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Unformatted text preview: Chapter 7 The Chemical Potential and Phase Equilibria c circlecopyrt 2010 by Harvey Gould and Jan Tobochnik 7 December 2010 We discuss the nature of the chemical potential by considering some simple models and simulations. We then discuss the role of the chemical potential in understanding phase transitions with a focus on the van der Waals equation of state. We also discuss chemical reactions and the law of mass action. 7.1 Meaning of the chemical potential Although the chemical potential plays a role analogous to that of temperature and pressure, un- derstanding the nature of the chemical potential is more difficult. We know that, if two systems are at different temperatures and are then placed in thermal contact, there will be a net transfer of energy from one system to the other until the temperatures of the two systems become equal. If there is a movable wall between two systems at different pressures, then the wall will move so as to change the volume of each system to make the pressures equal. Similarly, if two systems are initially at different chemical potentials and are then allowed to exchange particles, there will be a net transfer of particles from the system at the higher chemical potential to the one at the lower chemical potential until the chemical potentials become equal. You are asked to derive this result in Problem 7.1. Problem 7.1. Chemical equilibrium Assume that two systems A and B are initially in thermal and mechanical equilibrium, but not in chemical equilibrium, that is, T A = T B , P A = P B , but A negationslash = B . Use reasoning similar to that used in Section 2.13 to show that particles will be transferred from the system at the higher chemical potential to the system at the lower chemical potential. An easy way to remember the 355 CHAPTER 7. THE CHEMICAL POTENTIAL AND PHASE EQUILIBRIA 356 N A A ( N A ) ln A ( N A ) A /kT N B B ( N B ) ln B ( N B ) B /kT A B 1 1 9 1287 7.16 1287 2 9 2.20 1 . 90 8 792 6.68 . 51 7128 3 45 3.81 1 . 45 7 462 6.14 . 57 20790 4 165 5.11 1 . 20 6 252 5.53 . 65 41580 5 495 6.21 1 . 03 5 126 4.84 . 75 62370 6 1287 7.16 . 90 4 56 4.03 . 90 72072 7 3003 8.01 . 81 3 21 3.05 1 . 12 63063 8 6435 8.77 . 73 2 6 1.79 1 . 52 38610 9 12870 9.46 1 1 12870 Table 7.1: The number of states of subsystems A and B such that the composite Einstein solid has a total number of particles N = N A + N B = 10 with E A = 8 and E B = 5. The number of microstates of each macrostate is determined using (4.3). Neither N A nor N B can equal zero, because each subsystem has a nonzero energy and thus each subsystem must have at least one particle. The quantity /kT in columns 4 and 8 is determined by computing the ratio ln / N , with N = 1. The most probable macrostate corresponds to N A 6. The ratio /kT is the same (to two decimal places) for both subsystems for this macrostate. The fraction of microstates associatedtwo decimal places) for both subsystems for this macrostate....
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