Thermo & Stat Mech - Spring 2006 Class 9 1 Thermodynamics and Statistical Mechanics Change of Phase

Thermo & Stat Mech - Spring 2006 Class 92 Thermodynamic Potentials We know that for an isolated system,  S ≥ 0. Therefore, any processes in an isolated system can only increase entropy, and the system will be in equilibrium when it reaches maximum entropy. But what of a system that is not isolated?

Thermo & Stat Mech - Spring 2006 Class 93 Helmholtz Function Suppose a system is in contact with a reservoir at temperature T. The system undergoes a process, and Q is transferred from reservoir to system.  S 0 is the entropy change of the reservoir, and  S is the entropy change of the system.  S 0 +  S ≥ 0

Thermo & Stat Mech - Spring 2006 Class 94 Work so,  S 0 +  S ≥ 0 becomes

Thermo & Stat Mech - Spring 2006 Class 95 Helmholtz Function W ≤ –  F (Constant T) If work is zero for the process,  F ≤ 0, or F f ≤ F i System tends to go to lowest F. At stable equilibrium when dF = 0

Thermo & Stat Mech - Spring 2006 Class 96 Gibbs Function If contact with the reservoir keeps both temperature and pressure constant, the system goes to the lowest value of the Gibbs function. As before, T  S ≥ Q, but in addition, W = P  V. Then, Q =  U + P  V, or  U + P  V – Q = 0  U + P  V – T  S ≤ 0

Thermo & Stat Mech - Spring 2006 Class 97 Gibbs Function  U + P  V – T  S ≤ 0 (U f + PV f – TS f ) – (U i + PV i – TS i ) ≤ 0  G ≤ 0 G f ≤ G i Constant T and P. Stable equilibrium when dG = 0

Thermo & Stat Mech - Spring 2006 Class 98 Gibbs Function If non-mechanical work is done by the system, at constant T and P, then as with F, W nm ≤  G

Thermo & Stat Mech - Spring 2006 Class 99 Phase Transition A phase transition, as from a liquid to a vapor, usually takes place at constant temperature and pressure. Therefore the system will go to the state of lowest Gibbs function. Let us see how the specific Gibbs function changes with temperature. g  is the vapor and g  is the liquid.

Thermo & Stat Mech - Spring 2006 Class 910 Gibbs Function at Transition

Thermo & Stat Mech - Spring 2006 Class 911 Gibbs Function dG = – SdT + VdP G(T, P)

Thermo & Stat Mech - Spring 2006 Class 912 Transition

Thermo & Stat Mech - Spring 2006 Class 913 Transition

Thermo & Stat Mech - Spring 2006 Class 914 Real Substance

Thermo & Stat Mech - Spring 2006 Class 915 Transition

Thermo & Stat Mech - Spring 2006 Class 916 Clausius-Clapeyron Equation

Thermo & Stat Mech - Spring 2006 Class 917 Enthalpy and Latent Heat du = đq – Pdv At transition, u 2 – u 1 = l 12 – P(v 2 – v 1 ) l 12 = (u 2 + Pv 2 ) – (u 1 + P v 1 ) l 12 = h 2 – h 1

Thermo & Stat Mech - Spring 2006 Class 918 Enthalpy and Latent Heat

Thermo & Stat Mech - Spring 2006 Class 919

Thermo & Stat Mech - Spring 2006 Class 920 Regelation

Thermo & Stat Mech - Spring 2006 Class 921 Problem Consider a sealed steel container completely filled with water at 0ºC and pressure of one atmosphere. Lower the temperature to – 1ºC. What happens? Water starts to freeze, but tries to expand. That raises pressure, so freezing point is lowered. How much?

Thermo & Stat Mech - Spring 2006 Class 922 Freezing Problem

Thermo & Stat Mech - Spring 2006 Class 923 Freezing Problem How much freezes? Call the fraction that freezes x.

Thermo & Stat Mech - Spring 2006 Class 924 Freezing Problem

Thermo & Stat Mech - Spring 2006 Class 925 Freezing Problem

Thermo & Stat Mech - Spring 2006 Class 926 Freezing Answer x = = 1.7% Not much!