Thermodynamics, Statistical Physics, and Quantum Mechanics

118 SOLUTIONS

since the initial and final temperatures are thesame. Substitutinginto

(S.4.26.1) we find

4.27 Venting (Moscow Phys-Tech)

The air surrounding thechamber may bethought of as avery large reservoir

of gas at a constant pressure andtemperature Theprocess ofventing

is adiabatic, so we can assumethatthere is no energydissipation. Wethen

findthat theenergy of the gas admitted to the chamberequals the sum

of its energy in the reservoir plus the work done by the gas of the

reservoir at to expel the gas into thechamber. This may be calculated

by considering the process of filling a cylinder bypushing apiston back,

where the piston offers aresistantforce of A being the cross section

of the cylinder. ThetotalenergyEis thengiven by

where is thevolume of the gasneeded to fill the volumeof the chamber

V (notethatVdoes not coincidewith because the temperature of the

gas in the chamberTpresumably is not the same as see FigureS.4.27).

On the other hand.

where is the heat capacity of the gas, is the heat capacity per
molecule, and is the number ofmolecules. From (S.4.27.1) and
(S.4.27.2), we have