Please print this sheet, sign the statement at the bottom, and use it as cover page when you submit your homework. Remember to always include a complete explanation of your reasoning, and to show all calculations.

• 1: Ideal gas expansion. I
  An ideal gas has pressure equation of state pV = NkT and its internal energy is given by E = (f/2) NkT, where f is a constant that depends on the gas. If N particles of this gas are contained in a thermally insulated cylinder whose volume is allowed to increase from V to 2V by sliding a piston gradually until the volume reaches the desired value, what is the change in the entropy of the gas? What is the change in temperature? This is an adiabatic transformation, for which its pressure and volume are related by \(pV^\gamma = \) constant, where \(\gamma\) is a parameter characteristic of the type of gas, related to f.

• 2: Ideal gas expansion. II
  An ideal gas has pressure equation of state pV = NkT and its internal energy is given by E = (f/2) NkT, where f is a constant that depends on the gas. If N particles of this gas are contained in a thermally insulated box of volume V and the gas is allowed to flow freely through a hole until it fills a box of total volume 2V, what is the change in the entropy of the gas? What is the change in temperature?
   
• 3: Relationship between heat capacities
  Show that the constant-pressure and constant-volume heat capacities of a fluid are related by \(C_p^~ = C_V^~ + (TV/\kappa_T^~)\,\alpha^2\), with \(\kappa_T^~\) the isothermal compressibility of the system, and α its thermal expansion coefficient. [You may refer to the lecture notes on Second-Order Quantities and Relationships. Include all details of the calculations not spelled out there.] 

You are allowed, in fact encouraged, to discuss these problems and how to solve them with other students in the class, but you are not allowed to copy or use in any way anyone else's written solution to any of these problems. This includes solutions that may be provided to you by a student who took the course previously, or solutions you may find online or in print. This policy will hold for all assignments in this class.

Aside from oral discussions I may have had with other students in the class, the solutions to this homework set I am submitting are entirely my own.

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