Thermodynamics and Statistical Mechanics Open Systems and Chemical

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

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. d. U = Td. S – Pd. V + mdn Thermo & Stat Mech - Spring 2006 Class 13 2

Chemical Potential d. U = Td. S – Pd. V + mdn In this equation, m is the chemical energy per kilomole, and dn is the change in the number of kilomoles. Thermo & Stat Mech - Spring 2006 Class 13 3

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 13 4

Chemical Potential Thermo & Stat Mech - Spring 2006 Class 13 5

Chemical Potential Thermo & Stat Mech - Spring 2006 Class 13 6

Chemical Potential Thermo & Stat Mech - Spring 2006 Class 13 7

More Than One Component Thermo & Stat Mech - Spring 2006 Class 13 8

Gibbs Function Thermo & Stat Mech - Spring 2006 Class 13 9

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 13 10

Equilibrium Conditions The change in entropy for the combined system is given by, DS 0 = DS 1 + DS 2, where DS 1 and DS 2 are given by the expression on the previous slide. Then, Thermo & Stat Mech - Spring 2006 Class 13 11

Equilibrium Conditions Since the two systems are interacting only with each other, we have, DU 2 = – DU 1 DV 2 = – DV 1 Dn 2 = – Dn 1 Thermo & Stat Mech - Spring 2006 Class 13 12

Equilibrium Conditions Thermo & Stat Mech - Spring 2006 Class 13 13

Equilibrium Conditions When the two systems come to equilibrium, S 0 will be a maximum. That means that DS 0 will be zero for any small variations of U 1, V 1, or n 1. That is possible only if the coefficients of DU 1, DV 1, and Dn 1 are all zero. Thermo & Stat Mech - Spring 2006 Class 13 14

Equilibrium Conditions Thermo & Stat Mech - Spring 2006 Class 13 15

Approach to Equilibrium To examine the approach to equilibrium, we shall replace DU 1 by Q 1. To do so, use DU 1 = Q 1 – P 1 DV 1 + m 1 Dn 1. Then, Thermo & Stat Mech - Spring 2006 Class 13 16

Approach to Equilibrium DS 0 > 0, so each term must be positive. If T 1 > T 2 , DQ 1 < 0 If P 1 > P 2 , DV 1 > 0 If m 1 > m 2 , Dn 1 < 0 Thermo & Stat Mech - Spring 2006 Class 13 17

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. Thermo & Stat Mech - Spring 2006 Class 13 18