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 paramagnet, multiplicity and entropy, the stat. phys. definition of, how to get from the multiplicity to the equation of state. he test is an open textbook exam (but no open HW solutions!). I recommend to list all essential equations you won t have time to read the textbook!
(a) roblem One mole of a monatomic ideal gas goes through a quasistatic three-stage cycle (-, -, -) shown in the Figure. and are given. (a) () Calculate the work done by the gas. Is it positive or negative? (b) () sing two methods (Sackur-etrode eq. and dq/), calculate the entropy change for each stage and for the whole cycle, ΔS total. Did you get the expected result for ΔS total? Explain. (c) () What is the heat capacity (in units ) for each stage? const (isobaric process) δw ( ) ( ) > const (isochoric process) δw W d const (isothermal process) δ < δ W total δw d ( ) + + δw >
roblem (cont.) (b) Sackur-etrode equation: S, f f f f ΔS + + i i i i (,, ) + + k f ( m) const (isobaric process) ΔS const (isochoric process) const (isothermal process) ΔS ΔS Δ S cycle as it should be for a quasistatic cyclic process (quasistatic reversible), because S is a state function.
roblem (cont.) (b) δ Q d S - for quasi-static processes const (isobaric process) δ Q C d ΔS Cd const (isochoric process) d δ Q δ Q C d C d ΔS const (isothermal process) d d d δq δwo ΔS Δ S cycle
roblem (cont) (c) δqcd Let s express both δq and d in terms of d : const (isobaric process) C C C + + const (isochoric process) C C const (isothermal process), d while δq C t home: recall how these results would be modified for diatomic and polyatomic gases.
roblem One mole of a monatomic ideal gas goes through a quasistatic three-stage cycle (-, -, -) shown in the Figure. rocess - is adiabatic;,, and are given. (a) () For each stage and for the whole cycle, express the work δw done on the gas in terms of,, and. Comment on the sign of δw. (b) () What is the heat capacity (in units ) for each stage? (c) () Calculate δq transferred to the gas in the cycle; the same for the reverse cycle; what would be the result if δq were an exact differential? (d) () sing the Sackur-etrode equation, calculate the entropy change for each stage and for the whole cycle, ΔS total. Did you get the expected result for ΔS total? Explain. (a) const (isobaric process) δ W ( ) < const (isochoric process) δw adiabatic process δ W γ γ ( ) d d γ γ γ γ γ / >
roblem (cont.) adiabatic process δ Q (c) const (isobaric process) const (isochoric process) ( ) ( ) > C δ Q ( ) ( ) [ ] < γ γ γ δ C Q ( ) + + γ δ δ δ Q Q Q For the reverse cycle: Q Q reverse δ δ If δq were an exact differential, for a cycle δq should be zero.
roblem (cont.) const (isobaric process) ( ) ( ) f k S,, + + + + Δ i f i f i f i f S ΔS const (isochoric process) S Δ γ γ ΔS Δ S cycle as it should be for a quasistatic cyclic process (quasistatic reversible), because S is a state function. Sackur-etrode equation: δq (quasistatic adiabatic isentropic process) (d)
roblem Calculate the heat capacity of one mole of an ideal monatomic gas C() in the quasistatic process shown in the Figure. and are given. Start with the definition: C δ Q d δq d δw d + ( )d we need to find the equation of Q d this process C ( ) δ + ( ) d d () 4 d d d + d C( ) + ( ) + / d d d d C( ) 4
roblem (cont.) Does it make sense? Sconst adiabat C/ C( ) 4 C ( ) at / the line touches an isotherm const isotherm.. / /8 / /8 / C ( ) at /8 the line touches an adiabat
roblem 4 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. t the beginning of the experiment the energy in the Einstein solid S is 4 ε and the energy in the paramagnet is -4 ε. (a) (4) sing a schematic drawing of the Einstein solid, give an example of a microstate which corresponds to the macrostate S 4 ε. (b) (4) sing a schematic drawing of the paramagnet, give an example of a microstate which corresponds to the macrostate -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 S now? Draw an example of a microstate in which you might find the solid. What is the value of now? Draw an example of a microstate in which you might find the paramagnet.
roblem 4 (cont.) wo-state paramagnet Einstein solid E + μ E - μ ε ε (a) S 4 ε (b) -4 ε E + μ E - μ (c) ( 4 + ) Ω S Ω S Ω! 4!! 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 spinup and spin-down levels. he avoid confusion, consider the number of energy quanta ε available for the system.
(d) roblem 4 (cont.) S 4ε const In equilibrium, the multiplicity is maximum. he two-state paramagnet can absorb only multiples of ε. wo options: ε is transferred from S to, and 4ε is transferred from S to. ε transfer S, q, 4, 4ε transfer S, q, 4, 4!!! Example of one of the equilibrium microstates: Ω Ω S Ω Ω S Ω 4!!! 4!!! hus, the equilibrium situation corresponds to the transfer of ε from the Einstein solid to the two-state paramagnet S S Ω 6 4 E + μ E - μ ε ε ote that the equilibrium condition / / holds if both systems have only quadratic degrees of freedom.
roblem Consider a system whose multiplicity / Ω,, f f is described by the equation: where is the internal energy, is the volume, is the number of particles in the system, f is the total number of degrees of freedom, f() is some function of. (a) () Find the system s entropy and temperature as functions of. re these results in agreement with the equipartition theorem? Does the expression for the entropy makes sense when? (b) () Find the heat capacity of the system at fixed volume. (c) () ssume that the system is divided into two sub-systems, and ; subsystem holds energy and volume, while the sub-system holds - and -. Show that for an equilibrium macropartition, the energy per molecule is the same for both sub-systems. (a) S k f ( ) + k k Ω k f + f k S k f, - in agreement with the equipartition theorem f k ( ) ( ) When,, and S - - doesn t make sense. his means that the expression for Ω holds in the classical limit of high temperatures, it should be modified at low.
roblem (cont.) ( ) ( ) /,, f Ω ( ) ( ) ( ) / /,, Ω ( ) ( ) ( ) ( ) / / / / Ω ( ) (b) [ ] k f f k S d ds d Q C,,, δ (c)
roblem 6 () he ES (electron spin resonance) set-up can detect the minimum difference in the number of spin-up and spin-down electrons in a two-state paramagnet -. he paramagnetic sample is placed at K in an external magnetic field. he component of the electron s magnetic moment along is ± μ ± 9.x -4 J/. Find the minimum total number of electrons in the sample that is required to make this detection possible. E E exp E E k μ μ exp k + + exp μ μ + exp k μ k exp k 4 μ 9. exp exp.8 k.99 - the high- limit +.4 4,4