ENPH/PHYS 242 – Fall 2014 Problem Set 7 Due Friday October 31, 9:20 am 1. Thermal excitation in a gravitational system A crazy physicist installs a cylindrical tube in the CN tower, reaching from the bottom to highest visitor’s platform at a height of 450m. The tube is evacuated and then cooled down to a very low temperature of 1.0K. Finally, a very small amount of helium (atomic mass: 4amu*) is filled into the tube and we wait until thermal equilibrium is reached. We find that there are on average 1.00 10 helium atoms in the bottom 1.00cm of the tube. a) How many helium atoms are on average in the top 1.00cm of the tube? b) How many helium atoms are there in the tube in total? We now increase the density of states at the bottom of the tube by expanding the tube into a tank. The tank is also cylindrical and its diameter is larger than the diameter of the rest of the tube by a factor 2.89. It has a height of about half a meter. d) What is now the ratio of the number of atoms in the bottom centimetre of the expended tube, compared to the number of atoms in the top centimetre of the tube? e) If we were to turn the whole setup upside down, so we now have an increased density of states in the top centimetre. What is then the ratio of atoms in the bottom compared to the top centimetre of the setup? 2. Maxwell-Boltzmann Distribution Calculate the typical velocity and kinetic energy at room temperatures (293K) for hydrogen atoms (mass: 1.0amu) and for nitrogen molecules (mass: 28amu)? By ‘typical velocity’ we mean the most common velocity (this in turn means that you need to calculate where the maximum of the distribution is). Consider the Maxwell-Boltzmann distribution as function of velocity (calculate the typical velocity and associated kinetic energy) AND as function of energy (calculate only the typical energy). * 1amu (atomic mass unit) of the mass of a atom 931.5 M V⁄ 1.6605 10 kg © W. Rau