1.  One of your professor’s astrophysics colleagues got bored during COVID lockdown, and accidentally got some magnets stuck up his nose while trying to invent a better anti-coronavirus mask. (True story, believe it or not.) This got your professor thinking about magnetism.

The definitions of enthalpy and Gibbs free energy give special treatment to mechanical (expansion-compression) work. Analogous quantities can be defined for other kinds of work. For example, magnetic work is quantified by +p0 H dM , where p0  is the permeability of free space (a constant of nature that is  the same for all systems under all circumstances), M is the system’s magnetic moment, and H is the magnetic field. Suppose we consider magnetic work, but not mechanical (expansion-compression) work, in the system.

(a) Write down the relation analogous to the “thermodynamic identity” for the system.

(b) Define the analogues of enthalpy and Gibbs free energy for this magnetic system. Explain under what conditions (i.e. with what variables held constant) each of these quantities would be useful.

(c) Write down 2 different Maxwell relations that each involve a partial derivative with respect to H.

(d) Write down 3 different Maxwell relations that each involve a partial derivative with respect to M.

2. Your professor also got bored in lockdown, but since she’s obsessed with finding dark matter, she built a “bubble chamber” particle detector in her basement. Essentially this is a tank of liquid kept at a     specific temperature so that it is almost, but not quite, hot enough to boil. When a dark matter particle   wanders into the tank, it might interact with the liquid and deposit just enough energy to make a bubble form. (This is a simplified version of the PICO detector at SNOLAB: seehttp://www.picoexperiment.com/ if you’re curious.)

(a) Consider a spherical vapour bubble of Ng molecules, surrounded by Nl = N – Ng molecules of liquid. Suppose the entire system of vapour + liquid is in equilibrium. Write a formula for the total Gibbs free energy of the system, in terms of N, Ng, and the chemical potentials of the liquid and vapour (μl  , μg).

(b) Re-write Ng  in terms of the volume per molecule in the gas (vg ) and the radius of the bubble (r), and then re-write your formula from part (a) accordingly.

The surface forming the boundary between any two phases of a substance, e.g. the surface of a bubble forming the boundary between liquid and gas, has an additional Gibbs free energy Gboundary  associated  with it. This extra energy is the product of the surface’s area A and tension σ:  Gboundary  = A σ .

(c) Modify your expression for the total Gibbs free energy of the system to include Gboundary , written in terms of r and σ .

(d) The non-zero equilibrium radius of the bubble, if one exists, is called the “critical radius” rc . Express it in terms of σ, vg, and (μl - μg). (Hint: consider the dependence of G on r. Look for a non-zero value of r at which G reaches an extremum.)

3. Suppose the Helmholtz energy for a particular substance was found to be

F = A ∙ T ∙ exp(α ∙ T) – n R T [ ln(V − B) + D / V ]

where A, B, D, and α are constants. Assume n is always fixed.

(a) Find an expression for the substance’s pressure P (in terms of its volume V, temperature T, n, and various constants), assuming the temperature is held fixed.

(b) In what limit (of the values of some of the constants) does the behaviour of this substance’s pressure approach that of an ideal gas?

(c) Find an expression for the substance’s total energy U, in terms of T and constants, assuming the volume (instead of the temperature) is held fixed.

Now consider a different substance, with Gibbs free energy

G = n R T ∙ ln(P / P°) − a(T) ∙ P

where P° is the reference pressure, and a(T) is a positive-valued function of the temperature. Assume n is always fixed.

(d) Find an expression for the substance’s volume (in terms of its temperature, pressure, n, and a), assuming the temperature is held fixed.

(e) Find an expression for the substance’s entropy, assuming the pressure (instead of the temperature) is held fixed.

(f) Find an expression for the substance’s heat capacity at constant pressure.