Question

Suppose both the compression and the expansion were adiabatic. What would be the values of pp and TT when the gas reached its original volume?

Express your answers in terms of some or all of the variables V0V0, T0T0, p0p0, γγ, and RR separated by a comma.

Term 1: The correct answer does not depend on: P0P0, γγ.

A cylinder contains nn moles of an ideal gas that is at initial temperature T0T0, volume V0V0, and pressure p0p0. The gas has a heat capacity ratio γγ. The gas undergoes the following two thermodynamic processes: (1) It is compressed adiabatically to a volume V=V0/2V=V0/2.(2) It is isothermally expanded back to its original volume. In terms of γγ and numerical factors, derive an expression for (a) the ratio p/p0p/p0, where pp is the pressure at the volume V=V0/2V=V0/2, and (b) the ratio of the final kinetic energy to the initial average translational kinetic energy, Kav/K0avKav/Kav0, of the gas atoms after the initial compression. In terms of T0T0,γγ,nn, and the gas constant RR (c) derive an expression for the work done by the gas in the second process. Include the proper sign of work. (d) After the gas has been restored to its initial volume, does it have the same internal energy that it started with before the initial compression? Explain.
Term 2: The correct answer does not depend on: γγ.

Answer 1