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Thermodynamics and Statistical Mechanics

Thermodynamics and Statistical Mechanics. Open Systems and Chemical Potential. Diffusive Interaction.

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Thermodynamics and Statistical Mechanics

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

  2. 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 + mdn Thermo & Stat Mech - Spring 2006 Class 13

  3. Chemical Potential • dU = TdS – PdV + 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

  4. 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

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

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

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

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

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

  10. Equilibrium Conditions Consider two systems, A1 and A2, that can interact thermally, mechanically, and diffusively. For either system, Thermo & Stat Mech - Spring 2006 Class 13

  11. Equilibrium Conditions The change in entropy for the combined system is given by, DS0 = DS1 + DS2, where DS1 and DS2 are given by the expression on the previous slide. Then, Thermo & Stat Mech - Spring 2006 Class 13

  12. Equilibrium Conditions Since the two systems are interacting only with each other, we have, DU2 = – DU1 DV2 = – DV1 Dn2 = – Dn1 Thermo & Stat Mech - Spring 2006 Class 13

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

  14. Equilibrium Conditions • When the two systems come to equilibrium, S0 will be a maximum. That means that DS0 will be zero for any small variations of U1, V1, or n1. That is possible only if the coefficients of DU1, DV1, and Dn1 are all zero. Thermo & Stat Mech - Spring 2006 Class 13

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

  16. Approach to Equilibrium • To examine the approach to equilibrium, we shall replace DU1 by Q1. To do so, use • DU1 = Q1 – P1DV1 + m1Dn1. Then, Thermo & Stat Mech - Spring 2006 Class 13

  17. Approach to Equilibrium • DS0 > 0, so each term must be positive. If T1 > T2 , DQ1 < 0 If P1 > P2 , DV1 > 0 If m1 > m2 , Dn1 < 0 Thermo & Stat Mech - Spring 2006 Class 13

  18. 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

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