Slide 1Slide 2Slide 3Slide 4Slide 5Lecture 32—Grand canonical ensembleLecture 32—Grand canonical ensembleChapters 9, Wednesday April 2Chapters 9, Wednesday April 2ndnd•Systems with variable particle number•Conditions for equilibrium•Revised 1st and 2nd laws•Chemical and Gibbs potentials•Calculating the chemical potential•ExamplesReading: Reading: All of chapter 9 (pages 188 - 207)All of chapter 9 (pages 188 - 207)Homework 9 due Wed. Apr. 9th at 5pmHomework 9 due Wed. Apr. 9th at 5pmAssigned problems, Assigned problems, Ch. 9Ch. 9: 2, 4, 6, 8, 10: 2, 4, 6, 8, 10Chapter 9: systems with variable #s of Chapter 9: systems with variable #s of particlesparticles1ABdNdN=-Chemical reaction:C D CD+ �London – van der Waals’ interactionLondon – van der Waals’ interactionLennard-Jones potentialLennard-Jones potentialrr00EEtotaltotal = = KK + + VV = constant = constantRepulsion occurs when the negative electron Repulsion occurs when the negative electron clouds eventually come into contact. It is a clouds eventually come into contact. It is a very hard interaction, almost like the collision very hard interaction, almost like the collision between two billiard balls, i.e. between two billiard balls, i.e. VV increases increases very rapidly.very rapidly.London – van der Waals’ interactionLondon – van der Waals’ interactionEEtotaltotal = = KK + + VV = constant = constant•So, the addition of particles to a system results in the addition of kinetic energy to the system which almost immediately dissipates to the entire system.•Therefore, the addition of particles changes the internal energy of the system. The change in the internal energy dU is proportional to the number of particles dN that are added. The proportionality constant is called the ‘chemical potential’.dUdU = = TdSTdS – – PdVPdV + + dNdNThus, we need to modify the combined 1Thus, we need to modify the combined 1stst and 2 and 2ndnd laws: laws:The combined 1The combined 1stst and 2 and 2ndnd Laws Laws, , ,, , ,; ;V N S N S VV N S N S VdU TdS PdV dNU U UdS dV dNS V NU U UT PS V Nmm= - +� � �� � � � � �= + +� � � � � �� � �� � � � � �� � �� � � � � �� = - = =� � � � � �� � �� � � � � �If more than one type of particle (If more than one type of particle (constituentconstituent) ) is added, thenis added, then1, ,1mj jjjjS V NkdU TdS PdV dNUk m jNmm== - +� ��= = � �� �� ���
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