Chapter 3 Non-ideal Gases 1 2 CHAPTER 3. NON-IDEAL GASES Contents 3 Non-ideal Gases 1 3.1 Statistical Mechanics of Interacting Particles . 5 3.1.1 The partition function . 5 3.1.2 Cluster expansion . 6 3.1.3 Low density approximation . 7 3.1.4 Equation of state . 8 3.2 The Virial Expansion . 8 3.2.1 Virial coefficients . .8 3.2.2 Hard core potential . 10 3.2.3 Square-well potential . 12 3.2.4 Lennard-Jones potential . 13 3.2.5 The Sutherland potential . 17 3.2.6 Comparison of models . 20 3.2.7 Corresponding states . 21 3.2.8 Quantum gases – the special case(s) of helium . 22 3.3 Thermodynamics . 26 3.3.1 Throttling . 26 3.3.2 Joule-Thomson coefficient . 27 3.3.3 Connection with the second virial coefficient . 28 3.3.4 Inversion temperature . 29 3.4 Van der Waals Equation of State . 31 3.4.1 Approximating the Partition Function . 31 3.4.2 Van der Waals Equation . 32 3.4.3 Estimation of van der Waals Parameters . 34 3.4.4 Virial Expansion . 35 3.5 Other Phenomenological Equations of State . 35 3.5.1 The Dieterici equation . 35 3.5.2 The Berthelot equation . 37 3.5.3 The Redlich-Kwong equation . 37 3.6 Hard Sphere Gas . 38 3 4 CONTENTS 3.6.1 Possible approaches . 38 3.6.2 Hard Sphere Equation of state . 39 3.6.3 Virial Expansion . 40 3.6.4 Virial Coefficients . 41 3.6.5 Carnahan and Starling procedure . 42 3.6.6 Pad´eapproximants . 45 3.7 Bridge to the next chapter . 47 3.7.1 Van der Waals model . 49 3.7.2 Hard sphere model . 49 3.1. STATISTICAL MECHANICS OF INTERACTING PARTICLES 5 This chapter is devoted to considering systems where the interactions be- tween particles can no longer be ignored. We note that in the previous chapter we did indeed consider, albeit briefly, the effects of interactions in fermion and in boson gases. This chapter is concerned more with a system- atic treatment of interatomic interactions. Here the quantum aspect is but a complication and most of the discussions will thus take place within the context of a classical description. 3.1 Statistical Mechanics of Interacting Par- ticles 3.1.1 The partition function We are now considering gases where the interactions between the particles cannot be ignored. Our starting point is that everything can be found from the partition function. We will work, initially, in the classical framework where the energy function of the system is p2 H (p , q ) = i + U (q , q ) . (3.1.1) i i 2m i j i i<j X X Because of the interaction term U (qi, qj) the partition function can no longer be factorised into the product of single-particle partition functions. The many-body partition function is 1 p2/2m+ U(q ,q ) kT 3N 3N Z = e− ( i i i<j i j )/ d p d q (3.1.2) N!h3N Z P P where the factor 1/N! is used to account for the particles being indistinguish- able. While the partition function cannot be factorised into the product of single-particle partition functions, we can factor out the partition function for the non-interacting case since the energy is a sum of a momentum-dependent term (kinetic energy) and a coordinate-dependent term (potential energy). The non-interacting partition function is N V p2/2m kT 3N Z = e− ( i i )/ d p (3.1.3) id N!h3N Z P where the V factor comes from the integration over the qi. Thus the inter- acting partition function is 1 U(q ,q ) kT 3N Z = Z e−( i<j i j )/ d q. (3.1.4) id V N Z P 6 CONTENTS The “correction term” is referred to as the configuration integral. We denote this by Q 1 U(q ,q ) kT 3N Q = e−( i<j i j )/ d q. (3.1.5) V N Z P (Different authors have different pre-factors such as V or N!, but that is not important.) The partition function for the interacting system is then 1 V N Z = Q (3.1.6) N! Λ3 and the attention now focuses on evaluation/approximation of the configu- ration integral Q. 3.1.2 Cluster expansion We need a quantity in terms of which to perform an expansion. To this end we define U(q ,q )/kT f = e− i j 1, (3.1.7) ij − which has the property that fij is only appreciable when the particles are close together. In terms of this parameter the configuration integral is 1 Q = (1 + f ) d3N q (3.1.8) V N ij i i<j Z Y where the exponential of the sum has been factored into the product of exponentials. Next we expand the product as: (1 + fij) = 1 + fij + fijfkl + ... (3.1.9) i<j i<j i<j Y X X Xk<l The contributions to the second term are significant whenever pairs of parti- cles are close together. Diagrammatically we may represent the contributions to the second term as: Contributions to the third term are significant either, if i, j, k, l are distinct, when pairs i – j and k – l are simultaneously close together or, if j = k in the sums, when triples i, j, l are close together. The contributions to the third term may be represented as: The contributions to the higher order terms may be represented in a similar way. The general expansion in this way is called a “cluster expansion” for obvious reasons. 3.1. STATISTICAL MECHANICS OF INTERACTING PARTICLES 7 3.1.3 Low density approximation In the case of a dilute gas, we need only consider the effect of pairwise interactions – the first two terms of Eq. (3.1.9). This is because while the probability of two given particles being simultaneously close is small, the probability of three atoms being close is vanishingly small. Then we have (1 + f ) 1 + f (3.1.10) ij ≈ ij i<j i<j Y X so that, within this approximation, 1 Q = 1 + f d3N q V N ij i ( i<j ) Z X (3.1.11) 3N = 1 + fij d qi. i<j Z X There are N(N 1)/2 terms in the sum since we take all pairs without regard to order. And− for large N this may be approximated by N 2/2. Since the particles are identical, each integral in the sum will be the same, so that N 2 Q = 1 + f d3r . (3.1.12) 2V 12 12 Z The V N in the denominator has now become V since the integration over N 1 i, j = 1, 2 gives a factor V − in the numerator. Finally,6 then, we have the partition function for the interacting gas: 2 N U(r)/kT 3 Z = Z 1 + e− 1 d r (3.1.13) id 2V − Z and on taking the logarithm, the free energy is the sum of the non-interacting gas free energy and the new term 2 N U(r)/kT 3 F = F kT ln 1 + e− 1 d r . (3.1.14) id − 2V − Z 8 CONTENTS Since U(r) may be assumed to be spherically symmetric, in spherical polars we can integrate over the angular coordinates: ... d3r 4π r2 ... dr (3.1.15) → Z Z to give 2 N 2 U(r)/kT F = F kT ln 1 + 4π r e− 1 dr . (3.1.16) id − 2V − Z In this low density approximation the second term in the logarithm, which accounts for pairwise interactions, is much less than the first term. — Oth- erwise the third and higher-order terms would also be important. But if the second term is small then the logarithm can be expanded. Thus we obtain 2 N 2 U(r)/kT F = F 2πkT r e− 1 dr. (3.1.17) id − V − Z A more rigorous treatment of the cluster expansion technique, including the systematic incorporation of the higher-order terms, is given in the article by Mullin [1]. 3.1.4 Equation of state The pressure is found by differentiating the free energy: ∂F p = − ∂V T,N (3.1.18) 2 N N 2 U(r)/kT = kT kT 2π r e− 1 dr. V − V 2 − Z We see that the effect of the interaction U(r) can be regarded as modifying the pressure from the ideal gas value. The net effect can be either attractive or repulsive; decreasing or increasing the pressure. This will be examined, for various model interaction potentials U(r). However before that we considered a systematic way of generalising the gas equation of state. 3.2 The Virial Expansion 3.2.1 Virial coefficients At low densities we know that the equation of state reduces to the ideal gas equation. A systematic procedure for generalising the equation of state 3.2. THE VIRIAL EXPANSION 9 would therefore be as a power series in the number density N/V . Thus we write p N N 2 N 3 = + B (T ) + B (T ) + .... (3.2.1) kT V 2 V 3 V th The B factors are called virial coefficients; Bn is the n virial coefficient. By inspecting the equation of state derived above, Eq. (3.1.18), we see that it is equivalent to an expansion up to the second virial coefficient. And the second virial coefficient is given by ∞ 2 U(r)/kT B (T ) = 2π r e− 1 dr (3.2.2) 2 − − Z0 which should be “relatively” easy to evaluate once the form of the interpar- ticle interaction U(r) is known.