Written 4-hour exam in Thermodynamics and Statistical ... - dirac

1**Written** 4-**hour** **exam** **in****Thermodynamics** **and** **Statistical** MechanicsThursday 10/01-12, 10.00-14.00The **exam** consists of three problems on two pages altogether (+ this front page). The weights of theproblems are given, **and** with**in** a problem sub-questions have the same weight.Writ**in**g aids **and** a simple calculator (that is one which can not do graphs or symbolic manipulations)are allowed. The “Golden sheet” (3 pages) are **in**cluded with this **exam**. No further help**in**g aids areallowed.

3Problem 1 (40 %)In the follow**in**g we consider a system with 6 places **and** 2 particles (each place can maximal beoccupied by one particle). The energy of the system depends on which places are occupied, **in** thefollow**in**g way: 1 place exists which contributes 0ǫ if occupied, 2 places exists which contributes 1ǫ ifoccupied, **and** 3 places exists which contributes 2ǫ if occupied (ǫ > 0). Figure 1 shows three **exam**plesof configurations **and** the correspond**in**g total energy.The system is assumed to be **in** equilibrium, **and** **in** good thermal contact with a heat bath at a giventemperature, T.2ǫ1ǫ 0ǫ 2ǫ 1ǫ 0ǫ 2ǫ 1ǫ 0ǫE = 4ǫ E = 1ǫ E = 1ǫFigure 1: Three **exam**ples of configurations with correspond**in**g energy. The 6 r**in**gsillustrates the possible places **and** the filled circles the two particles.a) State the partition function for the system, furthermore show that there are 15 configurations **in**total.b) Calculate the mean energy for the system as function of temperature. Furthermore, calculate thehigh **and** low temperature limits of the mean energy.c) Calculate the entropy of the system as function of temperature. Furthermore, calculate the high**and** low temperature limits of the entropy.The system is now changed **in** such a way that 2 places exists which each contributes with 0ǫ ifoccupied, 3 places exists which each contributes 1ǫ if occupied, **and** 4 places exists which eachcontributes 2ǫ if occupied (total of 9 places).d) For the changed system, state (without explicit calculation) the high **and** low temperature limitof the entropy, **and** low temperature limit of the mean energy.Problem 2 (40 %)a) Show that the follow**in**g general relation holdsIn the follow**in**g a mono atomic ideal gas is considered.b) Calculate κ S **and** κ T for a mono-atomic ideal gas.κ S = κ T − TVα2 PC P. (1)Consider a system as sketched on figure 2. The system consists of an outer isolat**in**g box, with anisolat**in**g piston. The box is partitioned by an **in**ner wall which is a good heat conductor. The boxis filled with mono-atomic ideal gas. The system is considered to be **in** thermal equilibrium.

4V 2 ,N 2 V 1 ,N 1P 2 ,T 2 P 1 ,T 1S 2 S 1Figure 2: Illustration of the system for problem 2.We now def**in**e the “externally seen” (that is **in** volume 1) adiabatic compressibility as (where thesecond equality sign follows from the “fake cha**in** rule”):( )κ S,1 = − 1 ( ) ∂V1= 1 ∂S total∂P 1 V( ) 1(2)V 1 ∂P 1 S totalV 1 ∂S total∂V 1 P 1where S total = S 1 +S 2 .In the follow**in**g it is used that the entropy of an mono-atomic ideal gas can be written as:[ ]S = Nk B ln V (k BT) 3/2+5/2 . (3)Nc) Calculate κ S,1 .H**in**t: First show that the total entropy of the system can be written as:(S total = N 1 k B ln V ( ) (3/21 P1 V 1+5/2)+N 2 k B ln V ( ) ) 3/22 P1 V 1+5/2N 1 N 1 N 2 N 1. (4)d) Calculate κ S,1 **in** the limits N 1 ≫ N 2 **and** N 1 ≪ N 2 . Compare the results with the results fromsub-question b).Opgave 3 (20 %)Consider a system consist**in**g of an isolated conta**in**er which is separated **in**to two compartments(each with volume V 1 **and** V 2 ) by a heat conduct**in**g wall. The two compartments are filled withmono-atomic ideal gas (N 1 **and** N 2 particles).For an mono-atomic ideal gas, entropy can be written as:[ ]S = Nk B ln V (k BT) 3/2+5/2 . (5)Na) Discuss the fundamental criterion for equilibrium, **and** show explicitly thatT 1 = T 2 **in** equilibrium.Volume 2 is now considered as a heat bath, **and** the two compartments are assumed to have the sametempe-rature.b) Show that the change **in** total entropy, for the total system, com**in**g from a change **in** an **in**nerdegree of freedom **in** sub-volume 1, can be found as:dS total = − 1 T dF 1 (6)where F 1 is Helmholts free energy of sub-volume 1.End of the **exam**.