Given the quantum mechanical energy levels of a particle in a two-dimensional box are h2 y 1,2,3 8ma2(s^+s) calculate the density of states ω(e)de and use this to find the translational partition function of the two-dimensional ideal gas. derive the function using an alternative method [hint: q'eeny may be expressed using an integral] (i) (ii) iii find the equation of state, the energy e, the heat capacity ca and the entropy. (note: a → area, compare with v → volume in 3d case)

I) a gas contained within a piston-cylinder assembly undergoes two processes, a and b, between the same end states, 1 and 2, where p1 = 10 bar, v1 = 0.1 m3 , u1 = 400 kj and p2 = 1 bar, v2 = 1.0 m3 , u2 = 200 kj: ' process a: process from 1 to 2 during which the pressure-volume relation is pv = constant. ' process b: constant-volume process from state 1 to a pressure of 2 bar, followed by a linear pressure-volume process to state 2. kinetic and potential energy effects can be ignored. for each of the processes a and b, a) sketch the process on a p-v diagram. b) evaluate the work, in kj. c) evaluate the heat transfer, in kj.(ans : qb,12 = -65.0 kj)