a) You measure the energies of the 1000 ideal gas (non-interacting) molecules in this 1-D vessel and find that 500 have the ground state energy, 300 have the 1st excited

state energy and 200 have the 2nd excited state energy. Write y for a molecule in this gas consistent with these measurements. b) Use the equation for expectation values to calculate the average energy of molecules in this gas. Is this consistent with calculating the average energy directly from the information given in part a? c) Immediately after measuring that a molecule is in the 1st excited state you measure its momentum. What are the possible values you measure and their probabilities? d) You adiabatically (no change in populations of the different energy levels, which is equivalent to no heat exchange with the surroundings) compress this gas so that the cylinder now has width L/3. What is the change in total energy of the system? How much work was done on the gas? Does the temperature change? Does the amplitude of p change? Explain how these results are consistent with what you learn in the thermodynamics part of PChem.