Results of. Midterm 1. Points < Grade C D,F C B points.

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Pearl McGee
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1 esults of Midterm points Grade C D,F oints A C <0 # of students

2 roblem (a) const (isobaric process) One mole of a monatomic ideal gas goes through a quasistatic three-stage cycle (-, -, -) shown in the Figure. and are given. (a) (0) Calculate the work done by the gas. Is it positive or negative? (b) (0) Using two methods (ackur-etrode eq. and dq/), calculate the entropy change for each stage and for the whole cycle, total. Did you get the expected result for total? Explain. (c) (5) What is the heat capacity (in units ) for each stage? W ( ) ( ) 0 > const (isochoric process) W 0 const (isothermal process) W d d W total W + W ( ) + 0 > < 0

3 cycle 5 roblem (cont.) (b) ackur-etrode equation: ( U,, ) + U + k f ( ) f f f + + i i i f i const (isobaric process) 5 const (isochoric process) const (isothermal process) 0 as it should be for a quasistatic cyclic process (quasistatic reversible), because is a state function.

4 roblem (cont.) (b) d const (isobaric process) d Q d Q C d C d 5 const (isochoric process) d U d Q d Q C d C d const (isothermal process) d U d O d 0 d Q W cycle 5 0

5 roblem (cont) (c) dqcd Let s express both Q and d in terms of d : const (isobaric process) C C C const (isochoric process) C C const (isothermal process), d 0 while dq 0 C

6 roblem 0 0 electrons form a two-state paramagnet. he system is placed in an external magnetic field. he component of the electron s magnetic moment along is ± µ ± 9.x0-4 J/. (a) (5) At 00K, find the ratio / using oltzmann distribution. Calculate the entropy of the system, make reasonable approximations. (b) (5). epeat the same for 0.K (a) E E exp E E k µ µ 9. 0 exp exp k.8 0 (, ) 4 00 ( ) the high- limit We ve obtained the formula for in this case: ( ), k k ( ) ( ) (, ) k.8 0 J/K J/K ( )

7 roblem (cont.) (b) (, ) k µ exp exp k his is the high- limit, we can follow two paths: ( ) ( ) [ << ] µ k 6.74 (, ) J/K.8 0 J/K +, µ k k k e cosh µ k µ k tanh [ x + ].8 0 J/K x 8 µ k µ k x >>

8 roblem You are in possession of an Einstein solid with three oscillators and a two-state paramagnet with four spins. he magnetic field in the region of the paramagnet points up and is carefully tuned so that µ ε, where µ is the energy of a spin pointing down, -µ is the energy of a spin pointing up, and ε is the energy level separation of the oscillators. At the beginning of the experiment the energy in the Einstein solid U is 4 ε and the energy in the paramagnet U is -4 ε. (a) (4) Using a schematic drawing of the Einstein solid, give an example of a microstate which corresponds to the macrostate U 4 ε. (b) (4) Using a schematic drawing of the paramagnet, give an example of a microstate which corresponds to the macrostate U -4 ε. (c) (8) Considering that the system comprises the solid and the paramagnet, calculate the multiplicity of the system assuming that the solid and paramagnet cannot exchange energy. (d) (4) ow let the solid and paramagnet exchange energy until they come to thermal equilibrium. ote that because this system is small, there will be large fluctuations around thermal equilibrium, but let s assume that the system is not fluctuating at the moment. What is the value of U now? Draw an example of a microstate in which you might find the solid. What is the value of U now? Draw an example of a microstate in which you might find the paramagnet.

9 roblem (cont.) wo-state paramagnet Einstein solid E + µ ε ε E - µ (a) U 4 ε (b) (c) U -4 ε E + µ Ω Ω Ω E - µ ( 4 + ) 4 5 Most of the confusion came from the fact that we usually measure the energy of an oscillator in the Einstein solid from its ground state (which is / ε above the bottom of the potential well), whereas for the two-state paramagnet we ve chosen the zero energy in the middle of the energy gap between spin-up and spindown levels. he avoid confusion, consider the number of energy quanta ε available for the system.

10 roblem (cont.) (d) U 4 ε const In equilibrium, the multiplicity is maximum. he two-state paramagnet can absorb only multiples of ε. wo options: ε is transferred from to, and 4ε is transferred from to. ε transfer 4 4, q, 4, Ω Ω Ω 4 4ε transfer, q 0, 4, Ω Ω Ω 4 hus, the equilibrium situation corresponds to the transfer of ε from the Einstein solid to the two-state paramagnet Example of one of the equilibrium microstates: 6 E + µ ε ε E - µ

Lecture 9 Overview (Ch. 1-3)

Lecture 9 Overview (Ch. 1-3) Lecture 9 Overview (Ch. -) Format of the first midterm: four problems with multiple questions. he Ideal Gas Law, calculation of δw, δq and ds for various ideal gas processes. Einstein solid and two-state