N ∂T Q = , V,N N λ3 needs to be modified. In the coming sections we will consider different systems with and where V is the volume and λ is the de Broglie with out potentials which can be grouped in thermal wavelength given by to mixing and non mixing systems as relevent h to the Gibbs paradox. The systems are clas- λ = 1 . sified as (2πm kT ) 2 1. Free classical particles The entropy is V 3 2. Particles in a potential without Gibbs S = N k ln 3 + paradox λ 2

3. Particles in a potential with Gibbs para- which is not extensive. Introducing the Gibbs correction factor 1 in the N particle parti- dox N! tion function we get In these sections we will connect the effect N of potentials on entropy of mixing by finding 1 V QN = the extensive nature of entropy for different N! λ3 physical systems. and entropy becomes V 5 S = N k ln + (1) 4 Free classical particles N λ3 2 Free particles can be which is extensive.

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4.2 Massive particles with a Using this partition function if we calculate relativistic formulation the entropy we will get it as not extensive but intensive. Introducing the Gibbs correction For relativistic particles the energy is factor the entropy becomes

p 2 2 2 4 p = p c + m c V S = Nk ln 3 + 4 . For massive particles mc2 pc and the par- Nλr tition function is which is clearly extensive. N The above calculations show that all types of V 2 Q = e−β m c , N λ3 free particles exhibit the Gibbs paradox if the Gibbs correction factor is not used. 1 where β = kT , and the entropy is V 2 3 N S = Nk ln e−β m c + Nk + m c2 λ3 2 T 5 Particles in a potential

which is not extensive. With the Gibbs cor- In the coming two sections we will check what rection factor the entropy becomes will be the eﬀect of potentials in the system to the extensive character of entropy and V 5 S = N k ln + thereby to the Gibbs paradox. N λ3 2

which is same as in Eq.(1). 5.1 Particles in a potential without Gibbs paradox 4.3 Ultra relativistic 1. Particles in a harmonic potential 2 Here pc mc and the partition function is Consider an array of equally spaced N N harmonic oscillators along a ﬁnite length " 3# 8πV kT L with one particle per harmonic oscil- QN = . h3 c lator site. The Hamiltonian for a single particle is The de Broglie thermal wavelength is p2 1 hc H = + Kx2 λr = 1 . 2m 2 2π 3 kT So where K is the spring constant given V N by K = mω2, where m is the mass of Q = . N 3 the particles and ω its angular frequency. λr

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The single particle partition function is The N particle partition function is

√ N ∞ 2 L/2 2 3 Z px Z Kx 1 πm 1 −β −β 4 Q1 = e 2m dpx e 2 dx QN = Γ 1 (2 kT ) . 4 4 h −∞ −L/2 2 hK r ! kT βKL2 = Erf The entropy is ~ω 8 √ 3 1 πm 3 4 S = Nk + ln Γ 1 (2 kT ) In the thermodynamic limit N, L → ∞ 4 4 2 h K 4 and N/L is finite, the Erf(∞) = 1, so which is extensive. for the infinite length the single particle partition function reduces to 3. Anharmonic oscillator For an anharmonic oscillator the Hamil- kT Q = tonian is 1 ω ~ p2 H = + Cq2 − gq3 − fq4 and then 2m where C, g and f are positive constants Q = (Q )N . N 1 but their values are very small. The N particle partition function is So the entropy is " 1 #N 2 kT 2m πkT S = N k 1 + ln , QN = (X) , ~ω C h which is extensive even without Gibbs where correction factor. So there is no entropy 3kT f 5g2 X = 1 + + + .. . of mixing, and in this case the entropy 4 C2 4C3 is simply additive. The harmonic potential bounds the particles in the system. The entropy is When two such systems are in contact " " 1 ## no particle flow happens from system to 2m 2 πkT S = Nk 1 + ln + Y, the other. Thus the harmonic potential C h makes the system non mixing. where 2. Quartic oscillator 3 f 5g2 For a quartic oscillator the Hamiltonian Y = Nk kT 2 + 3 + .. . is 2 C 4C p2 1 H = + Kq4. 2m 2 The entropy is extensive.

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4. Electron in a magnetic ﬁeld Consider a system of localized particles of spin half with magnetic moment µB, in the presence of an external magnetic ﬁeld of strength B. The N particle partition function is given by

N QN = [2 cosh (βµBB)] .

The entropy

S =Nk ln [2 cosh (βµBB)]

− Nk [βµBB tanh (βµBB)]

is extensive. In all the above calculations we have taken particles which are localized by some potentials which makes the system non mixing so that there is no Gibbs paradox. There is another way to dif- ferentiate between mixing and non mixing systems or extensive and non extensive systems. The technique is to draw the single particle phase space trajectories. We can see in Figure 1 that the phase space diagrams of the above lo- Figure 1: Single particle phase space dia- calized systems have closed trajectories. grams This then implies that the systems are non mixing (but the converse is not true height L in a uniform gravitational ﬁeld. as, for example, in chaotic systems). The total energy of the system is

2 2 2 5.2 Particles in a potential px + py + pz E = + mgz, with Gibbs paradox 2m 1. Non-relativistic free particles in a where mgz is the gravitational potential gravitational ﬁeld energy with z as the height of the par- Consider a collection of N particles of ticle. The N particle partition function mass m enclosed in a vertical cylinder of is

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where AkT N Q = 1 − e−βmgL , 2r2 4 2e−βεr0 N mgλ3 X = 0 + − . βε (βε)3 (βε)3 where A is the area of cross section of the cylinder. In the limit when L → ∞ the partition function becomes " 3 # 2πm A = −NkT ln V 4πX AkT N β Q = . N mgλ3 and the entropy The entropy is " # 3 Y 2πm3 AkT 5 S = Nk + +Nk ln V 4πX , S = N k ln + 2 X β mgλ3 2 which is not extensive. With the Gibbs where correction factor the entropy is changed 2 3 2k(r0) T 12kT to Y = + ε ε 3 2 2 6kT 2k T r0 AkT 7 − e−βεr0 + . S = N k ln + ε ε2 Nmgλ3 2 which is now extensive. If the potential energy is very large the particles will S is not extensive. With the Gibbs cor- drop to the ground and will become sta- rection factor the entropy is tionary. " # 5 Y 2πm3 V S = Nk + +Nk ln 4πX . 2. Diatomic molecule with inter- 2 X β N particle potential Here we consider a system of diatomic We now get an extensive entropy. molecules with the 2-particle Hamilto- nian given by 3. Diatomic dipoles in external electric ﬁeld p2 + p2 H = 1 2 + ε|r − r | Consider a system of diatomic molecules 2m 12 0 with electric dipole moment µ and mo- where ε and r0 are positive constants and ment of inertia I placed in external elec- r12 = |~r1 −~r2| is the distance between the tric ﬁeld of strength E. The energy of a two particles. molecule is given by 3 2πm 2 2 2 p pθ pφ Q1 = V 4πX E = + + − µE cos θ. β 2m 2I 2I sin2 θ

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The Nparticle partition function is is not extensive. Introducing the Gibbs correction factor the entropy becomes V I 2 sinh(βµE)N QN = " 3 # 3 2 3 2 λ β~ βµE 2πmk V 9 2πk S =Nk ln T 2 and the entropy h N C 11 8π2k2T 2IV + Nk S =Nk ln sinh(βµE) 2 µEλ3 NµE 7 which is now extensive. − coth(βµE) + Nk T 2 5. Free electrons in a magnetic field is not extensive. With the Gibbs correc- Free electrons in a uniform magnetic tion factor the entropy is field B~ follow a helical path with an axis 8π2k2T 2IV parallel to the field direction, say, the z S =Nk ln sinh(βµE) axis. The projection on the x, y plane is µENλ3 a circle. The energy associated with the NµE 9 − coth(βµE) + Nk circular motion is quantized in units of T 2 e~B mc and the energy associated with the which is now extensive. linear motion along the z axis is taken as continuous. The total energy of such 4. Particles with a potential of the particles is 1 2 form C |r2 − r1| 2 e B 1 p2 Consider a system with Hamiltonian E = ~ j + + z 2mc 2 2m p2 + p2 1 H = 1 2 + C|r − r |2 2m 2 2 1 where j = 0, 1, 2, 3 ... . The N particle This Hamiltonian approximates a non partition function is interacting diatomic molecule. The N N V βµeff B particle partition function is QN = 3 λ sinh(βµeff B) " 3 #N 3 2 2πm 2π where µeff = e /2mc and the entropy is QN = V . ~ hβ βC V βµeff B S =Nk ln 3 The entropy λ sinh(βµeff B) " 3 # 3 2 1 2πmk 9 2πk + Nk βµeff B coth(βµeff B) + . S =Nk ln VT 2 2 h C 9 The entropy is not extensive. By intro- + Nk 2 ducing the Gibbs correction factor for

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the N particle partition function the en- for systems which shows the Gibbs paradox, tropy becomes and for systems which do not show the Gibbs paradox V βµeff B S =Nk ln 3 0 Nλ sinh(βµeff B) QN ∝ V φ(T ) 3 + Nk βµeff B coth(βµeff B) + 2 indicating the volume independence and only temperature dependence. In our analysis we found that in the case of free particles, the which is extensive. The above system is Gibbs correction is essential to overcome the the Landau diamagnetic system in which Gibbs paradox. In the cases of particles con- electrons circulate in a helical path and fined in a potential like harmonic, quartic, collide with the walls, and bounce to and anharmonic potentials, and static electrons fro and behave as free particles. in a magnetic field we found that confinement From the above examples we find that makes the system localized and does not al- even though these systems are under cer- low particles to flow and so there is no Gibbs tain potentials they are non localized paradox. For the systems like free electrons in and such systems shows the Gibbs para- a magnetic field, non-relativistic free particles dox. in gravitational field and diatomic dipoles in an external electric potential, even if there is a potential there is no confinement, and the Conclusions system is non localized which allows flow of We have attempted to find the dependence particles. of potentials and their effect on the Gibbs paradox by studying the extensive nature of entropy. It is found that there are potentials which do not allow mixing and for such sys- Acknowledgement tems there is no need of the Gibbs correction factor in the N particle partition function. There are some potentials which allow mix- The authors wish to acknowledge the valu- ing and hence there is a necessity of the Gibbs able suggestions provided by the unknown correction term for the entropy to be exten- referee, which helped us a lot to modify this sive. We found that if the equation for the article to the present form. T. P. Suresh and partition function is volume and temperature Lisha Damodaran wish to acknowledge the dependent it can be written as University Grants Commission for the assis- tance given under the Faculty Development QN ∝ V φ(T ) Programme.

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References [6] Roger Bowley and Mariana Sanchez, In- troductory Statistical Mechanics, Ox- [1] J. W. Gibbs, Elementary Principles in ford University Press, Second Edition Statistical Mechanics, Yale University (1999). Press (1902), reprinted by Dover, New [7] R. P. Feynman, Statistical mechan- York, 1960. ics; A set of Lectures, The Ben- [2] Stephen J Blundell and Katherine M jamin/Cummings Publishing company, Blundell, Concepts in Thermal Physics, Inc. (1982). Second Edition (2010). [8] Herbert B. Callen, Thermodynamics [3] R. K. Pathria and Paul D. Beale, Statis- and an Introduction to Thermo statistical Mechanics, Third Edition, Buttor- tics, Wiley, 2nd Edition (2005). worth (2011). [9] Daniel C. Mattis, Statistical Mechanics [4] Kerson Huang, Statistical Mechanics, made simple, World Scientiﬁc (2003). Second edition, John Wiley and Sons (1987). [5] Silvio R. A. Salinas, Introduction to Sta- tistical Physics, Springer (2001).

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