1. (20 points) In introducing the concept of entropy and the 2nd Law of Thermodynamics, we mentioned that some processes that occur spontaneously are not explained by a change in the energy of the system. The converse is also true. There are processes that are perfectly acceptable energetically, but they are never observed. An example is the scenario where all of the gas molecules in a room spontaneously end up being located in only half the volume of the room. Based on experience, such a scenario is utterly unlikely, so none of us spend time worrying that all the air around us will suddenly disappear because all the molecules spontaneously move to a different part of the overall room volume. In this problem, we want to calculate how unlikely this scenario would be in everyday life. We also want to consider whether our intuition about the unlikely nature of this possibility extends to nanoscale systems. The analysis will be based on counting microstates of the system (that is, the number of possible configurations of the system). We will consider that the system of the room is isolated and that the gas inside the room is ideal so there is no interaction between the particles. In this case, we don't need to worry about any changes in number of microstates associated with the momentum (velocity), so we only need to count configurations associated with the position of the particles. We consider the case of an ideal gas at T = 298 K, pressure of 0.85 bar (8.5 x 104 Pa, a typical value in Albuquerque), and filling a moderate-sized room with volume of 160 m. 1a. (6 points) Using the ideal gas equation, find the total number of gas molecules in the room