Formulas n in n. n! r(v)=(- 1)! = e-"x-da, where v > 0. sin(k)ek dk 0. yk2 -yk2 consider n impurity atoms. each impurity can be in one of two states, with respectively. the microstates are specified by the set of occupations energies 0 and e, numbers n; }, where n; = 0 or 1 and i= . the total energy of the system is e en1 where n1 is the total number of the excited impurities. (a) in the micro-canonical ensemble: i. find the number of distinct microstates (e en1, n) ii. find the entropy s(e, n) iii. find the internal energy e(t, n) in the limit of n1, n> 1. as a function of t and n. (b) in the canonical ensemble: i. find the canonical partition function. ii. find the internal energy e (t, n).

The particle in a two-dimensional well is a useful model for the motion of electrons around the indole ring (3), the conjugated cycle found in the side chain of tryptophan. we may regard indole as a rectangle with sides of length 280 pm and 450 pm, with 10 electrons in the conjugated p system. as in case study 9.1, we assume that in the ground state of the molecule each quantized level is occupied by two electrons. (a) calculate the energy of an electron in the highest occupied level. (b) calculate the frequency of radiation that can induce a transition between the highest occupied and lowest unoccupied levels. 9.27 electrons around the porphine ring (4), the conjugated macrocycle that forms the structural basis of the heme group and the chlorophylls. we may treat the group as a circular ring of radius 440 pm, with 20 electrons in the conjugated system moving along the perimeter of the ring. as in exercise 9.26, assume that in the ground state of the molecule quantized each level is occupied by two electrons. (a) calculate the energy and angular momentum of an electron in the highest occupied level. (b) calculate the frequency of radiation that can induce a transition between the highest occupied and lowest unoccupied levels.

An open cart is moving along a straight frictionless horizontal track. when rain starts falling vertically into the cart, what happens to the speed of the cart?

4. if you were an astronaut on the moon, what would you experience? what would you see from your perspective?