Thermo & Stat Mech - Spring 2006 Class 13 1 Thermodynamics and Statistical Mechanics Open Systems and Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 132 Diffusive Interaction If particles are added to a system, the energy of the system can change, because of the chemical potential of the added particles in their new environment. A term is needed to account for this effect. dU = TdS – PdV +  dn

Thermo & Stat Mech - Spring 2006 Class 133 Chemical Potential dU = TdS – PdV +  dn In this equation,  is the chemical energy per kilomole, and dn is the change in the number of kilomoles.

Thermo & Stat Mech - Spring 2006 Class 134 Chemical Potential Suppose 2.0 ×10 -5 kilomoles of acid is added to a 1.0 liter of water at room temperature. The temperature of the water rises 0.15ºC. From this data the chemical potential of the acid in water can be calculated.

Thermo & Stat Mech - Spring 2006 Class 135 Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 136 Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 137 Chemical Potential

Thermo & Stat Mech - Spring 2006 Class 138 More Than One Component

Thermo & Stat Mech - Spring 2006 Class 139 Gibbs Function

Thermo & Stat Mech - Spring 2006 Class 1310 Equilibrium Conditions Consider two systems, A 1 and A 2, that can interact thermally, mechanically, and diffusively. For either system,

Thermo & Stat Mech - Spring 2006 Class 1311 Equilibrium Conditions The change in entropy for the combined system is given by,  S 0 =  S 1 +  S 2, where  S 1 and  S 2 are given by the expression on the previous slide. Then,

Thermo & Stat Mech - Spring 2006 Class 1312 Equilibrium Conditions Since the two systems are interacting only with each other, we have,  U 2 = –  U 1  V 2 = –  V 1  n 2 = –  n 1

Thermo & Stat Mech - Spring 2006 Class 1313 Equilibrium Conditions

Thermo & Stat Mech - Spring 2006 Class 1314 Equilibrium Conditions When the two systems come to equilibrium, S 0 will be a maximum. That means that  S 0 will be zero for any small variations of U 1, V 1, or n 1. That is possible only if the coefficients of  U 1,  V 1, and  n 1 are all zero.

Thermo & Stat Mech - Spring 2006 Class 1315 Equilibrium Conditions

Thermo & Stat Mech - Spring 2006 Class 1316 Approach to Equilibrium To examine the approach to equilibrium, we shall replace  U 1 by Q 1. To do so, use  U 1 = Q 1 – P 1  V 1 +  1  n 1. Then,

Thermo & Stat Mech - Spring 2006 Class 1317 Approach to Equilibrium  S 0 > 0, so each term must be positive. If T 1 > T 2,  Q 1 < 0 If P 1 > P 2,  V 1 > 0 If  1 >  2,  n 1 < 0

Thermo & Stat Mech - Spring 2006 Class 1318 Approach to Equilibrium When two systems interact, Heat flows from the hotter to the cooler. The system at higher pressure expands at the expense of the other. Particles flow from the system of higher chemical potential to the other system.