4/17/2014PHY 770 Spring 2014 -- Lecture 231 PHY 770 -- Statistical Mechanics 12:00 * - 1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course.

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4/17/2014PHY 770 Spring Lecture 231 PHY Statistical Mechanics 12:00 * - 1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course Webpage: Lecture 23 Review and perspective  Comments about some homework problems  Treatment of multicomponent systems * Partial make-up lecture -- early start time

4/17/2014PHY 770 Spring Lecture 232 Signup schedule for presentations available. Please list the title of your presentation when you sign up.

4/17/2014PHY 770 Spring Lecture 233 Comments about homework problems:

4/17/2014PHY 770 Spring Lecture 234 HW #16 -- continued 7.2 A Brownian particle of mass m is attached to a harmonic spring with force constant k, and is driven by an external force F(t). The particle is constrained to move in one dimension. The Langevin equation is (a)Find the equilibrium correlation function directly from the solution of the Langevin equation assuming that and,, etc. (b)Find the equilibrium correlation function from the fluctuation-dissipation theorem relation

4/17/2014PHY 770 Spring Lecture 235 HW #16 -- continued We can argue that since the random force averages to 0, the correlation function depends only on the homogeneous solution. Note: we are assuming the over-damped case

4/17/2014PHY 770 Spring Lecture 236 HW #16 -- continued

4/17/2014PHY 770 Spring Lecture 237 HW #16 -- continued

4/17/2014PHY 770 Spring Lecture 238 HW #16 -- continued Re(z) Im(z) x x x

4/17/2014PHY 770 Spring Lecture 239 Comments about homework problems: A dilute gas of density n is contained in a cubic box and is in equilibrium with the walls at temperatureT. Find the number of particles per unit area per unit time which collide with the walls and have magnitude of velocity greater than v 0.

4/17/2014PHY 770 Spring Lecture 2310 HW #18 -- continued A

4/17/2014PHY 770 Spring Lecture 2311 Treatment of multicomponent systems including chemical reactions (Sect of 3 rd Ed. Reichl) Summary of thermodynamic potentials (note X  V, Y  -P) PotentialVariablesTotal DiffFund. Eq. US,X,N i HS,Y,N i AT,X,N i GT,Y,N i  T,X,  i

4/17/2014PHY 770 Spring Lecture 2312 Derivative relationships of thermodynamic potentials

4/17/2014PHY 770 Spring Lecture 2313 Properties of the Gibb’s free energy

4/17/2014PHY 770 Spring Lecture 2314 Properties of the Gibb’s free energy -- Gibbs phase rule and phase equilibria Consider a system at uniform temperature T and pressure P with n chemical constituents. How many distinct phases r of this system can exist in equilibrium? liquid solid1 solid2

4/17/2014PHY 770 Spring Lecture 2315 Properties of the Gibb’s free energy -- Gibbs phase rule and phase equilibria -- continued From Schwabl, Statistical Mechanics

4/17/2014PHY 770 Spring Lecture 2316 Properties of the Gibb’s free energy -- multicomponent ideal gas

4/17/2014PHY 770 Spring Lecture 2317 Multicomponent ideal gas and possible chemical reactions -- continued

4/17/2014PHY 770 Spring Lecture 2318 Multicomponent ideal gas and possible chemical reactions -- continued

4/17/2014PHY 770 Spring Lecture 2319 Multicomponent ideal gas and possible chemical reactions -- continued

4/17/2014PHY 770 Spring Lecture 2320 Multicomponent ideal gas and possible chemical reactions -- continued

4/17/2014PHY 770 Spring Lecture 2321 Multicomponent ideal gas and possible chemical reactions -- continued