Mathematically, there are quite a few similarity between the system considered here and the ideal gas system considered in class. The position of a two-dimensional diatomic molecule with fixed distance between the two atoms can be described by the three coordinates (x, y,0), where I and y are the Cartesian coordinates of the center-of-mass of the molecule and 0 gives the orientation of the molec- ular axis with respect to the r-axis. The conjugate momenta are denoted by (P.x, Py, pe). Physically, pc and py are the linear center-of-mass momenta and pe is the angular momen- tum of the molecule about its center-of-mass. The energy e of the molecule is p. + PP 21' E + (6) 2m where I is the moment of inertia about the center-of-mass. (a) For a system of N non-interacting two-dimensional diatomic molecules confined to a two-dimensional area A, use the microcanonical ensemble to calculate the entropy S(N, E, A). (b) Using the entropy, derive the equations of state of the system, i.e., the equations that give the pressure and the energy per particle as functions of the temperature and density. (c) Calculate the constant volume (actually, it is better to say constant area) specific heat per particle, defined through 1 (BE(N,T, A) (N, (7) N ar (@F TA = NA Note for part (b): In two dimensions, pressure is "force per unit length that is required to confine the particles to an area A. It is given by a formula analogous to the three- dimensional formula, namely P=T () as aa (8) NE