Thermodynamics and Statistical Mechanics - Quiz 2 with 4 Problems | PHYS 4420, Quizzes of Physics

Download Thermodynamics and Statistical Mechanics - Quiz 2 with 4 Problems | PHYS 4420 and more Quizzes Physics in PDF only on Docsity! 1 PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS SPRING 2006 Quiz 2 Friday, April 21, 2006 NAME: __________ANSWERS____________________ To receive credit for a problem, you must show your work, or explain how you arrived at your answer. Credit Grade 1 15% 2 20% 3 50% 4 15% Total 100% 1. (15%) Suppose you get involved in a game in which six coins are tossed simultaneously. The object of the game is not to guess how many heads, or tails will show up in a toss. Instead, you are to guess what the distribution will be; i.e. will all six be the same, or will there be one of one type and five of the other, or two of one type and four of the other, or three of each. (One head and five tails is the same distribution as one tail and five heads.) In such a game, what distribution has the greatest probability of occuring, and what is that probability? The probability of getting six of one kind and none of the other equals the probability of getting six heads plus the probability of getting no heads (six tails). The probability of getting n heads is given by: nNnqp nNn N nP    )!(! ! )( Then, 031.0 64 2 2 1 !0!6 !6 2 2 1 2 1 )!06(!0 !6 2 1 2 1 )!66(!6 !6 )0,6( 6060666                                  P 188.0 64 12 2 1 !1!5 !6 2 2 1 2 1 )!16(!1 !6 2 1 2 1 )!56(!5 !6 )1,5( 6161565                                  P 469.0 64 30 2 1 !2!4 !6 2 2 1 2 1 )!26(!2 !6 2 1 2 1 )!46(!4 !6 )2,4( 6262464                                  P 313.0 64 20 2 1 !3!3 !6 2 1 2 1 )!36(!3 !6 )3,3( 6363                     P . There is no second distribution. Most probable distribution: ______4 and 2______ Probability = ___0.469__________ 2 2. (20%) For a certain system, there is only one accessible state and it has energy,        0 ln V V NkTs where V0 is a constant. a) (10%) What is the partition function for this system? N N V V V V N V V NkT kTV V NkT s V V eeeeeZ s                                                  0 lnlnln 1 ln 0000   N V V Z        0 b) (10%) Use the result of part a) to find the average pressure for this system as a function of temperature and volume.   V kTN VNVN V NkT V V V NkT V Z NkTP T T N 2 0 0 lnlnln ln                         V kTN P 2  or, )1lnlnln()1ln(ln 0 NVNVNNkTNZNkTF , and V kTN V F P 2     , as above.