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# Thermodynamics and Statistical Mechanics PHYS 4230

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This 5 page Class Notes was uploaded by Mrs. Peter Toy on Friday October 30, 2015. The Class Notes belongs to PHYS 4230 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/232122/phys-4230-university-of-colorado-at-boulder in Physics 2 at University of Colorado at Boulder.

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Date Created: 10/30/15

Physics 4230 Fall 2008 Conceptual outline for lectures 1719 106 108 and 1010 Precise de nition of heat For an in nitesimal process we de ned heat to be Q MS lt1 where T is the temperature and d5 is the change in entropy This is more satisfying than the de nition we adopted earlier where we de ned work and said that heat lumps together all other energy changes This definition says what heat is rather than de ning it by what it isn t We motivated this definition of heat by looking the definition of temperature 1 BS 7 j 2 T 6U NV What this equation does is de ne T in terms of a change in entropy ds and a change in energy dU where both changes occur as constant number N and volume V Keeping this in mind we can write 1 d5 T EA 3 The crucial step and this is really a de nition is to identify the heat with dU so Q dU Given our intuition about heat and work this is reasonable since this is an energy change that occurs without changing V or the number N This then gives us the definition Q TdS Cartoon picture of heat and work We considered the example of a quantum ideal gas where the configurations microstates can be visualized as particles occupying a bunch of energy levels We developed a rough cartoon picture that helps illustrate the fundamental difference between heat and work Work corresponds to a process that shifts energy levels around without reshuffling particles among the levels Heat on the other hand corresponds to a process that keeps the energy levels xed but shuf es the particles around among the levels and changes the total energy What we can say rigorously beyond the rough cartoon is that if the energy levels don7t shift in a process the work was zero It s harder to use this cartoon picture to make rigorous statements about heat The reason is that heat is really de ned in terms of an entropy or multiplicity change and these cartoon pictures generally show just a single microstate before and after the process Therefore its hard to say what actually happened to the multiplicity Thermodynamic Identity We supposed the energy is a function only of entropy volume and number of particles U US VN This is indeed true for all the examples we7ve considered so far Assuming for the moment that the number of particles is constant we considered an infinitesimal change in the state of the system dUQWTdSiPdV 4 All we did here is take the first law of thermodynamics and put in our de nitions of heat and work This equation is sometimes referred to as the thermodynamic identity Its really just a restatement of the first law for infinitesimal processes It also holds for quasistatic processes if we get the total change in energy by adding up all the differentials First de nition of pressure We used the thermodynamic identity to relate T and P to partial derivatives of U T ltgtVN 5 EU P 7ltWgt5N 6 The first expression is just the definition of T again just taking the inverse of both sides The second equation can be viewed as a definition of pressure Derivation of ideal gas law Using our formula for the entropy of a monatomic ideal gas Sackur Tetrode equation we solved for U US V N From this we calculated P using the definition above and we found 8U N kT P77ltWgtswi V i 7 We therefore deiived the ideal gas law Alternatively we can view this as a check that our definition of P is consistent with the ideal gas law To get this result we need to use U 32NkT which we proved in class a while ago by plugging the ideal gas entropy into the definition of temperature Second de nition of pressure We considered two systems in contact that can exchange both energy and volume and showed that if the total entropy is maximum then in equilibrium we must have BSA 7 853 BVAgtUANA 7 lt8V3gtUBNB 8 Since the pressure of the two systems should be equal when they can exchange volume this led us to de ne 85 P 7TltWgtMN7 9 where we put the factor of T in front to make the units come out right We showed using this de nition that if PA gt PB then the volume of A will increase while the volume of B decreases because this increases the total entropyi This is all assuming volume changes occurring at constant energy This is a property we expect pressure to have so it looks like this de nition makes sense Two de nitions of pressure are the same We showed that the two de nition of P are the same by relating the partial derivatives of S 5U VN and U US V N Zero temperature is special We discussed how a system at zero temperature is an energy acceptor77 that can absorb energy from other systems but never give up energy This follows directly from the thermodynamic de nition of temperature lT BS8UVN just because heat always ows from hotter systems to colder systems This discussion implies that a quantum system in its ground state lowest energy state must be at zero temperature Thatls because such a system can absorb energy from other hotter systems but has no energy to give up to other systems Physics 4230 Fall 2008 Counting microstates in the twostate paramagnet The aim of these notes is to go through the calculation of 9MA for the twostate paramagnet using Stirlingls approximation Letls rst recall the situation we want to consider We imagine our paramagnet is made out of two subsystems which we call A and B Subsystem A has NA dipoles and magnetization MA while subsystem B has NB dipoles and magnetization MB Therefore there are N NA NB total dipoles and the total magnetization is M MA MB Given xed values of NA N3 and M 9MA is the total number of microstates of the combined system where the magnetization of system A is equal to MA In class we wrote down a general formula for 9MA 9MA QMMA 39QBM MA ltltNAZY34Agtgt 39 ltltNBZXJB7MAgtgt 1 Here QAMA is the number of microstates of subsystem A given that it has magnetization MA and similarly for 93 M 7 M A Now to make life simpler we consider the case NA NB subsystems have the same number of dipoles and also M 0 zero total magnetization In this case plugging in the de nition of the notation nlkln 7 Ml we have 2 9MA NA yr NAEMA I NAEMA This is an exact formula for 9MA which we could plot in Mathematica etc However it is more useful to derive an approximate form that is valid when NA is very large 6 NA gtgt 1 To do this we need Stirling s approximation for the factorial which can be written for N gtgt 1 as lanlenN7Nln27rNlenNiN 3 For N very large the logarithm of the square root is much smaller than the other terms and it can be dropped We now use this to nd an approximate form for 9MA First we write IanA 2lt1nNA171n 4 NA MA NA MA 7 71n 7 2 2 To apply Stirlingls approximation we need to be sure all the numbers of which we take the factorials above are actually large We7ve already assumed NA is large We make the further assumption that lMAl ltlt NA which we will be able to justify later This means that the numbers NA i MA2 are also large and we can apply Stirlingls approximation all the factorials Using Stirlingls approximation we get ianA 7 2NA1nNA 7 NA 7 1nNA 7 MA2 7 W 5 7 W mm 7 MW 7 W lt6 7 2NA1nNA 7 1nNA 7 MA2 7 1nNA 7 MA2 7 In the second line above we combined the terms not involving logarithms and noticed that they all cancel out This is still not really simple enough to be that useful But remember that we already assumed lMAl ltlt NA This suggests that we should make a taylor series expansion in powers of MA Let7s consider the following expression Inltwgtlnlltgtltmgti mltgt1nlt17gt7 ltsgt We can think of the last term above as lnl i z where z is the small quantity MANA We use the Taylor series expansion 1 2 lnl i z m ix 7 Ex 9 where we have kept terms up through order 12 but thrown out terms like 13 and higher If we put this into Eq 8 above we have the approximate result 1M3 ln 1731 10 MA N 1nNA2 i N7 7 Next we substitute this for the two logarithms involving MA in Eq Throughout the following manipulations we drop any terms of order M3 or higher MA 1M3 N NA MA 1nQMA N 2NA1nNA 7 lnNA2 7 N7 7 EN7 11 NA 7 MA MA 1 Mi 7lt 2 1nNA27 NA 72 12 N NA Jr MA MA Mi Mi N 2NA1nNA 7 1nNA2 7 T 7 m 7 m 13 NA 7 MA MA Mi Mi lt 2 llnlNAgH 2 2NA4NAl 14 M31 7 2NA1nNA 7 NA 1nNA2 7 m 15 M2 7 ZlNAlnNA7NAlnNANA1n27iAl 16 ZNA M31 7 2NA1n27N7A1 17 Notice that the terms linear in MA cancelled out 7 that s why we had to keep the quadratic termsi Now lets use this to get 9MA instead of just its logarithm mMA elanA m e2NA1n2e7MgNA e1n22NAe7M NA 22NAE7MgNA 18 Well discuss this expression in class Physics 4230 Fall 2008 Conceptual outline for lectures 1718 106 and 108 Precise de nition of heat For an in nitesimal process we de ned heat to be Q MS lt1 where T is the temperature and d5 is the change in entropy This is more satisfying than the de nition we adopted earlier where we de ned work and said that heat lumps together all other energy changes This definition says what heat is rather than de ning it by what it isn t We motivated this definition of heat by looking the definition of temperature 1 BS 7 7 lt2 T 6U NV What this equation does is de ne T in terms of a change in entropy ds and a change in energy dU where both changes occur as constant number N and volume V Keeping this in mind we can write 1 d5 T m 3 The crucial step and this is really a de nition is to identify the heat with dU so Q dU Given our intuition about heat and work this is reasonable since this is an energy change that occurs without changing V or the number N This then gives us the definition Q TdS Cartoon picture of heat and work We considered the example of a quantum ideal gas where the configurations microstates can be visualized as particles occupying a bunch of energy levels We developed a rough cartoon picture that helps illustrate the fundamental difference between heat and work Work corresponds to a process that shifts energy levels around without reshuffling particles among the levels Heat on the other hand corresponds to a process that keeps the energy levels xed but shuf es the particles around among the levels and changes the total energy What we can say rigorously beyond the rough cartoon is that if the energy levels don7t shift in a process the work was zero It s harder to use this cartoon picture to make rigorous statements about heat The reason is that heat is really de ned in terms of an entropy or multiplicity change and these cartoon pictures generally show just a single microstate before and after the process Therefore its hard to say what actually happened to the multiplicity Thermodynamic Identity We supposed the energy is a function only of entropy volume and number of particles U US VN This is indeed true for all the examples we7ve considered so far Assuming for the moment that the number of particles is constant we considered an infinitesimal change in the state of the system dUQWTdSiPdV 4 All we did here is take the first law of thermodynamics and put in our de nitions of heat and work This equation is sometimes referred to as the thermodynamic identity Its really just a restatement of the first law for infinitesimal processes It also holds for quasistatic processes if we get the total change in energy by adding up all the differentials First de nition of pressure We used the thermodynamic identity to relate T and P to partial derivatives of U T ltgtVN lt5gt P 7ltgt5N 6 The first expression is just the definition of T again just taking the inverse of both sides The second equation can be viewed as a definition of pressure Derivation of ideal gas law Using our formula for the entropy of a monatomic ideal gas Sackur Tetrode equation we solved for U US V N From this we calculated P using the definition above and we foun 6U N kT P77ltWgtswi V i 7 We therefore deiived the ideal gas law Alternatively we can view this as a check that our definition of P is consistent with the ideal gas law To get this result we need to use U 32NkT which we proved in class a while ago by plugging the ideal gas entropy into the definition of temperature

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