2/4/2014PHY 770 Spring Lecture 61 PHY Statistical Mechanics 12:30-1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course Webpage: Lecture 6 -- Chapter 4 Review of Thermodynamics – continued 1.Thermodynamics of phase transitions 2.Phase equilibria; Clausius-Clapeyron equation

2/4/2014PHY 770 Spring Lecture 62

2/4/2014PHY 770 Spring Lecture 63

2/4/2014PHY 770 Spring Lecture 64

2/4/2014PHY 770 Spring Lecture 65 Thermodynamics of phase equilibria and transitions Initially, focus on one component systems with multiple phases Example: water Phase I: Solid Phase II: Liquid Phase III: Gas

2/4/2014PHY 770 Spring Lecture 66 Phase diagram of water (

2/4/2014PHY 770 Spring Lecture 67 Comments about phase transitions at constant T and P (or more generally T and Y).  G I (T,Y)=G II (T,Y) and  I (T,Y)=  II (T,Y)  Coexistence line defined by Y I-II coexist (T).  In this case, the coexistance of three phases is only possible at a point in the Y,T plane.  Phase transition can be first order when G has discontinuous first derivatives  Phase transition can be second order when G has continuous first derivatives

2/4/2014PHY 770 Spring Lecture 68

2/4/2014PHY 770 Spring Lecture 69 Thermodynamic description of the equilibrium between two forms “phases” of a material under conditions of constant T and P

2/4/2014PHY 770 Spring Lecture 610 Example of phase diagram :

2/4/2014PHY 770 Spring Lecture 611 Phase diagram for water:

2/4/2014PHY 770 Spring Lecture 612 piston T,P A B Consider a two phase system at constant T and P Note: A and B denote different phases (previously I and II)

2/4/2014PHY 770 Spring Lecture 613 piston T,P A B

2/4/2014PHY 770 Spring Lecture 614 P T g A > g B g A < g B

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2/4/2014PHY 770 Spring Lecture 616 Example – liquid – solid transition in water G T P constant Liquid Solid Melting T m

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2/4/2014PHY 770 Spring Lecture 618 More details

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2/4/2014PHY 770 Spring Lecture 622 Approximate liquid-gas vaporization curve from Clausius- Clapeyron equation: P

2/4/2014PHY 770 Spring Lecture 623 The van der Waals equation of state -- More realistic than the ideal gas law; contains some of the correct attributes for liquid-gas phase transitions.

2/4/2014PHY 770 Spring Lecture 624 Some motivation for van der Waals equation of state Johannes Diderik van der Waals – Ph. D. Thesis written in 1873 (The Netherlands) -- received Nobel Prize in 1910 General potential between two particles:

2/4/2014PHY 770 Spring Lecture 625 Substancea ( eV m 3 )b ( m 3 ) water SO CO O2O Ar N2N H2H He From: Baierlein, Thermal Physics

2/4/2014PHY 770 Spring Lecture 626 P/P c V/V c T/T c =1 T/T c >1 T/T c <1 Gas Liquid

2/4/2014PHY 770 Spring Lecture 627 V/V c P/P c Decreasing values of T

2/4/2014PHY 770 Spring Lecture 628 P   unphysical

2/4/2014PHY 770 Spring Lecture 629 

2/4/2014PHY 770 Spring Lecture 630 Behavior of the thermodynamic potentials for the van der Waals equation of state

2/4/2014PHY 770 Spring Lecture 631 Behavior of the thermodynamic potentials for the van der Waals equation of state

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2/4/2014PHY 770 Spring Lecture 633 Gibbs chemical potential at coexistence point liquid gas

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2/4/2014PHY 770 Spring Lecture 635 Analysis of gas-liquid coexistence conditions

2/4/2014PHY 770 Spring Lecture 636 Analysis of gas-liquid coexistence conditions -- continued

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2/4/2014PHY 770 Spring Lecture 638 P/P c V/V c liquid gas liquid & gas P-V diagram for van der Waals material at T/T c =0.9 

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2/4/2014PHY 770 Spring Lecture 640 Phase diagram for van der Waals material P/P c T/T c 1 1 gas liquid