EK424 - HW 2 (1 00 points) Please show your steps, write clearly, use units, box your final answer, and submit your homework 1. properly through gradescope. We will deduct points ifyou do not do these things. 1 mole of an ideal gas is at an initial state of T = 200K and P = 1atm. For each of the following processes, calculate q, w, AU, and .55. Note: for ideal gas, CF = Cl, + R a. The gas is heated to 300K at constant pressure. I. b. The gas is heated to 300K at constant volume (V = Vinitial}. c. The gas undergoes a reversible isothermal expansion at 200 K until the pressure is half of its initial value {T = Tinitial) OIThis fictional...thing... in the figure on the right has fixed amounts of water, air, and a marble (incompressible glass), all enclosed in a sealed isolated box resting on your kitchen table. Assume the marble has an adiabatic outer layer, and that the walls of the vessel are also adiabatic. Initially the marble is suspended above the water (don't 7W ask how this is possible, just roll with it). When released, it falls through a m: / o / g a the air into the water and comes to rest at the bottom of the vessel, causing the water and air (but not the marble)to change temperature slightly. The process is complete when the system returns to an equilibrium state. a. Does the temperature of the air and water increase or decrease? Please explain briefly. gar/1% For the following, please state how q, w, and AU change for each Figure talcum system, and briefly explain why b. The System is the marble c. The system is the water + air d. The system is the whole box Calculating the entropy of dipoles in a eld. You have a solution of dipolar molecules with a positive charge at the head and a negative charge at the tail. When there is no electric field applied to l' the solution, the dipoles point north {n}, east (e), west (w), or south (5) with equal probabilities. The probability distribution is shown in Figure a. However when you apply a field to the solution, you now observe a different distribution, with more heads pointing north, as shown in Figure b. You may keep your answers short in \"a"and "d" ofthis problem. in} Electric Field Absenl [bl Electric Field Present What is the polarity of the applied field? (In which direction does the field have its most positive pole?) Calculate the entropy of the system in the absence of the field. c. Calculate the entropy ofthe system in the applied field. d. Does the system become more ordered or disordered when the field is applied? 4. Please rank the S distributions from highest to lowest entropy. Note, the markings are accurate and all the info our need is oiven here. A B C IIIII 12345 state i What does an entropy of 0 tell you about a system? Please refer to both forms of the entropy equation when explaining your answer. 6. '13- Consider a lattice with N sites and n green particles. Consider another lattice, adjacent to the first, with M sites and m red particles. Assume that the green and red particles cannot switch lattices. This is state A. a. What is the total number of configurations Wu of the system in state A? b. Now assume that all N+M sites are available to all the green and red particles. The particles remain distinguishable by their color. This is state B. Now what is the total number of configurations W3 of the system? Now take N = M and n = m for the following two problems. c. Using Stirling's approximation, what is the ratio W's/WE? d. Which state, A or B, has the greatest entropy? Calculate the entropy difference given by: H'. _\\s = 5453 = k In(:') "B |n class, we discussed the fundamental thermodynamic equation by starting with U(S,V,N) and taking a differential. However, as briefly mentioned in class, additional variables can be added if needed to deal with an extra force not covered by variables S, V, and N. For example, in a Langmuir trough, there is a membrane pressure (11) exerted by surface tension effects from surfactants that changes with area A, thus yielding U(S,V,N,A), and an extra dU=....1TdA term in the differential form. Let's say you wanted to If you wanted to model the change in energy for a rubber band. What "extra variable" do you need to make U a function of? Please also write the full differential for dU with this extra term included