A long time ago, the specific heat at constant volume of gases (cv) was determined indirectly from cp measurements and from a special measurement of the cp/cv ration, which is described below. A small amount of gas is held in a rigid bottle fitted with a valve. Initially (state 1), the valve is closed, the gas temperature is atmospheric (Ta), and the gas pressure is slightly above atmospheric (P1). The valve is then opened and closed immediately (state 2); during this incident, enough of the bottled gas escapes and the pressure inside the bottle drops to the atmospheric level (Pa). The experimenter also notices that during the same incident, temperature T2 of the bottled gas drops, T2 < Ta. Waiting so that the bottled gas comes to thermal equilibrium with the atmosphere (state 3), the experimenter finds that the gas pressure climbs to a final level P3 above atmospheric.
Show that if the expansion of the gas that remains trapped in the bottle is modeled as reversible and adiabatic, and if the gas is modeled as ideal, the ration cp/cv is a function of only P1/Pa and P3/Pa. In other words, show that the ratio of specific heats can be calculated from the measurement of three pressures, P1, P3 and Pa.