# Is there only one trajectory to extract work from an isothermal isochoric system?

So I was reading about Helmholtz free energy, and how it represents the maximum amount of useful non-PV work that can be extracted from an isothermal isochoric system.

I was trying to better understand what kind of trajectory through phase space such a system would take as we extract work from it. Ie, if we start at a particular thermodynamic state (N, T, S, P, V, U) with non-zero Helmholtz free-energy, and start isothermally isochorically extracting work until the Helmholtz free-energy is zero, does that define a unique path between those two points in phase space? Are there non-unique possible termini at which the Helmholtz free energy is zero?

Any thoughts appreciated.

Thanks.

• Are you allowing cases in which electrochemical reaction occurs? – Chet Miller May 14 at 13:59
• @ChetMiller I don't have a sense for how that changes things, but maybe for simplicity we can just say no electrochemical reactions. – user49404 May 14 at 15:55
• I’m trying to understand what you include as non-PV work. – Chet Miller May 14 at 16:22
• Ah ok. I don't have a clear sense of the forms non-PV work can take, but turning a crank (shaft work) might be a simple example to start with. – user49404 May 14 at 16:59
• I can visualize shaft work in a continuous flow system, but find it hard to visualize for a constant volume closed system. – Chet Miller May 14 at 17:21

If we start at a particular thermodynamic state (N, T, S, P, V, U) with non-zero Helmholtz free-energy, and start isothermally isochorically extracting work until the Helmholtz free-energy is zero, does that define a unique path between those two points in phase space?

In my thinking, this really boils down to "is the universe deterministic at the macroscopic scale?" My answer is therefore: yes, if you define the initial conditions precisely enough. If your system is, say, a battery (not quite isothermal, but essentially isochoric, and capable of doing (non-PV) electrical work), then "the battery is 50% charged" likely is not a sufficiently complete description of the initial state, but "here is the chemical composition at each point inside the battery" likely is.

Are there non-unique possible termini at which the Helmholtz free energy is zero?

In my thinking, this boils down to "is there a unique equilibrium point?" My answer is related to that above: with a sufficiently complete definition of the total properties of the system (total amount of each element) then there is precisely one equilibrium state: the state in which the contents are homogeneous and at chemical equilibrium (this assumes that the hydrostatic pressure variation associated with any gravitational field present isn't significant enough to compromise homogeneity). Note that defining the equilibrium point requires less information than defining the trajectory by which it is reached, since each equilibrium state is the endpoint of an infinite number of unique trajectories (distinct initial states can converge to the same equilibrium point).

I was trying to better understand what kind of trajectory through phase space such a system would take as we extract work from it.

I think that this depends on how you interpret the constraint "isothermal and isochoric"

• The most restrictive possible interpretation would be "every point in the system is at a specified temperature and specific volume." In this case the only system properties which can change are chemical compositions, and the only possible relaxation processes are chemical reactions and diffusion/mixing (isothermal heat transfer is also possible, but is not, in my view, "relaxation" since it is reversible).
• The loosest possible interpretation would be "the system boundary is at a specified temperature (although internal temperatures may vary), and the total system volume is constant (although internal densities may vary)." In this case the constraints aren't very restrictive and essentially any type of relaxation process (including chemical reactions and diffusion plus heat transfer and expansion, which were prohibited above) could happen inside the system as it evolves.