For an arbitrary system, this quantity is given by the derivative with respect to the thermodynamic beta of the partition function, and the Helmholtz free energy is obtained by subtracting temperature times entropy from the average value of this quantity.
In a Fermi gas, this quantity is proportional to h-bar over 2 m times the number of particles raised to the five-thirds power, over the volume raised to the two-thirds power. For a system that obeys the van der Waals equation of state, this quantity contains an additive factor of the particle number squared times the a constant over volume. For an arbitrary system, this quantity is given by the derivative with respect to the thermodynamic beta of the partition function, and the Helmholtz free energy is obtained by subtracting temperature times entropy from the average value of this quantity. For a closed system the change in this quantity is given by the work done by the system plus the heat, and for an ideal gas it is given by the three-halves times the number of particles times Boltzmann's constant times temperature. For 10 points, identify this quantity which represents the total amount of kinetic and potential energy in a gas.
In a Fermi gas, this quantity is proportional to h-bar over 2 m times the number of particles raised to the five-thirds power, over the volume raised to the two-thirds power.
{internal energy} [prompt on partial answer]