A system consitst of n weakly interacting particles each of...

A system consists of N weakly interacting particles, each of which can be in one of two states with energies (e1) or (e2), where (e1) is less than (e2). Calculate the mean energy and heat capacity of the system. Also, determine the high and low temperature limits of E and Cv(T).

Transcript

00:01 First one's first question is that is equal to summation of i is equal to 0 to n e power minus beta ei and beta is equal to 1 divided by kb into t for two energy states e1 and e2 the partition function is given by z is equal to e power minus beta e1 plus e power minus beta e2 lon of z will be equal to lon of e power minus beta e1 plus e power minus beta e2 mean energy is given by e is equal to minus 1 by z into dou z divided by dou beta which is equal to dou by dou beta of lon z so e will be equal to minus dou by dou beta into e power minus beta e1 plus e power minus beta e2 divided by e power minus beta e1 plus e power minus beta e2 which is equal to e 1 into e power minus beta e1 plus e 2 into e power minus beta e2 divided by e power minus beta e1 plus e power minus beta e2 so e will be equal to e1 e power minus beta e1 plus e 2 into e power minus beta e2 divided by e power minus beta e1 plus e power minus beta e2 and cv will be equal to 1 by kbt square into e square minus e the whole square e square will be equal to 1 by z dou square z by dou beta square so cv will be equal to e 1 square e power minus 2 beta e1 plus e 1 square plus e 2 square into e power minus beta into e 1 plus e 2 plus e 2 square e power minus beta e2 minus e 1 square e power minus beta e1 minus 2 e 1 e 2 e power minus beta into e 1 plus e 2 minus e 2 square e power minus beta e2 the whole divided by kbt square into e power minus beta e1 plus e power minus beta e2 the whole square which is equal to e 1 square minus 2 e 1 e 2 plus e 2 square into e power minus beta into e 1 plus e 2 divided by kbt square e power minus beta e1 plus e power minus beta e2 the whole square which is equal to e 1 minus e 2 the whole square e power minus beta into e 1 plus e 2 divided by kbt square e power minus beta e1 plus e power minus beta e2 the whole square in the low temperature limit beta tends to infinity so we can write limit beta tends to infinity mean free path will be equal to e 1 plus e 2 divided by 2 and limit beta tends to infinity cv will be equal to e 1 minus e 2 the whole square divided by 4 kbt square in the high temperature limit beta tends to 0 so we can write limit beta tends to 0 mean free path will be equal to e 1 plus e 2 divided by 2 and limit beta tends to 0 cv will be equal to e 1 minus e 2 the whole square divided by 4 kbt square...

Question

A system consists of N weakly interacting particles, each of which can be in one of two states with energies (e1) or (e2), where (e1) is less than (e2). Calculate the mean energy and heat capacity of the system. Also, determine the high and low temperature limits of E and Cv(T).

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