{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# ch7 - Advanced Topics in Equilibrium Statistical Mechanics...

This preview shows pages 1–4. Sign up to view the full content.

Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 7. Critical Phenomena The subject of critical phenomena , i.e., the study of the equilibrium and nonequi- librium properties of systems near a critical point (e.g., a gas-liquid critical point), has a long history, but many notable recent advances (culminating in a Nobel prize to Robert Wilson in 1982). A. Mean-Field Theory: The VDW ﬂuid The simplest theories to construct of critical phenomena are so-called mean-field theories —approaches that average the environment around a given molecule, thus decoupling that molecule from its neighbors and making the statistical mechanics calculation simple, like that of an ideal gas . Indeed the Van der Waals EOS for a ﬂuid can be “derived” using this sort of reasoning: p + a N V 2 [ V Nb ] = Nk B T and is one of the simplest MF theories of the gas-liquid critical point. Let’s begin by rewriting the equation in terms of ρ = N V rather than V : ( p + 2 )(1 ) = ρk B T. Qualitatively, we have: plane: pT plane: 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
plane: Let’s now locate the critical point in the VDW model. Clearly ∂p ∂ρ T = 0 , 2 p ∂ρ 2 T = 0 , VDW equation give three equations that can give p c , T c , ρ c in terms of a, b . Simpler is to write out the VDW equation as abρ 3 + 2 ( bp + k B T ) ρ + p = 0 or ρ 3 1 b ρ 2 + p a + k B T ab ρ p ab = 0 . Above T c , this has 1 real, 2 imaginary roots; these merge to 3 equal real roots at T c . Thus, at ( T c , p c ): ρ 3 1 b ρ 2 + p c a + k B T c ab ρ p c ab = 0 = ( p p c ) 3 = p 3 3 ρ 2 ρ c + 3 ρρ 2 c ρ 3 c ρ c = 1 3 b , p c = a 27 b 2 , k B T c = 8 a 27 b , Z c = β c p c ρ c = 3 8 Next, we introduce reduced variables: ρ r = ρ/ρ c , p r = p/p c , T r = T/T c which allows the VDW equation to be rewritten as: ( p r + 3 ρ 2 r )(3 ρ r ) = 8 T r ρ r . The absence of any material-dependent parameters is the basis for the familiar Law of Corresponding States ”, which you have undoubtedly studied in your thermo classes. Next, let’s zoom in on the critical region by further changing variables and expanding the VDW equation: t T T c T c = T r 1 “temp difference” ψ ρ ρ c ρ c = ρ r 1 “order parameter” (density difference) π p p c p c = p r 1 “pressure difference” 2
[ π + 1 + 3( ψ + 1) 2 ][3 ( ψ + 1)] = 8( t + 1)( ψ + 1) This reduces to: π = 4 t 1+ ψ 1 1 2 ψ + 3 2 ψ 3 1 1 1 2 ψ π 4 t ( 1 + 3 2 ψ + . . . ) + 3 2 ψ 3 + . . . for | t | 1, | ψ | 1. Next, we do a thermodynamic integration to get the Helmholtz f.e.: write A V A V , A V = A V ( ρ, T ) Helmholtz f.e. per unit volume (intensive) p = ∂A ∂V N,T = A V V ∂A V ∂V N,T = A V V ∂ρ ∂V N ∂A V ∂ρ T = A V + ρ ∂A V ∂ρ T p c ( π + 1)= A V + (1 + ψ ) ∂A V ∂ψ t or in terms of a dimensionless f.e. density F A V /p c : F + (1 + ψ ) ∂F ∂ψ t = 1 + π ( t, ψ ) Integrate this ODE subject to F ( t, 0) F 0 ( t ) .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern