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: 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.] 

• 2: Ideal gas chemical potential
  Using the pressure and energy equations of state for an ideal gas, the definitions of the main thermodynamic potentials, and the expression for the entropy of an ideal gas derived in class (see also the lecture notes L04) show all the steps leading to an expression for the chemical potential of an ideal gas as a function of V, T.

• 3: Adiabatic constant
  The pressure and volume for an ideal gas undergoing a reversible adiabatic transformation are related by \(pV^\gamma =\) constant, where the adiabatic constant \(\gamma\) depends on the type of gas. Show that \(\gamma\) is equal to the ratio between heat capacities, \(C_p/C_V^~\).

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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