Thermodynamics, Statistical Physics, and Quantum Mechanics

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On the other hand, at equilibrium, there is no net force acting on the piston

(seeFigure S.4.23), and we have

Substituting(S.4.23.2)into(S.4.23.1)gives

We can also use energyconservation in this thermally insulatedsystem.

Then thework done to the gas equals its energy change For an ideal

gas

where is theheat capacity of one mole of the gas (for amonatomic gas,
The work done to the gas

where is the distance thepiston moves,wheredownward
is positive. From(S.4.23.4) and(S.4.23.5), wehave

Solving (S.4.23.3) and(S.4.23.6) yields

We may checkthat if i.e.,that the piston was initially

balanced,(S.4.23.7)gives and

4.24 Spring Cylinder (Moscow Phys-Tech)

For athermallyinsulatedsystem (no heattransfer), theenergy change
is given by

THERMODYNAMICS AND STATISTICAL PHYSICS