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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 Volume 32, Number 3, Article Number: 4. www.physedu.in Physics Education 2 Jul - Sep 2016 [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 mix- ing. 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 Volume 32, Number 3, Article Number: 4. www.physedu.in Physics Education 3 Jul - Sep 2016 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. Volume 32, Number 3, Article Number: 4. www.physedu.in Physics Education 4 Jul - Sep 2016 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 sys- tem 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 = m!2, where m is the mass of Q = : N 3 the particles and ! its angular frequency. λr Volume 32, Number 3, Article Number: 4. www.physedu.in Physics Education 5 Jul - Sep 2016 The single particle partition function is The N particle partition function is p N 1 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 ~! 8 p 3 1 πm 3 4 S = Nk + ln Γ 1 (2 kT ) In the thermodynamic limit N, L ! 1 4 4 2 h K 4 and N=L is finite, the Erf(1) = 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 ! ~ 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) ; ~! 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 poten- tial 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. Volume 32, Number 3, Article Number: 4. www.physedu.in Physics Education 6 Jul - Sep 2016 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.
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