Please look into the problems below:

Alright, people, let's do this one last time! Consider again a paramagnet system of N spins in a magnetic field B. We will use the symbol to denote a microstate of this system, which ammounts to determining all N individual spins oi = +1 (i = 1, . .., N). The Hamiltonian/energy function for this system is a function of the N individual spins, N E(J) = H(loi})=-UB> oi i=1 Let our paramagnet be a closed system in thermal contact with a large reservoir of constant tem- perature T.(a) Use the Boltzmann factor to find the probability distribution of just the first spin. That is, find Pi(1, T) and P1(4, T). Hint (highlight to reveal): [We already did this using the micro-canonical approach in Problem 6.1. You should get the same answer here but the derivation itself should be much more straightforward.] Hint (highlight to reveal): [We can consider "just the first spin" as a closed sub-system.] (b) Find the canonical partition function Z(T, N) for the N-spin paramagnet. Hint (highlight to reveal): [We learned in Problem 6.3 that the canonical particion function of a combined system is the product of the canonical partition functions of the individual elements. You can break up the N-spin paramagnet into A individual close systems and use this result.] (c) Using the canonical partition function, show that the expectation value of energy is (E) = -NuB tanh MB KBT . Then find the heat capacity Cv. [Supplementary Part (Not for Credit): Find the standard deviation of the energy OF = V(E2) - (E)2 for this system.] Hint (highlight to reveal): There are actually two consistent but distinct ways of finding OF using Z!]