How Does Helmholtz Free Energy Determine Maximum Work in a Thermodynamic System?

[Feynmanfan]

Let’s say we have a 1 mol system in a state A (Pa,Va,Ta are given) and we take it to a state B(Pb,Vb are given) . We want to know what’s the maximum work we can give to a reversible work source. Considering the process is carried out attached to a 150 K heat source, the only data available is this:

1. equation of adiabatic reversible curve
2. Cp (heat capacity at a given constant pressure (Pa )
3. Alpha (expansion coefficient at a given constant pressure (Pa )

I’ve translated the problem into this: find A Helmholtz free energy change between points A and B. However I don’t know how to get dF=-SdT-PdV from the experimental coefficients mentioned above.

I’d appreciate your help very much.

 

[Feynmanfan]

Just a hint is enough. I don't have any sample problems nor thermo books so I don't know where to start from

 

[wais]

Thermodynamic potentials are important concepts in thermodynamics that help us understand the behavior of a system in terms of its energy and entropy. In this problem, we are given a 1 mol system in state A with known values of pressure, volume, and temperature, and we are asked to determine the maximum work that can be extracted from this system when it is taken to state B, while being attached to a 150 K heat source.

To solve this problem, we can use the Helmholtz free energy (F) as our thermodynamic potential. The Helmholtz free energy is defined as F = U - TS, where U is the internal energy of the system, T is the temperature, and S is the entropy. It is a useful quantity because it takes into account both the energy and entropy of a system.

To find the change in Helmholtz free energy between states A and B, we can use the fact that F is a state function, meaning it depends only on the initial and final states and not the path taken to get there. Therefore, we can calculate the change in F by integrating the differential form of F, which is given by dF = -SdT - PdV.

In this problem, we are provided with the equation of an adiabatic reversible curve, which can be used to determine the change in entropy (ΔS) between states A and B. The equation of an adiabatic reversible curve is given by ΔS = Cp ln(Tb/Ta) - R ln(Vb/Va), where Cp is the heat capacity at constant pressure and R is the gas constant. We also know the values of Cp and the expansion coefficient (α) at constant pressure, which can be used to calculate the change in volume (ΔV) between states A and B.

Using these values, we can solve for the change in F between states A and B by integrating the differential form of F, dF = -SdT - PdV. This will give us the maximum work that can be extracted from the system when it is taken from state A to B while being attached to a 150 K heat source.

In summary, to solve this problem, we need to use the Helmholtz free energy as our thermodynamic potential and integrate the differential form of F using the given experimental coefficients. This will give us the maximum work that can be extracted from the system when it is