Helmholtz free energy *first draft

\( F=U-TS \) is that weird formula that we wanted to minimize. But why?

Note: This is my first attempt, I am not sure if its valid to imagine a system being built at constant temperature T.

To answer why, one must understand what U and TS are in this expression and the conditions our little crystal is in. It is in a huge temperature bath at T and our crystal has constant volume so no work can be done on or by it. Hence the only interaction with the outside system is thermal.

In general U in the Helmholtz free energy equation is just the internal energy of the system, I think that we made an approximation in class as we plugged in U at T=0 K.

If you want to be persuaded that in class the U we used is the energy of the system at T=0K read the following:

*Explanation: The above holds since we did not use any fermi-dirac statistics to calculate U instead we plugged in f(T)=1 in \( U= \int ED(E)f(T,E)dE \). Where in \(f(T)=\frac{1}{e^(E-\mu)/kT+1}\) by definition the fermi energy is greater than any occupied state making the exponent negative hence \(lim_{T\Rightarrow0}f(T)=1 \)

How can TS be interpreted?

First, \(Q=TdS\) by definition hence if a system is at constant temperature T and you raise its entropy from 0 to S you get \( \int_0^STds=TS \)

Therefore TS can be thought as the total heat added to a system at temperature T as it is raised from 0 to S entropy.

Now we combine these two quantities to get \(F=U-TS \)

Minimizing this quantity will allow us to have a minimum energy U internally while extracting as much heat from the system as we can!

I think that I need to think more and maybe redo all of it