Consider again the lattice of N spin-1/2 particles in an external homogeneous magnetic field, where each particle has two possible states: spin "down" with energy e = 0 and spin "up" with energy e = 1/2". The microstate of the system is specified by the energy states of all the particles, i.e. the list (e1, e2...e_n), while the macrostate is specified by the total energy of the system E, i.e. the sum E=∑_(i=1)^N¦e_i .

1. Calculate the entropy of the system as a function of the total energy, S(E), by assuming that both the number of particles in the "down" state N-n and the number of particles in the "up" state n are large, i.e. N-n»1 and n»1

2. Calculate the temperature of the system by using the relation T^(-1)=∂S/∂E

3. Calculate the heat capacity of the system, C=∂E/∂T

4. Calculate the partition function Z for the system.

5. By assuming the Boltzmann distribution, calculate the average energy of the system.

6. Calculate the heat capacity of the system by using the average energy, C=∂E/∂T.