Consider a solid of volume v containing n sites (n> > 1). each site is predisposed and susceptible to trap only one particle. a trapped particle can take two energies: or-e2, with ε,» ε.> 0, we neglect the interactions between particles trapped by two different sites. the solid is in contact with a thermostat set at temperature t and a reservoir of particles of chemical potential, u. a) calculate the grand canonical ensemble of the solid b) calculate the average number of trapped particles , and the average energy of the solid as a function of tand h. (< ..> denotes average quantity) c) calculate and the average numbers of trapped particles in the energy levels-81 and -e2, respectively, as a function of t and u. d) calculate the average number, , of empty sites (no trapped particles). e) let us apply the above results to the case of electrons of magnetic moment, m, which are subjected to a magnetic field, b. the energy levels 1 and 2 are thus: 哟=-eo + mb and-e2 =-60-mb where > 0. level 1 is the level occupied by an electron of spin opposite to the magnetic field (zeeman energy +mb) whereas level 2 is the level occupied by an electron of spin parallel to the magnetic field. using the results above, calculate the average magnetic moment of the occupied traps (sites) by the electrons. determine the high and low temperature limits of the average moment. comment? ? problem consider a classical perfect gas of n particles, each of mass m. the gas is contained in a cylinder of cross- section a and infinite height and is placed in a gravitational field. the gas is in equilibrium. noting that the energy is expressed as: e-p'/2m +mgl, where p is the momentum of a particle in 3-dimension, g the amplitude of the gravitational field, and i denotes the height from the ground. (a) calculate the partition function of one particle (b) calculate the partition function of n-particle system. (c) calculate the average energy of the system (d) calculate the heat capacity at constant volume (e) calculate the canonical entropy (f) calculate the free energy