University:
Massachusetts Institute of Technology
Course:
8.333 | Statistical Mechanics I: Statistical Mechanics of Particles
Academic year:

2021
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492

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7
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96XGbl6o

V.E Mean Field Theory of Condensation In principle, all properties of the interacting system, including phase separation, are contained within the thermodynamic potentials that can be obtained by evaluating Z(T, N ) or Q(T, µ). Phase transitions, however, are characterized by discontinuities in various state functions and must correspond to the appearance of singularities in the par tition functions. At ﬁrst glance, it is somewhat surprising that any singular behavior should emerge from computing such well behaved integrals (for short-ranged interactions) as Z(T, N, V ) = Z QN 3 3  d p~i d q~i exp −β N !h3N i=1 N X i=1 p2i 2m −β X i 0 (see the discussion after eq.(I.72)). The coordinates of the critical point are thus obtained from solving the following coupled equations,          ∂P ∂v      ∂ 2P    ∂v 2 kB T a − 2 v−b v kB T 2a =− + 3 =0, 2 v (v − b) P = T = T 2kB T 6a − =0 v4 (v − b)3 where v = V /N is the volume per particle. The solution to these equations is (V.77) a 27b2 vc = 3b .     k T = 8a B c 27b      Pc = (V.78) Naturally, the critical point depends on the microscopic Hamiltonian (e.g. on the 2 body interaction) via the parameters a and b. However, we can scale out such dependencies by measuring P , T , and v in units of Pc , Tc and vc . Setting Pr = P/Pc , vr = v/vc , and Tr = T /Tc , a reduced version of the van der Waals equation is obtained as Pr = 3 8 Tr − 2 3 vr − 1/3 vr . (V.79) We have thus constructed a universal (material independent) equation of state. Since the original van der Waals equation depends only on two two parameters, eqs.(V.78) predict a universal dimensionless ratio, Pc vc 3 = = 0.375. 8 kB Tc (V.80) Experimentally, this ratio is found to be in the range of 0.28 to 0.33. The van der Waals equation is thus not a good candidate for the putative universal equation of state. We can attempt to construct a generalized equation empirically by using three inde pendent critical coordinates, and finding Pr ≡ pr (vr , Tr ) from the collapse of experimental data. Such an approach has some limited success in describing similar gases, e.g. the sequence of noble gases Ne, Xe, Kr, etc. However, different types of gases (e.g. diatomic, ionic, etc.) show rather different behaviors. This is not particularly surprising given the differences in the underlying Hamiltonians. We know rigorously from the virial expansion that perturbative treatment of the interacting gas does depend on the details of the mi croscopic potential. Thus the hope of finding a universal equation of state for all liquids and gases has no theoretical or experimental justification; each case has to be studied separately starting from its microscopic Hamiltonian. It is thus quite surprising that the collapse of experimental data does in fact work very well in the vicinity of the critical point, as described in the next section. V.H Critical Point Behavior To account for the universal behavior of gases close to their critical point, let us examine the isotherms in the vicinity of (Pc , vc , Tc ). For T ≥ Tc , a Taylor expansion of P (T, v) in the vicinity of vc , for any T gives, P (T, v) = P (T, vc ) + ∂P ∂v (v −vc ) + T Since ∂P/∂v|T and ∂ 2 P/∂v 2 T 1 ∂ 2P 2 ∂v 2 (v −vc )2 + T 1 ∂ 3P 6 ∂v 3 (v −vc )3 + · · · . (V.81) T are both zero at Tc , the expansion of the derivatives around the critical point gives P (T, vc ) = Pc + α(T − Tc ) + O (T − T )2 ∂P = −a(T − Tc ) + O (T − Tc )2 , ∂v T,vc ∂ 2P ∂v 2 T,vc 3 ∂ P ∂v 3 = b(T − Tc ) + O (T − T )2 , (V.82) = −c + O [(T − Tc )] , Tc ,vc where a, b, and c are material dependent constants. The stability condition, δP δv ≤ 0, requires a > 0 (for T > Tc ) and c > 0 (for T = Tc ), but provides no information on the sign of b. If an analytical expansion is possible, the isotherms of any gas in the vicinity of its critical point must have the general form, b c P (T, v) = Pc +α(T −Tc )−a(T −Tc )(v −vC )+ (T −Tc )(v −vc )2 − (v −vc )3 +· · · . (V.83) 2 6 Note that the third and ﬁfth terms are of comparable magnitude for (T − Tc ) ∼ (v − vc )2 . The fourth (and higher order terms) can then be neglected when this condition is satisﬁed. • The analytic expansion of eq.(V.83) results in the following predictions for behavior close to the critical point: (a) The gas compressibility diverges on approaching the critical point from the high tem perature side, along the critical isochore (v = vc ) as, lim κ(T, vc ) = − T →Tc+ 1 ∂P vc ∂v −1 = T 1 . vc a(T − Tc ) (V.84) (b) The critical isotherm (T = Tc ) behaves as c P = Pc − (v − vc )3 + · · · . 6 (V.85) (c) Eq.(V.83) is manifestly inapplicable to T < Tc . However, we can try to extract infor mation for the low temperature side by applying the Maxwell construction to the unstable isotherms. Actually, dimensional considerations are suﬃcient to show that on approaching Tc from the low temperature side, the speciﬁc volumes of the coexisting liquid and gas phases approach each other as lim (vgas − vliquid ) ∝ (Tc − T )1/2 . T →Tc− (V.86) The liquid–gas transition for T < Tc is accompanied by a discontinuity in density, and the release of latent heat L. This kind of transition is usually referred to as ﬁrst order or discontinuous. The diﬀerence between the two phases disappears as the line of ﬁrst order transitions terminates at the critical point. The singular behavior at this point is attributed to a second order or continuous transition. Eq.(V.83) follows from no more than the constraints of mechanical stability, and the assumption of analytical isotherms. Although there are some unknown coeﬃcients, eqs.(V.85)–(V.86) predict universal forms for the singularities close to the critical point and can be compared to experimental data. The experimental results do indeed conﬁrm the general expectations of singular behavior, and there is universality in the results for diﬀerent gases which can be summarized as (a) The compressibility diverges on approaching the critical point as lim κ(T, vc ) ∝ (T − Tc )−γ , T →Tc+ with γ ≈ 1.3 . (V.87) (b) The critical isotherm behaves as (P − Pc ) ∝ (v − vc )δ , with δ ≈ 5.0 . (V.88) (c) The diﬀerence between liquid and gas densities vanishes close to the critical point as lim (ρliquid − ρgas ) ∝ lim (vgas − vliquid ) ∝ (Tc − T )β , T →Tc− T →Tc− with β ≈ 0.3 . (V.89) These results clearly indicate that the assumption of analyticity of the isotherms, leading to eq.(V.83), is not correct. The exponents δ, γ, and β appearing in eqs.(V.88)– (V.89) are known as critical indices. Understanding the origin, universality, and numerical values of these exponents is the a fascinating subject explored in the modern theory of critical phenomena.

Mean Field Theory of Condensation, Corresponding States, Critical Point Behavior (from L17 & L18)

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