1. (a) A single particle with mass m is confined to a box of length L in the x-direction. If the particle is moving with speed u, in the x-direction and bouncing elastically off the walls at either end, show that it exerts a pressure p on these walls of p= mv LA where A is the area of the walls. Using this result, show that the root-mean-square speed Urms of molecules in an ideal gas is given by 3p where p is the mass density of the gas. (b) At 26.5C and 1.57 kPa the density of a gas is 2.0210-5 g/cm. i. Find Urms for the gas molecules. ii. Find the number of molecules per unit volume of the gas and, using this, find the mass of the gas molecules and identify the gas. (c) 2.5 mol of the gas above is compressed isothermally to one tenth of it initial volume. Compute (i) the work done on the gas, (ii) the heat added to the gas and (iii) the change in entropy of the gas in this process. If you did not get an answer in the previous question, you may assume that the gas is diatomic.