Consider a system of N distinguishable particles which can exist in 3 energy states, with energies: e, 0, +6 (an example of such a system could be N distinguishable spin 1 particles in a magnetic eld). The system is in contact with a heat bath at fundamental temperature 7. When we ask for thermodynamic quantities for such a system, you can imagine we have an ensemble of similar systems and we are speaking about average quantities. (a) (1 point) What is the one particle partition function and what is the N particle partition function? (b) (2 points) Without doing any serious calculations estimate the average energy of the system in the limit of the two extremes 'T ) 0 and 1' > oo. Justify your answers with simple physics arguments. (c) (2 points) Derive the full expression (as a function of T) for the average energy of the system. (d) (2 points) Derive an expression for the fundamental entropy of this system as a function of T. (e) (3 points) By direct counting of states, calculate the entropy of this system at very low tem peratures (T > 0) and at very high temperatures (T ) 00). Show that your results are consistent with the limiting values (for T > 0 and T > 00) of your result in part (d). (Please note: even if you're stumped on one part of the above try to do the next part)