A molecular simulation sounds like an advanced subject, but it is really quite simple for hard spheres. Furthermore, modern software is readily available to facilitate performing simulations, after an understanding of the basis for the simulations has been demonstrated. This problem provides an opportunity to demonstrate that understanding. Suppose that four hard disks are bouncing in two dimensions around a square box. Let the diameters of the disks, σ, be 0.4 nm, masses be 40 g/mole, and length of the square box, L, be 5 nm. Start the four disks at (0.25L, 0.25L), (0.75L, 0.25L), (0.25L, 0.75L), (0.75L, 0.75L) and with initial velocities of (v, v/(1 + 2½)), (–v, v), (v/2½, -v/2½), (-v/2½, -v/2½), where v designates an arbitrary velocity.

(a) Compute v initially assuming a temperature of 298 K.

(b) Sum the velocities of all four particles (x and y separately). Explain the significance of these sums.

(c) Sketch the disks using arrows to show their directions. Make the sizes of the arrows proportional to the magnitudes of their velocities.

(d) Solve for the time of the first collision. Is it with a wall or between particles? Compute the velocities of all disks after the first collision.